CSS Values and Units Module Level 4

1. Introduction

The value definition field of each CSS property can contain keywords, data types (which appear between < and >), and information on how they can be combined. Generic data types (<length> being the most widely used) that can be used by many properties are described in this specification, while more specific data types (e.g., <spacing-limit>) are described in the corresponding modules.

1.1. Module Interactions

This module replaces and extends the data type definitions in [CSS21] sections 1.4.2.1, 4.3, and A.2.

2. Value Definition Syntax

The value definition syntax described here is used to define the set of valid values for CSS properties (and the valid syntax of many other parts of CSS). A value so described can have one or more components.

2.1. Component Value Types

Component value types are designated in several ways:

1. keyword values (such as auto, disc, etc.), which appear literally, without quotes (e.g. auto)
2. basic data types, which appear between < and > (e.g., <length>, <percentage>, etc.). For numeric data types, this type notation can annotate any range restrictions using the bracketed range notation described below.
3. types that have the same range of values as a property bearing the same name (e.g., <'border-width'>, <'background-attachment'>, etc.). In this case, the type name is the property name (complete with quotes) between the brackets. Such a type does not include CSS-wide keywords such as inherit, and also does not include any top-level comma-separated-list multiplier (i.e. if a property named pairing is defined as [ <custom-ident> <integer>? ]#, then <'pairing'> is equivalent to [ <custom-ident> <integer>? ], not [ <custom-ident> <integer>? ]#).
4. non-terminals that do not share the same name as a property. In this case, the non-terminal name appears between < and >, as in <spacing-limit>. Notice the distinction between <border-width> and <'border-width'>: the latter is defined as the value of the border-width property, the former requires an explicit expansion elsewhere. The definition of a non-terminal is typically located near its first appearance in the specification.

Some property value definitions also include the slash (/), the comma (,), and/or parentheses as literals. These represent their corresponding tokens. Other non-keyword literal characters that may appear in a component value, such as “+”, must be written enclosed in single quotes.

Commas specified in the grammar are implicitly omissible in some circumstances, when used to separate optional terms in the grammar. Within a top-level list in a property or other CSS value, or a function’s argument list, a comma specified in the grammar must be omitted if:

• all items preceding the comma have been omitted
• all items following the comma have been omitted
• multiple commas would be adjacent (ignoring white space/comments), due to the items between the commas being omitted.

example( first? , second? , third? )

All CSS properties also accept the CSS-wide keyword values as the sole component of their property value. For readability these are not listed explicitly in the property value syntax definitions. For example, the full value definition of border-color is <color>{1,4} | inherit | initial | unset (even though it is listed as <color>{1,4}).

Note: This implies that, in general, combining these keywords with other component values in the same declaration results in an invalid declaration. For example, background: url(corner.png) no-repeat, inherit; is invalid.

2.2. Component Value Combinators

Component values can be arranged into property values as follows:

• Juxtaposing components means that all of them must occur, in the given order.
• A double ampersand (&&) separates two or more components, all of which must occur, in any order.
• A double bar (||) separates two or more options: one or more of them must occur, in any order.
• A bar (|) separates two or more alternatives: exactly one of them must occur.
• Brackets ([ ]) are for grouping.

Juxtaposition is stronger than the double ampersand, the double ampersand is stronger than the double bar, and the double bar is stronger than the bar. Thus, the following lines are equivalent:

  a b   |   c ||   d &&   e f
[ a b ] | [ c || [ d && [ e f ]]]

For reorderable combinators (||, &&), ordering of the grammar does not matter: components in the same grouping may be interleaved in any order. Thus, the following lines are equivalent:

a || b || c
b || a || c

2.3. Component Value Multipliers

Every type, keyword, or bracketed group may be followed by one of the following modifiers:

• An asterisk (*) indicates that the preceding type, word, or group occurs zero or more times.
• A plus (+) indicates that the preceding type, word, or group occurs one or more times.
• A question mark (?) indicates that the preceding type, word, or group is optional (occurs zero or one times).
• A single number in curly braces ({A}) indicates that the preceding type, word, or group occurs A times.
• A comma-separated pair of numbers in curly braces ({A,B}) indicates that the preceding type, word, or group occurs at least A and at most B times. The B may be omitted ({A,}) to indicate that there must be at least A repetitions, with no upper bound on the number of repetitions.
• A hash mark (#) indicates that the preceding type, word, or group occurs one or more times, separated by comma tokens (which may optionally be surrounded by white space and/or comments). It may optionally be followed by the curly brace forms, above, to indicate precisely how many times the repetition occurs, like <length>#{1,4}.
• An exclamation point (!) after a group indicates that the group is required and must produce at least one value; even if the grammar of the items within the group would otherwise allow the entire contents to be omitted, at least one component value must not be omitted.

For repeated component values (indicated by *, +, or #), UAs must support at least 20 repetitions of the component. If a property value contains more than the supported number of repetitions, the declaration must be ignored as if it were invalid.

2.4. Combinator and Multiplier Patterns

There are a small set of common ways to combine multiple independent component values in particular numbers and orders. In particular, it’s common to want to express that, from a set of component value, the author must select zero or more, one or more, or all of them, and in either the order specified in the grammar or in any order.

All of these can be easily expressed using simple patterns of combinators and multipliers:

Note that all of the "any order" possibilities are expressed using combinators, while the "in order" possibilities are all variants on juxtaposition.

2.5. Component Values and White Space

Unless otherwise specified, white space and/or comments may appear before, after, and/or between components combined using the above combinators and multipliers.

Note: In many cases, spaces will in fact be required between components in order to distinguish them from each other. For example, the value 1em2em would be parsed as a single <dimension-token> with the number 1 and the identifier em2em, which is an invalid unit. In this case, a space would be required before the 2 to get this parsed as the two lengths 1em and 2em.

2.6. Property Value Examples

Below are some examples of properties with their corresponding value definition fields

3. Combining Values: Interpolation, Addition, and Accumulation

Some procedures, for example transitions and animations, combine two CSS property values. The following combining operations—on the two computed values V_a and V_B yielding the computed value V_result—are defined:

interpolation

Given two property values V_a and V_B, produces an intermediate value V_result at a distance of p along the interval between V_a and V_B such that p = 0 produces V_a and p = 1 produces V_B.

The range of p is (−∞, ∞) due to the effect of timing functions. As a result, this procedure must also define extrapolation behavior for p outside [0, 1].

addition

Given two property values V_a and V_B, returns the sum of the two properties, V_result. For addition that is not commutative (for example, matrix multiplication) V_a represents the first term of the operation and V_B represents the second.

Note: While addition can often be expressed in terms of the same weighted sum function used to define interpolation, this is not always the case. For example, interpolation of transform matrices involves decomposing and interpolating the matrix components whilst addition relies on matrix multiplication.

accumulation

Given two property values V_a and V_B, returns the result, V_result, of combining the two operands such that V_B is treated as a delta from V_a. For accumulation that is not commutative (for example, accumulation of mismatched transform lists) V_a represents the first term of the operation and V_B represents the second.

Note: For many types of animation such as numbers or lengths, accumulation is defined to be identical to addition.

A common case where the definitions differ is for list-based types where addition may be defined as appending to a list whilst accumulation may be defined as component-based addition. For example, the filter list values blur(2) and blur(3), when added together would produce blur(2) blur(3), but when accumulated would produce blur(5).

These operations are only defined on computed values. (As a result, it is not necessary to define, for example, how to add a <length> value of 15pt with 5em since such values will be resolved to their canonical unit before being passed to any of the above procedures.)

If a value type does not define a specific procedure for addition or is defined as not additive, its addition operation is simply V_result = V_a.

If a value type does not define a specific procedure for accumulation, its accumulation operation is identical to addition.

3.1. Representing Interpolated Values: the mix() notation

Interpolation of two values can be represented by the mix() functional notation, whose syntax is defined as follows:

mix( <percentage> ; <start-value> ; <end-value> )

<percentage>

Represents the interpolation point as progress from <start-value> to <end-value>.

<start-value>

The value at the “start” (0%) of the interpolation range.

<end-value>

The value at the “end” (100%) of the interpolation range.

Note: This functional notation uses semicolons to separate arguments rather than the more typical comma because the values themselves can contain commas.

A mix() notation is invalid if either its <start-value> or <end-value> is invalid if substituted in its place, or if the property using it is not animatable.

Define more precisely where mix() is allowed. Is it a top-level functional notation only? Or can it be nested more deeply in the grammar somehow? [Issue #6700]

What happens to values greater than 100% / less than 0%? Do we clamp them to the range? Interpolate past the range? Make them invalid? [Issue #6701]

3.2. Range Checking

Interpolation can result in a value outside the valid range for a property, even if all of the inputs to interpolation are valid; this especially happens when p is outside the [0, 1] range, but some easing functions can cause this to occur even within that range. If the final result after interpolation, addition, and accumulation is out-of-range for the target context the value is being used in, it does not cause the declaration to be invalid. Instead, the value must be clamped to the range allowed in the target context, exactly the same as math functions (see § 11.12 Range Checking).

Note: Even if interpolation results in an out-of-range value, addition/accumulation might "correct" the result and bring it back into range. Thus, clamping is only applied to the final result of applying all interpolation-related operations.

4. Textual Data Types

The textual data types include various keywords and identifiers as well as strings (<string>) and URLs (<url>). Aside from the casing of pre-defined keywords or as explicitly defined for a given property, no normalization is performed, not even Unicode normalization: the specified and computed value of a property are exactly the provided Unicode values after parsing (which includes character set conversion and escaping). [UNICODE] [CSS-SYNTAX-3]

CSS identifiers, generically denoted by <ident>, consist of a sequence of characters conforming to the <ident-token> grammar. [CSS-SYNTAX-3] Identifiers cannot be quoted; otherwise they would be interpreted as strings. CSS properties accept two classes of identifiers: pre-defined keywords and author-defined identifiers.

Note: The <ident> production is not meant for property value definitions—<custom-ident> should be used instead. It is provided as a convenience for defining other syntactic constructs.

All textual data types interpolate as discrete and are not additive.

4.1. Pre-defined Keywords

In the value definition fields, keywords with a pre-defined meaning appear literally. Keywords are CSS identifiers and are interpreted ASCII case-insensitively (i.e., [a-z] and [A-Z] are equivalent).

Value: collapse | separate

table { border-collapse: separate }

4.1.1. CSS-wide keywords: initial, inherit and unset

As defined above, all properties accept the CSS-wide keywords, which represent value computations common to all CSS properties.

The initial keyword represents the value specified as the property’s initial value. The inherit keyword represents the computed value of the property on the element’s parent. The unset keyword acts as either inherit or initial, depending on whether the property is inherited or not. All of these keywords are normatively defined in the Cascade module. [CSS3CASCADE]

Other CSS specifications can define additional CSS-wide keywords.

4.2. Author-defined Identifiers: the <custom-ident> type

Some properties accept arbitrary author-defined identifiers as a component value. This generic data type is denoted by <custom-ident>, and represents any valid CSS identifier that would not be misinterpreted as a pre-defined keyword in that property’s value definition. Such identifiers are fully case-sensitive (meaning they’re compared using the "identical to" operation), even in the ASCII range (e.g. example and EXAMPLE are two different, unrelated user-defined identifiers).

The CSS-wide keywords are not valid <custom-ident>s. The default keyword is reserved and is also not a valid <custom-ident>. Specifications using <custom-ident> must specify clearly what other keywords are excluded from <custom-ident>, if any—for example by saying that any pre-defined keywords in that property’s value definition are excluded. Excluded keywords are excluded in all ASCII case permutations.

When parsing positionally-ambiguous keywords in a property value, a <custom-ident> production can only claim the keyword if no other unfulfilled production can claim it.

Note: When designing grammars with <custom-ident>, the <custom-ident> should always be "positionally unambiguous", so that it’s impossible to conflict with any keyword values in the property.

4.3. Explicitly Author-defined Identifiers: the <dashed-ident> type

Some contexts accept both author-defined identifiers and CSS-defined identifiers. If not handled carefully, this can result in difficulties adding new CSS-defined values; UAs have to study existing usage and gamble that there are sufficiently few author-defined identifiers in use matching the new CSS-defined one, so giving the new value a special CSS-defined meaning won’t break existing pages.

While there are many legacy cases in CSS that mix these two values spaces in exactly this fraught way, the <dashed-ident> type is meant to be an easy way to distinguish author-defined identifiers from CSS-defined identifiers.

The <dashed-ident> production is a <custom-ident>, with all the case-sensitivity that implies, with the additional restriction that it must start with two dashes (U+002D HYPHEN-MINUS).

<dashed-ident>s are reserved solely for use as author-defined names. CSS will never define a <dashed-ident> for its own use.

.foo {
  --fg-color: blue;
}

@color-profile --foo { src: url(https://example.com/foo.icc); }
.foo {
  color: color(--foo 1 0 .5 / .2);
}

4.4. Quoted Strings: the <string> type

Strings are denoted by <string> and consist of a sequence of characters delimited by double quotes or single quotes. They correspond to the <string-token> production in the CSS Syntax Module [CSS-SYNTAX-3].

content: "this is a 'string'.";
content: "this is a \"string\".";
content: 'this is a "string".';
content: 'this is a \'string\'.'

It is possible to break strings over several lines, for aesthetic or other reasons, but in such a case the newline itself has to be escaped with a backslash (\). The newline is subsequently removed from the string. For instance, the following two selectors are exactly the same:

a[title="a not s\
o very long title"] {/*...*/}
a[title="a not so very long title"] {/*...*/}

Since a string cannot directly represent a newline, to include a newline in a string, use the escape "\A". (Hexadecimal A is the line feed character in Unicode (U+000A), but represents the generic notion of "newline" in CSS.)

4.5. Resource Locators: the <url> type

The <url> type represents a URL, which is a pointer to a resource.

Typically, a <url> is written with the url() or src() functional notations:

<url> = url( <string> <url-modifier>* ) |
        src( <string> <url-modifier>* )

body { background: url("http://www.example.com/pinkish.gif") }

For legacy reasons, a url() can be written without quotation marks around the URL itself, in which case it is specially-parsed as a <url-token> [CSS-SYNTAX-3]. Because of this special parsing, url() is only able to specify its URL literally; src() lacks this special parsing rule, and so its URL can be provided by functions, such as var().

background: url("http://www.example.com/pinkish.gif");
background: url(http://www.example.com/pinkish.gif);

background: src("http://www.example.com/pinkish.gif");
--foo: "http://www.example.com/pinkish.gif";
background: src(var(--foo));

--foo: "http://www.example.com/pinkish.gif";
background: url(var(--foo));

Some CSS contexts (such as @import) also allow a <url> to be represented by a bare <string>, without the function wrapper. In such cases the string behaves identically to a url() function containing that string.

@import url("base-theme.css");
@import "base-theme.css";

4.5.1. Relative URLs

In order to create modular style sheets that are not dependent on the absolute location of a resource, authors should use relative URLs. Relative URLs (as defined in [URL]) are resolved to full URLs using a base URL. RFC 3986, section 3, defines the normative algorithm for this process. For CSS style sheets, the base URL is that of the style sheet itself, not that of the styled source document. Style sheets embedded within a document have the base URL associated with their container.

Note: For HTML documents, the base URL is mutable.

When a <url> appears in the computed value of a property, it is resolved to an absolute URL, as described in the preceding paragraph. The computed value of a URL that the UA cannot resolve to an absolute URL is the specified value.

body { background: url("tile.png") }

http://www.example.org/style/basic.css

http://www.example.org/style/tile.png

4.5.1.1. Fragment URLs

To work around some common eccentricities in browser URL handling, CSS has special behavior for fragment-only urls.

If a url()’s value starts with a U+0023 NUMBER SIGN (#) character, parse it as per normal for URLs, but additionally set the local url flag of the url().

When matching a url() with the local url flag set, ignore everything but the URL’s fragment, and resolve that fragment against the current document that relative URLs are resolved against. This reference must always be treated as same-document (rather than cross-document).

When serializing a url() with the local url flag set, it must serialize as just the fragment.

4.5.2. Empty URLs

If the value of the url() is the empty string (like url("") or url()), the url must resolve to an invalid resource (similar to what the url about:invalid does).

Note: This matches the behavior of empty urls for embedded resources elsewhere in the web platform, and avoids excess traffic re-requesting the stylesheet or host document due to editing mistakes leaving the url() value empty, which are almost certain to be invalid resources for whatever the url() shows up in. Linking on the web platform does allow empty urls, so if/when CSS gains some functionality to control hyperlinks, this restriction can be relaxed in those contexts.

4.5.3. URL Modifiers

The url() function supports specifying additional <url-modifier>s, which change the meaning or the interpretation of the URL somehow. A <url-modifier> is either an <ident> or a functional notation.

This specification does not define any <url-modifier>s, but other specs may do so.

Note: A <url> that is either unquoted or not wrapped in url() notation cannot accept any <url-modifier>s.

5. Numeric Data Types

Numeric data types are used to represent quantities, indexes, positions, and other such values. Although many syntactic variations can exist in expressing the quantity (numeric aspect) in a given numeric value, the specified and computed value do not distinguish these variations: they represent the value’s abstract quantity, not its syntactic representation.

The numeric data types include <integer>, <number>, <percentage>, and various dimensions including <length>, <angle>, <time>, <frequency>, and <resolution>.

Note: While general-purpose dimensions are defined here, some other modules define additional data types (e.g. [css-grid-1] introduces fr units) whose usage is more localized.

The precision and supported range of numeric values in CSS is explicitly undefined, and can vary based on the property or other context a value is used in. However, within the CSS specifications, infinite precision and range is assumed. When a value cannot be explicitly supported due to range/precision limitations, it must be converted to the closest value supported by the implementation, but how the implementation defines "closest" is explicitly undefined as well.

If an <angle> must be converted due to exceeding the implementation-defined range of supported values, it must be clamped to the nearest supported multiple of 360deg.

5.1. Range Restrictions and Range Definition Notation

Properties can restrict numeric values to some range. If the value is outside the allowed range, then unless otherwise specified, the declaration is invalid and must be ignored. Range restrictions can be annotated in the numeric type notation using CSS bracketed range notation—[min,max]—within the angle brackets, after the identifying keyword, indicating a closed range between (and including) min and max. For example, <integer [0,10]> indicates an integer between 0 and 10, inclusive.

Note: CSS values generally do not allow open ranges; thus only square-bracket notation is used.

CSS theoretically supports infinite precision and infinite ranges for all value types; however in reality implementations have finite capacity. UAs should support reasonably useful ranges and precisions. Range extremes that are ideally unlimited are indicated using ∞ or −∞ as appropriate. For example, <length [0,∞]> indicates a non-negative length.

If no range is indicated, either by using the bracketed range notation or in the property description, then [−∞,∞] is assumed.

Note: At the time of writing, the bracketed range notation is new; thus in most CSS specifications any range limitations are described only in prose. (For example, “Negative values are not allowed” or “Negative values are invalid” indicate a [0,∞] range.) This does not make them any less binding.

5.2. Integers: the <integer> type

Integer values are denoted by <integer>.

When written literally, an integer is one or more decimal digits 0 through 9 and corresponds to a subset of the <number-token> production in the CSS Syntax Module [CSS-SYNTAX-3]. The first digit of an integer may be immediately preceded by - or + to indicate the integer’s sign.

5.2.1. Computation and Combination of <integer>

Unless otherwise specified, the computed value of a specified <integer> is the specified abstract integer.

Interpolation of <integer> is defined as V_result = round((1 - p) × V_a + p × V_b); that is, interpolation happens in the real number space as for <number>s, and the result is converted to an <integer> by rounding to the nearest integer, with values halfway between adjacent integers rounded towards positive infinity.

Addition of <integer> is defined as V_result = V_a + V_b

5.3. Real Numbers: the <number> type

Number values are denoted by <number>, and represent real numbers, possibly with a fractional component.

When written literally, a number is either an integer, or zero or more decimal digits followed by a dot (.) followed by one or more decimal digits and optionally an exponent composed of "e" or "E" and an integer. It corresponds to the <number-token> production in the CSS Syntax Module [CSS-SYNTAX-3]. As with integers, the first character of a number may be immediately preceded by - or + to indicate the number’s sign.

The value <zero> represents a literal number with the value 0. Expressions that merely evaluate to a <number> with the value 0 (for example, calc(0)) do not match <zero>; only literal <number-token>s do.

5.3.1. Computation and Combination of <number>

Unless otherwise specified, the computed value of a specified <number> is the specified abstract number.

Interpolation of <number> is defined as V_result = (1 - p) × V_a + p × V_b

Addition of <number> is defined as V_result = V_a + V_b

5.4. Numbers with Units: dimension values

The general term dimension refers to a number with a unit attached to it; and is denoted by <dimension>.

When written literally, a dimension is a number immediately followed by a unit identifier, which is a CSS identifier. It corresponds to the <dimension-token> production in the CSS Syntax Module [CSS-SYNTAX-3]. Like keywords, unit identifiers are ASCII case-insensitive.

CSS uses <dimension>s to specify distances (<length>), durations (<time>), frequencies (<frequency>), resolutions (<resolution>), and other quantities.

5.4.1. Compatible Units

When serializing computed values [CSSOM], compatible units (those related by a static multiplicative factor, like the 96:1 factor between px and in, or the computed font-size factor between em and px) are converted into a single canonical unit. Each group of compatible units defines which among them is the canonical unit that will be used for serialization.

When serializing resolved values that are used values, all value types (percentages, numbers, keywords, etc.) that represent lengths are considered compatible with lengths. Likewise any future API that returns used values must consider any values that represent distances/durations/frequencies/etc. as compatible with the relevant class of dimensions, and canonicalize accordingly.

5.4.2. Combination of Dimensions

Interpolation of compatible dimensions (for example, two <length> values) is defined as V_result = (1 - p) × V_a + p × V_b

Addition of compatible dimensions is defined as V_result = V_a + V_b

5.5. Percentages: the <percentage> type

Percentage values are denoted by <percentage>, and indicates a value that is some fraction of another reference value.

When written literally, a percentage consists of a number immediately followed by a percent sign %. It corresponds to the <percentage-token> production in the CSS Syntax Module [CSS-SYNTAX-3].

Percentage values are always relative to another quantity, for example a length. Each property that allows percentages also defines the quantity to which the percentage refers. This quantity can be a value of another property for the same element, the value of a property for an ancestor element, a measurement of the formatting context (e.g., the width of a containing block), or something else.

5.5.1. Computation and Combination of <percentage>

Unless otherwise specified (such as in font-size, which computes its <percentage> values to <length>), the computed value of a percentage is the specified percentage.

Interpolation of <percentage> is defined as V_result = (1 - p) × V_a + p × V_b

Addition of <percentage> is defined as V_result = V_a + V_b

5.6. Mixing Percentages and Dimensions

In cases where a <percentage> can represent the same quantity as a dimension in the same component value position, and can therefore be combined with them in a calc() expression, the following convenience notations may be used in the property grammar:

<length-percentage>

Equivalent to [ <length> | <percentage> ], where the <percentage> will resolve to a <length>.

<frequency-percentage>

Equivalent to [ <frequency> | <percentage> ], where the <percentage> will resolve to a <frequency>.

<angle-percentage>

Equivalent to [ <angle> | <percentage> ], where the <percentage> will resolve to an <angle>.

<time-percentage>

Equivalent to [ <time> | <percentage> ], where the <percentage> will resolve to a <time>.

Note: Specifications should never alternate <percentage> in place of a dimension in a grammar unless they are compatible.

Note: More <type-percentage> productions can be added in the future as needed. A <number-percentage> will never be added, as <number> and <percentage> can’t be combined in calc().

5.6.1. Combination of Percentage and Dimension Mixes

Interpolation of percengage-dimension value combinations (e.g. <length-percentage>, <frequency-percentage>, <angle-percentage>, <time-percentage> or equivalent notations) is defined as

• equivalent to interpolation of <length> if both V_a and V_b are pure <length> values
• equivalent to interpolation of <percentage> if both V_a and V_b are pure <percentage> values
• equivalent to converting both values into a calc() expression representing the sum of the dimension type and a percentage (each possibly zero) and interpolating each component individually (as a <length>/<frequency>/<angle>/<time> and as a <percentage>, respectively)

Addition of <percentage> is defined the same as interpolation except by adding each component rather than interpolating it.

5.7. Ratios: the <ratio> type

Ratio values are denoted by <ratio>, and represent the ratio of two numeric values. It most often represents an aspect ratio, relating a width (first) to a height (second).

When written literally, a ratio has the syntax:

<ratio> = <number [0,∞]> [ / <number [0,∞]> ]?

The second <number> is optional, defaulting to 1. However, <ratio> is always serialized with both components.

The computed value of a <ratio> is the pair of numbers provided.

If either number in the <ratio> is 0 or infinite, it represents a degenerate ratio (and, generally, won’t do anything).

If two <ratio>s need to be compared, divide the first number by the second, and compare the results. For example, 3/2 is less than 2/1, because it resolves to 1.5 while the second resolves to 2. (In other words, “tall” aspect ratios are less than “wide” aspect ratios.)

5.7.1. Combination of <ratio>

The interpolation of a <ratio> is defined by converting each <ratio> to a number by dividing the first value by the second (so a ratio of 3 / 2 would become 1.5), taking the logarithm of that result (so the 1.5 would become approximately 0.176), then interpolating those values. The result during the interpolation is converted back to a <ratio> by inverting the logarithm, then interpreting the result as a <ratio> with the result as the first value and 1 as the second value.

If either <ratio> is degenerate, the values cannot be interpolated.

start  = log(5);   // ≈ 0.69897
end    = log(1.5); // ≈ 0.17609
interp = 0.69897*.5 + 0.17609*.5; // ≈ 0.43753
final  = 10^interp; // ≈ 2.73

Note: Interpolating over the logarithm of the ratio means the results are scale-independent (5 / 1 to 300 / 200 would give the same results as above), that they’re symmetrical over "wide" and "tall" variants (interpolating from 1 / 5 to 2 / 3 would give a ratio approximately equal to 1 / 2.73 at the halfway point), and that they’re symmetrical over whether the width is fixed and the height is based on the ratio or vice versa. These properties are not shared by many other possible interpolation strategies.

Note: Due to the properties of logarithms, any log can be used; the example here uses base-10 log, but if, say, the natural log and e was used, the intermediate results would be different but the final result would be the same.

Addition of <ratio>s is not possible.

6. Distance Units: the <length> type

Lengths refer to distance measurements and are denoted by <length> in the property definitions. A length is a dimension.

For zero lengths the unit identifier is optional (i.e. can be syntactically represented as the <number> 0). However, if a 0 could be parsed as either a <number> or a <length> in a property (such as line-height), it must parse as a <number>.

Properties may restrict the length value to some range. If the value is outside the allowed range, the declaration is invalid and must be ignored.

While some properties allow negative length values, this may complicate the formatting and there may be implementation-specific limits. If a negative length value is allowed but cannot be supported, it must be converted to the nearest value that can be supported.

In cases where the used length cannot be supported, user agents must approximate it in the actual value.

There are two types of length units: relative and absolute. The specified value of a length is represented by its quantity and its unit. The computed value of a length is the specified length resolved to an absolute length, and its unit is not distinguished: it can be represented by any absolute length unit (but will be serialized using its canonical unit, px).

6.1. Relative Lengths

Relative length units specify a length relative to another length. Style sheets that use relative units can more easily scale from one output environment to another.

The relative units are:

Child elements do not inherit the relative values as specified for their parent; they inherit the computed values.

6.1.1. Font-relative Lengths: the em, ex, cap, ch, ic, rem, lh, rlh units

The font-relative lengths refer to the font metrics of the element on which they are used—or, in the case of rem and rlh, the metrics of the root element.

em unit

Equal to the computed value of the font-size property of the element on which it is used.

The rule:

h1 { line-height: 1.2em }

means that the line height of h1 elements will be 20% greater than the font size of h1 element. On the other hand:

h1 { font-size: 1.2em }

means that the font size of h1 elements will be 20% greater than the computed font size inherited by h1 elements.

ex unit

Equal to the used x-height of the first available font [CSS3-FONTS]. The x-height is so called because it is often equal to the height of the lowercase "x". However, an ex is defined even for fonts that do not contain an "x". The x-height of a font can be found in different ways. Some fonts contain reliable metrics for the x-height. If reliable font metrics are not available, UAs may determine the x-height from the height of a lowercase glyph. One possible heuristic is to look at how far the glyph for the lowercase "o" extends below the baseline, and subtract that value from the top of its bounding box. In the cases where it is impossible or impractical to determine the x-height, a value of 0.5em must be assumed.

cap unit

Equal to the used cap-height of the first available font [CSS3-FONTS]. The cap-height is so called because it is approximately equal to the height of a capital Latin letter. However, a cap is defined even for fonts that do not contain Latin letters. The cap-height of a font can be found in different ways. Some fonts contain reliable metrics for the cap-height. If reliable font metrics are not available, UAs may determine the cap-height from the height of an uppercase glyph. One possible heuristic is to look at how far the glyph for the uppercase “O” extends below the baseline, and subtract that value from the top of its bounding box. In the cases where it is impossible or impractical to determine the cap-height, the font’s ascent must be used.

ch unit

Equal to the used advance measure of the “0” (ZERO, U+0030) glyph in the font used to render it. (The advance measure of a glyph is its advance width or height, whichever is in the inline axis of the element.)

Note: This measurement is an approximation (and in monospace fonts, an exact measure) of a single narrow glyph’s advance measure, thus allowing measurements based on an expected glyph count.

Note: The advance measure of a glyph depends on writing-mode and text-orientation as well as font settings, text-transform, and any other properties that affect glyph selection or orientation.

In the cases where it is impossible or impractical to determine the measure of the “0” glyph, it must be assumed to be 0.5em wide by 1em tall. Thus, the ch unit falls back to 0.5em in the general case, and to 1em when it would be typeset upright (i.e. writing-mode is vertical-rl or vertical-lr and text-orientation is upright).

ic unit

Equal to the used advance measure of the “水” (CJK water ideograph, U+6C34) glyph found in the font used to render it.

This measurement is a typically an exact measure (in the few fonts with proportional fullwidth glyphs, an approximation) of a single fullwidth glyph’s advance measure, thus allowing measurements based on an expected glyph count.

In the cases where it is impossible or impractical to determine the ideographic advance measure, it must be assumed to be 1em.

rem unit

Equal to the computed value of font-size on the root element. When specified in the font-size property of the root element, or in a document with no root element, 1rem is equal to the initial value of the font-size property.

lh unit

Equal to the computed value of the line-height property of the element on which it is used, converting normal to an absolute length by using only the metrics of the first available font.

rlh unit

Equal to the computed value of line-height property on the root element, converting normal to an absolute length as above.

Note: Setting the height of an element using either the lh or the rlh units does not enable authors to control the actual number of lines in that element. These units only enable length calculations based on the theoretical size of an ideal empty line; the size of actual lines boxes may differ based on their content. In cases where an author wants to limit the number of actual lines in an element, the max-lines property can be used instead.

We can potentially add more typographic units, like cicero, didot, etc. They’re just absolute units, and so can be done with the existing units, but is there enough desire for them (potentially for printing use-cases) that it would be worth adding them? Or should we just wait for Houdini Custom Units?

Some user-agents allow users to apply additional restrictions to font sizes in a document, such as setting minimum font sizes to ensure readability. When used in the context of an element, these additional restrictions must be applied to the used value of these properties only; they must not affect the resolution of relative units.

When used outside the context of an element (such as in media queries), these units refer to the metrics corresponding to the initial values of the font and line-height properties. In this context, the units must apply any additional restrictions to the values, contrary to the normal behavior mentioned above.

When used in the value of the font-size property on the element they refer to, they resolve against the computed metrics of the parent element—or against the computed metrics corresponding to the initial values of the font and line-height properties, if the element has no parent. Additionally, when lh or rlh units are used in the value of the line-height property on the element they refer to, they resolve against the computed line-height and font metrics of the parent element—or the computed metrics corresponding to the initial values of the font and line-height properties, if the element has no parent. (The other font-relative units continue to resolve against the element’s own metrics when used in line-height.)

6.1.2. Viewport-percentage Lengths: the *vw, *vh, *vi, *vb, *vmin, *vmax units

The viewport-percentage lengths are relative to the size of the initial containing block—which is itself based on the size of either the viewport (for continuous media) or the page area (for paged media). When the height or width of the initial containing block is changed, they are scaled accordingly.

6.1.2.1. The Large, Small, and Dynamic Viewport Sizes

There are four variants of the viewport-percentage length units, corresponding to four (possibly identical) notions of the viewport size.

UA-default viewport

The UA-default viewport-percentage units (v*) are defined with respect to a UA-defined UA-default viewport size, which for any given document should be equivalent to the large viewport size, small viewport size, or some intermediary size.

Should the UA-default viewport size be required to correspond to the size of the initial containing block?

Note: Implementations that choose a size other than the large viewport size or small viewport size are encouraged to explain their choice to the CSSWG for consideration in future specification updates.

large viewport

The large viewport-percentage units (lv*) are defined with respect to the large viewport size: the viewport sized assuming any UA interfaces that are dynamically expanded and retracted to be retracted. This allows authors to size content such that it is guaranteed to fill the viewport, noting that such content might be hidden behind such interfaces when they are expanded.

The sizes of the large viewport-percentage units are fixed (and therefore stable) unless the viewport itself is resized.

For example, on phones, where screen real-estate is at a premium, browsers will often hide part or all of the title and address bar once the user starts scrolling the page. The large viewport-percentage units are sized relative to this larger everything-retracted space, so content using these units will fill the entire visible page when these UI elements are hidden. However, when these retractable elements are shown, they can obscure content that is sized or positioned using these units.

small viewport

The small viewport-percentage units (sv*) are defined with respect to the small viewport size: the viewport sized assuming any UA interfaces that are dynamically expanded and retracted to be expanded. This allows authors to size content such that it can fit within the viewport even when such interfaces are present, noting that such content might not fill the viewport when such interfaces are retracted.

The sizes of the small viewport-percentage units are fixed (and therefore stable) unless the viewport itself is resized.

An element that is sized as height: 100svh, for example, will fill the screen perfectly, without any of its content being obscured, when all the dynamic UI elements of the UA are shown.

Once those UI elements start being hidden, however, there will be extra space around the element. The small viewport-percentage units units are thus “safer” in general, but might not produce the most attractive layout once the user starts interacting with the page.

dynamic viewport

The dynamic viewport-percentage units (dv*) are defined with respect to the dynamic viewport size: the viewport sized with dynamic consideration of any UA interfaces that are dynamically expanded and retracted. This allows authors to size content such that it can exactly fit within the viewport whether or not such interfaces are present.

The sizes of the dynamic viewport-percentage units are not stable even while the viewport itself is unchanged. Using these units can cause content to resize e.g. while the user scrolls the page. Depending on usage, this can be disturbing to the user and/or costly in terms of performance.

The UA is not required to animate the dynamic viewport-percentage units while expanding and retracting any relevant interfaces, and may instead calculate the units as if the relevant interface was fully expanded or retracted during the UI animation. (It is recommended that UAs assume the fully-retracted size for this duration.)

Whether the expansion/retraction of a particular interface (A) changes the sizes of all of the viewport-percentage lengths (and the initial containing block) simultaneously or (B) contributes to the differences between the large viewport size and small viewport size is largely UA-dependent. However:

• Changes in interface that happen as a result of scrolling or other frequent page interactions that would disturb the user if they resulted in substantial layout changes must be categorized as the former (A).

• Changes in interface that have a sufficiently steady state that re-laying out the document into the adjusted space would be beneficial to the user must be categorized as the latter (B).

• Additionally, UAs may have some dynamically-shown interfaces that intentionally overlay content and do not cause any shifts in layout—and therefore have no effect on any of the viewport-percentage lengths. (Typically on-screen keyboards will fit into this category.)

In all cases, scrollbars are assumed not to exist. Note however that the initial containing block's size is affected by the presence of scrollbars on the viewport.

Level 3 assumes scrollbars never exist because it was hard to implement and only Firefox bothered to do so. This is making authors unhappy. Can we improve here?

6.1.2.2. The Various Viewport-relative Units

The viewport-percentage length units are:

vw unit

svw unit

lvw unit

dvw unit

Equal to 1% of the width of the UA-default viewport size, small viewport size, large viewport size, and dynamic viewport size, respectively.

In the example below, if the width of the viewport is 200mm, the font size of h1 elements will be 16mm (i.e. (8×200mm)/100).

h1 { font-size: 8vw }

vh unit

svh unit

lvh unit

dvh unit

Equal to 1% of the height of the UA-default viewport size, small viewport size, large viewport size, and dynamic viewport size, respectively.

vi unit

svi unit

lvi unit

dvi unit

Equal to 1% of the size of the large viewport size, small viewport size, and dynamic viewport size (respectively) in the direction of the root element’s inline axis.

vb unit

svb unit

lvb unit

dvb unit

Equal to 1% of the size of the initial containing block UA-default viewport size, small viewport size, large viewport size, and dynamic viewport size (respectively) in the direction of the root element’s block axis.

vmin unit

svmin unit

lvmin unit

dvmin unit

Equal to the smaller of *vw or *vh.

vmax unit

svmax unit

lvmax unit

dvmax unit

Equal to the larger of *vw or *vh.

Originally the (unprefixed) viewport units were defined relative to the viewport size in general. The dynamism of browser chrome shifting in and out during scrolling was invented later, and following Safari’s lead, most UAs mapped these units to the larger size. Defining it this way is prettier in many cases, but can also block critical content (such as toolbars, headers, and footers) in others. It’s therefore not entirely clear whether this is the best mapping.

In situations where there is no root element or it hasn’t yet been styled (such as when evaluating media queries), the *vi and *vb units use the initial value of the writing-mode property to determine which axis they correspond to.

6.2. Absolute Lengths: the cm, mm, Q, in, pt, pc, px units

The absolute length units are fixed in relation to each other and anchored to some physical measurement. They are mainly useful when the output environment is known. The absolute units consist of the physical units (in, cm, mm, pt, pc, Q) and the visual angle unit (pixel unit) (px):

h1 { margin: 0.5in }      /* inches  */
h2 { line-height: 3cm }   /* centimeters */
h3 { word-spacing: 4mm }  /* millimeters */
h3 { letter-spacing: 1Q } /* quarter-millimeters */
h4 { font-size: 12pt }    /* points */
h4 { font-size: 1pc }     /* picas */
p  { font-size: 12px }    /* px */

All of the absolute length units are compatible, and px is their canonical unit.

For a CSS device, these dimensions are anchored either

1. by relating the physical units to their physical measurements, or
2. by relating the pixel unit to the reference pixel.

For print media at typical viewing distances, the anchor unit should be one of the physical units (inches, centimeters, etc). For screen media (including high-resolution devices), low-resolution devices, and devices with unusual viewing distances, it is recommended instead that the anchor unit be the pixel unit. For such devices it is recommended that the pixel unit refer to the whole number of device pixels that best approximates the reference pixel.

Note: If the anchor unit is the pixel unit, the physical units might not match their physical measurements. Alternatively if the anchor unit is a physical unit, the pixel unit might not map to a whole number of device pixels.

Note: This definition of the pixel unit and the physical units differs from the earlier editions of CSS1 and CSS2. In particular, in previous versions of CSS the pixel unit and the physical units were not related by a fixed ratio: the physical units were always tied to their physical measurements while the pixel unit would vary to most closely match the reference pixel. (This unfortunate change was made because too much existing content relies on the assumption of 96dpi, and breaking that assumption broke the content.)

Note: Units are ASCII case-insensitive and serialize as lower case, for example 1Q serializes as 1q.

The reference pixel is the visual angle of one pixel on a device with a pixel density of 96dpi and a distance from the reader of an arm’s length. For a nominal arm’s length of 28 inches, the visual angle is therefore about 0.0213 degrees. For reading at arm’s length, 1px thus corresponds to about 0.26 mm (1/96 inch).

The image below illustrates the effect of viewing distance on the size of a reference pixel: a reading distance of 71 cm (28 inches) results in a reference pixel of 0.26 mm, while a reading distance of 3.5 m (12 feet) results in a reference pixel of 1.3 mm.

This second image illustrates the effect of a device’s resolution on the pixel unit: an area of 1px by 1px is covered by a single dot in a low-resolution device (e.g. a typical computer display), while the same area is covered by 16 dots in a higher resolution device (such as a printer).

7. Other Quantities

7.1. Angle Units: the <angle> type and deg, grad, rad, turn units

Angle values are <dimension>s denoted by <angle>. The angle unit identifiers are:

deg

Degrees. There are 360 degrees in a full circle.

grad

Gradians, also known as "gons" or "grades". There are 400 gradians in a full circle.

rad

Radians. There are 2π radians in a full circle.

turn

Turns. There is 1 turn in a full circle.

For example, a right angle is 90deg or 100grad or 0.25turn or approximately 1.57rad.

All <angle> units are compatible, and deg is their canonical unit.

Note: For legacy reasons, some uses of <angle> allow a bare 0 to mean 0deg. This is not true in general, however, and will not occur in future uses of the <angle> type.

7.2. Duration Units: the <time> type and s, ms units

Time values are dimensions denoted by <time>. The time unit identifiers are:

s

Seconds.

ms

Milliseconds. There are 1000 milliseconds in a second.

All <time> units are compatible, and s is their canonical unit.

Properties may restrict the time value to some range. If the value is outside the allowed range, the declaration is invalid and must be ignored.

7.3. Frequency Units: the <frequency> type and Hz, kHz units

Frequency values are dimensions denoted by <frequency>. The frequency unit identifiers are:

Hz

Hertz. It represents the number of occurrences per second.

kHz

KiloHertz. A kiloHertz is 1000 Hertz.

For example, when representing sound pitches, 200Hz (or 200hz) is a bass sound, and 6kHz (or 6khz) is a treble sound.

All <frequency> units are compatible, and hz is their canonical unit.

Note: Units are ASCII case-insensitive and serialize as lower case, for example 1Hz serializes as 1hz.

7.4. Resolution Units: the <resolution> type and dpi, dpcm, dppx units

Resolution units are dimensions denoted by <resolution>. The resolution unit identifiers are:

dpi

Dots per inch.

dpcm

Dots per centimeter.

dppx

x

Dots per px unit.

The <resolution> unit represents the size of a single "dot" in a graphical representation by indicating how many of these dots fit in a CSS in, cm, or px. For uses, see e.g. the resolution media query in [MEDIAQ] or the image-resolution property defined in [CSS3-IMAGES].

All <resolution> units are compatible, and dppx is their canonical unit.

Note that due to the 1:96 fixed ratio of CSS in to CSS px, 1dppx is equivalent to 96dpi. This corresponds to the default resolution of images displayed in CSS: see image-resolution.

@media (min-resolution: 2dppx) { ... }

8. Data Types Defined Elsewhere

Some data types are defined in their own modules. This example talks about some of the most common ones used across several specifications.

8.1. Colors: the <color> type

The <color> data type is defined in [CSS-COLOR-4]. UAs must interpret <color> as defined therein.

8.1.1. Combination of <color>

Interpolation of <color> is defined in CSS Color 4 § 13 Interpolation. Interpolation is done between premultiplied colors, as defined in CSS Color 4 § 13.2 Interpolating with alpha.

Addition of <color> is likewise defined as the independent addition of each component as a <number> in premultiplied space.

8.2. Images: the <image> type

The <image> data type is defined in [CSS3-IMAGES]. UAs that support CSS Images Level 3 or its successor must interpret <image> as defined therein. UAs that do not yet support CSS Images Level 3 must interpret <image> as <url>.

8.2.1. Combination of <image>

Note: Interpolation of <image> is defined in CSS Images 3 § 6 Interpolation.

Images are not additive.

8.3. 2D Positioning: the <position> type

The <position> value specifies the position of a object area (e.g. background image) inside a positioning area (e.g. background positioning area). It is interpreted as specified for background-position. [CSS3-BACKGROUND]

<position> = [
  [ left | center | right ] || [ top | center | bottom ]
|
  [ left | center | right | <length-percentage> ]
  [ top | center | bottom | <length-percentage> ]?
|
  [ [ left | right ] <length-percentage> ] &&
  [ [ top | bottom ] <length-percentage> ]
]

Note: The background-position property also accepts a three-value syntax. This has been disallowed generically because it creates parsing ambiguities when combined with other length or percentage components in a property value.

The canonical order when serializing is the horizontal component followed by the vertical component.

When specified in a grammar alongside other keywords, <length>s, or <percentage>s, <position> is greedily parsed; it consumes as many components as possible.

8.3.1. Combination of <position>

Interpolation of <position> is defined as the independent interpolation of each component (x, y) normalized as an offset from the top left corner as a <length-percentage>.

Addition of <position> is likewise defined as the independent addition each component (x, y) normalized as an offset from the top left corner as a <length-percentage>.

9. Functional Notations

A functional notation is a type of component value that can represent more complex types or invoke special processing. The syntax starts with the name of the function immediately followed by a left parenthesis (i.e. a <function-token>) followed by the argument(s) to the notation followed by a right parenthesis. White space is allowed, but optional, immediately inside the parentheses. Functions can take multiple arguments, which are formatted similarly to a CSS property value.

Some legacy functional notations, such as rgba(), use commas unnecessarily, but generally commas are only used to separate items in a list, or pieces of a grammar that would be ambiguous otherwise. If a comma is used to separate arguments, white space is optional before and after the comma.

background: url(http://www.example.org/image);
color: rgb(100, 200, 50 );
content: counter(list-item) ". ";
width: calc(50% - 2em);

The math functions are defined in § 11 Mathematical Expressions.

9.1. Toggling Between Values: toggle()

The toggle() expression allows descendant elements to cycle over a list of values instead of inheriting the same value.

em { font-style: toggle(italic, normal); }

ul { list-style-type: toggle(disc, circle, square, box); }

The syntax of the toggle() expression is:

toggle( <toggle-value># )

where <toggle-value> is any CSS value that is valid where the expression is placed, and that doesn’t contain any top-level commas. If any of the values inside are not valid, then the entire toggle() expression is invalid. The toggle() expression may be used as the value of any property, but must be the only component in that property’s value.

The toggle() notation is not allowed to be nested; nor may it contain attr() or calc() notations. Declarations containing such constructs are invalid.

background-position: 10px toggle(50px, 100px);
/* toggle() must be the sole value of the property */

list-style-type: toggle(disc, 50px);
/* 50px isn’t a valid value of 'list-style-type' */

To determine the computed value of toggle(), first evaluate each argument as if it were the sole value of the property in which toggle() is placed to determine the computed value that each represents, called C_n for the n-th argument to toggle(). Then, compare the property’s inherited value with each C_n. For the earliest C_n that matches the inherited value, the computed value of toggle() is C_n+1. If the match was the last argument in the list, or there was no match, the computed value of toggle() is the computed value that the first argument represents.

Note: This means that repeating values in a toggle() short-circuits the list. For example toggle(1em, 2em, 1em, 4em) will be equivalent to toggle(1em, 2em).

Note: That toggle() explicitly looks at the computed value of the parent, so it works even on non-inherited properties. This is similar to the inherit keyword, which works even on non-inherited properties.

Note: That the computed value of a property is an abstract set of values, not a particular serialization [CSS21], so comparison between computed values should always be unambiguous and have the expected result. For example, a Level 2 background-position computed value is just two offsets, each represented as an absolute length or a percentage, so the declarations background-position: top center and background-position: 50% 0% produce identical computed values. If the "Computed Value" line of a property definition seems to define something ambiguous or overly strict, please provide feedback so we can fix it.

If toggle() is used on a shorthand property, it sets each of its longhands to a toggle() value with arguments corresponding to what the longhand would have received had each of the original toggle() arguments been the sole value of the shorthand.

margin: toggle(1px 2px, 4px, 1px 5px 4px);

margin-top:    toggle(1px, 4px, 1px);
margin-right:  toggle(2px, 4px, 5px);
margin-bottom: toggle(1px, 4px, 4px);
margin-left:   toggle(2px, 4px, 5px);

margin-top:    toggle(1px, 4px);
margin-right:  toggle(2px, 4px, 5px);
margin-bottom: toggle(1px, 4px);
margin-left:   toggle(2px, 4px, 5px);

10. Attribute References: the attr() function

The attr() function substitutes the value of an attribute on an element into a property, similar to how the var() function substitutes a custom property value into a function.

attr() = attr( <wq-name> <attr-type>? , <declaration-value>?)

<attr-type> = string | url | ident | color | number | percentage |
              length | angle | time | frequency | flex | <dimension-unit>

The <dimension-unit> production matches a literal "%" character (that is, a <delim-token> with a value of "%") or an ident whose value is any of the CSS units for <length>, <angle>, <time>, <frequency>, or <flex> values (such as px or ms).

The arguments of attr() are:

<wq-name>

Gives the name of the attribute being referenced.

If no namespace is specified (just an identifier is given, like attr(foo)), the null namespace is implied. (This is usually what’s desired, as namespaced attributes are rare. In particular, HTML and SVG do not contain namespaced attributes.) As with attribute selectors, the case-sensitivity of <wq-name> depends on the document language.

If attr() is used in a property applied to an element, it references the attribute of the given name on that element; if applied to a pseudo-element, the attribute is looked up on the pseudo-element’s originating element.

<attr-type>

Specifies what kind of CSS value the attribute’s value will be interpreted into (the attr()’s substitution value) and what, if any, special parsing will be done to the value.

The possible values and their behavior are defined in § 10.1 attr() Types.

Defaults to string if omitted.

<declaration-value>

Specifies a fallback value for the attr(), which will be substituted instead of the attribute’s value if the attribute is missing or fails to parse as the specified type.

If the <attr-type> argument is string, defaults to the empty string if omitted; otherwise, defaults to the guaranteed-invalid value if omitted.

If a property contains one or more attr() functions, and those functions are syntactically valid, the entire property’s grammar must be assumed to be valid at parse time. It is only syntax-checked at computed-value time, after attr() functions have been substituted.

10.1. attr() Types

The behavior of the attr() function depends partially on the value of the <attr-type> argument:

string

The substitution value is a CSS string, whose value is the literal value of the attribute. (No CSS parsing or "cleanup" of the value is performed.)

No value triggers fallback.

url

The substitution value is a CSS <url> value, whose url is the literal value of the attribute. (No CSS parsing or "cleanup" of the value is performed.)

Note: If url() was syntactically capable of containing functions, attr(foo url) would be identical to url(attr(foo string)).

No value triggers fallback.

ident

The substitution value is a CSS <custom-ident>, whose value is the literal value of the attribute, with leading and trailing ASCII whitespace stripped. (No CSS parsing of the value is performed.)

If the attribute value, after trimming, is the empty string, there is instead no substitution value.

If the <custom-ident>’s value is a CSS-wide keyword or default, there is instead no substitution value.

color

Parse a component value from the attribute’s value. If the result is a <hex-color> or a named color ident, the substitution value is that result as a <color>.

Otherwise there is no substitution value.

number

Parse a component value from the attribute’s value. If the result is a <number-token>, the result is the substitution value.

Otherwise, there is no substitution value.

percentage

Parse a component value from the attribute’s value. If the result is a <percentage-token>, the result is the substitution value.

Otherwise, there is no substitution value.

length

angle

time

frequency

flex

Parse a component value from the attribute’s value. If the result is a <dimension-token> whose unit matches the given type, the result is the substitution value.

Otherwise, there is no substitution value.

<dimension-unit>

Parse a component value from the attribute’s value. If the result is a <number-token>, the substitution value is a dimension with the result’s value, and the given unit.

Otherwise, there is no substitution value.

Do we want to allow math functions as attr values for all the numeric types? And color functions for "color"? I think we do, but I’d have to check the contents to make sure they don’t contain further reference functions; foo="rgb(var(--red), 0, 0)" needs to be illegal for attr(foo color).

<stock>
  <wood length="12"/>
  <wood length="5"/>
  <metal length="19"/>
  <wood length="4"/>
</stock>

stock::before {
  display: block;
  content: "To scale, the lengths of materials in stock are:";
}
stock > * {
  display: block;
  width: attr(length em, 0px);
  height: 1em;
  border: solid thin;
  margin: 0.5em;
}
wood {
  background: orange url(wood.png);
}
metal {
  background: silver url(metal.png);
}

10.2. attr() Substitution

attr() and var() substitute at the same time, so I should probably rewrite substitute a var() to be more generally about "substitute a reference" and just use that for both of these functions.

attr() functions are substituted at computed-value time. If a declaration, once all attr() functions are substituted in, does not match its declared grammar, the declaration is invalid at computed-value time.

To substitute an attr():

1. If the attr() function has a substitution value, replace the attr() function by the substitution value.

2. Otherwise, if the attr() function has a fallback value as its last argument, replace the attr() function by the fallback value. If there are any var() or attr() references in the fallback, substitute them as well.

3. Otherwise, the property containing the attr() function is invalid at computed-value time.

11. Mathematical Expressions

The math functions (calc(), clamp(), sin(), and others defined in this chapter) allow numeric CSS values to be written as mathematical expressions.

A math function represents a numeric value, one of:

• <length>,

• <frequency>,

• <angle>,

• <time>,

• <flex>,

• <resolution>,

• <percentage>,

• <number>,

• <integer>

...or the <length-percentage>/etc mixed types, and can be used wherever such a value would be valid.

11.1. Basic Arithmetic: calc()

The calc() function is a math function that allows basic arithmetic to be performed on numerical values, using addition (+), subtraction (-), multiplication (*), division (/), and parentheses.

A calc() function contains a single calculation, which is a sequence of values interspersed with operators, and possibly grouped by parentheses (matching the <calc-sum> grammar), which represents the result of evaluating the expression using standard operator precedence rules (* and / bind tighter than + and -, and operators are otherwise evaluated left-to-right). The calc() function represents the result of its contained calculation.

Components of a calculation can be literal values (such as 5px), other math functions, or other expressions, such as attr(), that evaluate to a valid argument type (like <length>).

section {
  float: left;
  margin: 1em; border: solid 1px;
  width: calc(100% / 3 - 2 * 1em - 2 * 1px);
}

.fade {
  background-image: linear-gradient(silver 0%, white 20px,
                                    white calc(100% - 20px), silver 100%);
}

:root {
  font-size: calc(100vw / 35);
}

.aspect-ratio-box {
  --ar: calc(16 / 9);
  --w: calc(100% / 3);
  --h: calc(var(--w) / var(--ar));
  width: var(--w);
  height: var(--h);
}

11.2. Comparison Functions: min(), max(), and clamp()

The comparison functions of min(), max(), and clamp() compare multiple calculations and represent the value of one of them.

The min() or max() functions contain one or more comma-separated calculations, and represent the smallest (most negative) or largest (most positive) of them, respectively.

The clamp() function takes three calculations—a minimum value, a central value, and a maximum value—and represents its central calculation, clamped according to its min and max calculations, favoring the min calculation if it conflicts with the max. (That is, given clamp(MIN, VAL, MAX), it represents exactly the same value as max(MIN, min(VAL, MAX))).

.type {
  /* Set font-size to 10x the average of vw and vh,
     but don’t let it go below 12px. */
  font-size: max(10 * (1vw + 1vh) / 2, 12px);
}

.type {
  /* Force the font-size to stay between 12px and 100px */
  font-size: clamp(12px, 10 * (1vw + 1vh) / 2, 100px);
}

11.3. Stepped Value Functions: round(), mod(), and rem()

The stepped-value functions, round(), mod(), and rem(), all transform a given value according to another "step value", in different ways.

The round(<rounding-strategy>?, A, B) function contains an optional rounding strategy, and two calculations A and B, and returns the value of A, rounded according to the rounding strategy, to the nearest integer multiple of B either above or below A. The argument calculations can resolve to any <number>, <dimension>, or <percentage>, but must have the same type, or else the function is invalid; the result will have the same type as the arguments.

If A is exactly equal to an integer multiple of B, round() resolves to A exactly (preserving whether A is 0⁻ or 0⁺, if relevant). Otherwise, there are two integer multiples of B that are potentially "closest" to A, lower B which is closer to −∞ and upper B which is closer to +∞. The following <rounding-strategy>s dictate how to choose between them:

nearest

Choose whichever of lower B and upper B that has the smallest absolute difference from A. If both have an equal difference (A is exactly between the two values), choose upper B.

up

Choose upper B.

down

Choose lower B.

to-zero

Choose whichever of lower B and upper B that has the smallest absolute difference from 0.

If lower B would be zero, it is specifically equal to 0⁺; if upper B would be zero, it is specifically equal to 0⁻.

If <rounding-strategy> is omitted, it defaults to nearest.

CSSOM needs to specify how it rounds, and it’s probably good for CSS functions to round the same way by default. What behavior should be used? [Issue #5689]

Note: JavaScript and other programming languages sometimes separate out the rounding strategies into separate rounding functions. JS’s Math.floor() is equivalent to CSS’s round(down, ...); JS’s Math.ceil() is equivalent to CSS’s round(up, ...); JS’s Math.trunc() is equivalent to CSS’s round(to-zero, ...); and JS’s Math.round() is equivalent to CSS’s round(nearest, ...), or just round(...).

Note: The <rounding-strategy> keywords are the same as the keywords in block-step-size and have the same behavior. (block-step-size just lacks to-zero; since block sizes are always non-negative, to-zero and down would be identical.)

The modulus functions mod(A, B) and rem(A, B) similarly contain two calculations A and B, and return the difference between A and the nearest integer multiple of B either above or below A. The argument calculations can resolve to any <number>, <dimension>, or <percentage>, but must have the same type, or else the function is invalid; the result will have the same type as the arguments.

The two functions are very similar, and in fact return identical results if both arguments are positive or both are negative: the value of the function is equal to the value of A shifted by the integer multiple of B that brings the value between zero and B. (Specifically, the range includes zero and excludes B. More specifically, if B is positive the range starts at 0⁺, and if B is negative it starts at 0⁻.)

Their behavior diverges if the A value and the B step are on opposite sides of zero: mod() (short for “modulus”) continues to choose the integer multiple of B that puts the value between zero and B, as above (guaranteeing that the result will either be zero or share the sign of B, not A), while rem() (short for "remainder") chooses the integer multiple of B that puts the value between zero and -B, avoiding changing the sign of the value.

Note: mod() and rem() can also be defined directly in terms of other functions: mod(A, B) is equivalent to calc(A - sign(B)*round(down, A*sign(B), B)) (a hacky way to say "round(down) when B is positive, round(up) when B is negative), while rem(A, B) is equivalent to calc(A - round(to-zero, A, B)). (These expressions don’t always handle 0⁺ and 0⁻ correctly, though, because 0⁻ semantics aren’t commutative for addition.)

11.3.1. Argument Ranges

In round(A, B), if B is 0, the result is NaN. If A and B are both infinite, the result is NaN.

If A is infinite but B is finite, the result is the same infinity.

If A is finite but B is infinite, the result depends on the <rounding-strategy> and the sign of A:

nearest

to-zero

If A is positive or 0⁺, return 0⁺. Otherwise, return 0⁻.

up

If A is positive (not zero), return +∞. If A is 0⁺, return 0⁺. Otherwise, return 0⁻.

down

If A is negative (not zero), return −∞. If A is 0⁻, return 0⁻. Otherwise, return 0⁺.

In mod(A, B) or rem(A, B), if B is 0, the result is NaN. If A is infinite, the result is NaN.

In mod(A, B) only, if B is infinite and A is non-zero and has opposite sign to B, the result is NaN.

Note: All other "infinite B" cases are valid, and just return A immediately.

11.4. Trigonometric Functions: sin(), cos(), tan(), asin(), acos(), atan(), and atan2()

The trigonometric functions—sin(), cos(), tan(), asin(), acos(), atan(), and atan2()—compute the various basic trigonometric relationships.

The sin(A), cos(A), and tan(A) functions all contain a single calculation which must resolve to either a <number> or an <angle>, and compute their corresponding function by interpreting the result of their argument as radians. (That is, sin(45deg), sin(.125turn), and sin(3.14159 / 4) all represent the same value, approximately .707.) They all represent a <number>; sin() and cos() will always return a number between −1 and 1, while tan() can return any number between −∞ and +∞. (See § 11.9 Type Checking for details on how math functions handle ∞.)

The asin(A), acos(A), and atan(A) functions are the "arc" or "inverse" trigonometric functions, representing the inverse function to their corresponding "normal" trig functions. All of them contain a single calculation which must resolve to a <number>, and compute their corresponding function, interpreting their result as a number of radians, representing an <angle>. The angle returned by asin() must be normalized to the range [-90deg, 90deg]; the angle returned by acos() to the range [0deg, 180deg]; and the angle returned by atan() to the range [-90deg, 90deg].

The atan2(A, B) function contains two comma-separated calculations, A and B. A and B can resolve to any <number>, <dimension>, or <percentage>, but must have the same type, or else the function is invalid. The function returns the <angle> between the positive X-axis and the point (B,A). The returned angle must be normalized to the interval (-180deg, 180deg] (that is, greater than -180deg, and less than or equal to 180deg).

Note: atan2(Y, X) is generally equivalent to atan(Y / X), but it gives a better answer when the point in question may include negative components. atan2(1, -1), corresponding to the point (-1, 1), returns 135deg, distinct from atan2(-1, 1), corresponding to the point (1, -1), which returns -45deg. In contrast, atan(1 / -1) and atan(-1 / 1) both return-45deg, because the internal calculation resolves to -1 for both.

11.4.1. Argument Ranges

In sin(A), cos(A), or tan(A), if A is infinite, the result is NaN. (See § 11.9 Type Checking for details on how math functions handle NaN.)

In sin(A) or tan(A), if A is 0⁻, the result is 0⁻.

In tan(A), if A is one of the asymptote values (such as 90deg, 270deg, etc), the result must be +∞ for 90deg and all values a multiple of 360deg from that (such as -270deg or 450deg), and −∞ for -90deg and all values a multiple of 360deg from that (such as -450deg or 270deg).

Note: This is only relevant for units that can exactly represent the asymptotic values, such as deg or grad. rad cannot, and so whether the result is a very large negative or positive value can depend on rounding and precise details of how numbers are internally stored. It’s recommended you don’t depend on this behavior if using such units.

In asin(A) or acos(A), if A is less than -1 or greater than 1, the result is NaN.

In acos(A), if A is exactly 1, the result is 0.

In asin(A) or atan(A), if A is 0⁻, the result is 0⁻.

In atan(A), if A is +∞, the result is 90deg; if A is −∞, the result is -90deg.

In atan2(Y, X), the following table gives the results for all unusual argument combinations:

Note: All of these behaviors are intended to match the "standard" definitions of these functions as implemented by most programming languages, in particular as implemented in JS.

Note: The behavior of tan(90deg), while not constrained by JS behavior (because the JS function’s input is in radians, and one cannot perfectly express a value of π/2 in JS numbers), is defined so that roundtripping of values works; tan(atan(infinity)) yields +∞, tan(atan(-infinity)) yields −∞, atan(tan(90deg)) yields 90deg, and atan(tan(-90deg)) yields -90deg.

11.5. Exponential Functions: pow(), sqrt(), hypot(), log(), exp()

The exponential functions—pow(), sqrt(), hypot(), log(), and exp()—compute various exponential functions with their arguments.

The pow(A, B) function contains two comma-separated calculations A and B, both of which must resolve to <number>s, and returns the result of raising A to the power of B, returning the value as a <number>.

The sqrt(A) function contains a single calculation which must resolve to a <number>, and returns the square root of the value as a <number>. (sqrt(X) and pow(X, .5) are basically equivalent, differing only in some error-handling; sqrt() is a common enough function that it is provided as a convenience.)

The hypot(A, …) function contains one or more comma-separated calculations, and returns the length of an N-dimensional vector with components equal to each of the calculations. (That is, the square root of the sum of the squares of its arguments.) The argument calculations can resolve to any <number>, <dimension>, or <percentage>, but must have the same type, or else the function is invalid; the result will have the same type as the arguments.

The log(A, B?) function contains one or two calculations (representing the value to be logarithmed, and the base of the logarithm, defaulting to e), which must resolve to <number>s, and returns the logarithm base B of the value A, as a <number>.

The exp(A) function contains one calculation which must resolve to a <number>, and returns the same value as pow(e, A) as a <number>.

:root {
  --h6: calc(1rem * pow(1.5, -1));
  --h5: calc(1rem * pow(1.5, 0));
  --h4: calc(1rem * pow(1.5, 1));
  --h3: calc(1rem * pow(1.5, 2));
  --h2: calc(1rem * pow(1.5, 3));
  --h1: calc(1rem * pow(1.5, 4));
}

11.5.1. Argument Ranges

In pow(A, B), if A is negative and finite, and B is finite, B must be an integer, or else the result is NaN.

If A or B are infinite or 0, the following tables give the results:

In sqrt(A), if A is +∞, the result is +∞. If A is 0⁻, the result is 0⁻. If A is less than 0, the result is NaN.

In hypot(A, …), if any of the inputs are infinite, the result is +∞.

In log(A, B), if B is 1 or negative, B values between 0 and 1, or greater than 1, are valid. the result is NaN. If A is negative, the result is NaN. If A is 0⁺ or 0⁻, the result is −∞. If A is 1, the result is 0⁺. If A is +∞, the result is +∞.

In exp(A), if A is +∞, the result is +∞. If A is −∞, the result is 0⁺.

(See § 11.9 Type Checking for details on how math functions handle NaN and infinities.)

11.6. Sign-Related Functions: abs(), sign()

The sign-related functions—abs() and sign()—compute various functions related to the sign of their argument.

The abs(A) function contains one calculation A, and returns the absolute value of A, as the same type as the input: if A’s numeric value is positive or 0⁺, just A again; otherwise -1 * A.

The sign(A) function contains one calculation A, and returns -1 if A’s numeric value is negative, +1 if A’s numeric value is positive, 0⁺ if A’s numeric value is 0⁺, and 0⁻ if A’s numeric value is 0⁻.

Note: Both of these functions operate on the fully simplified/resolved form of their arguments, which may give unintuitive results at first glance. In particular, an expression like 10% might be positive or negative once it’s resolved, depending on what value it’s resolved against. For example, in background-position positive percentages resolve to a negative length, and vice versa, if the background image is larger than the background area. Thus sign(10%) might return 1 or -1, depending on how the percentage is resolved! (Or even 0, if it’s resolved against a zero length.)

11.7. Numeric Constants: e, pi

While the trigonometric and exponential functions handle many complex numeric operations, some reasonable calculations must be put together more manually, and many times these include well-known constants, such as e and π.

Rather than require authors to manually type out several digits of these constants, a few of them are provided directly:

e is the base of the natural logarithm, approximately equal to 2.7182818284590452354.

pi is the ratio of a circle’s circumference to its diameter, approximately equal to 3.1415926535897932.

Note: These keywords are only usable within a calculation, such as calc(pow(e, pi) - pi), or min(pi, 5, e). If used outside of a calculation, they’re treated like any other keyword: animation-name: pi; refers to an animation named "pi"; line-height: e; is invalid (not similar to line-height: 2.7, but line-height: calc(e); is).

11.7.1. Degenerate Numeric Constants: infinity, -infinity, NaN

When a calculation or a subtree of a calculation becomes infinite or NaN, representing it with a numeric value is no longer possible. To aid in serialization of these degenerate values, the additional math constants infinity (with the value +∞), -infinity (with the value −∞), and NaN (with the value NaN) are defined.

As usual for CSS keywords, these are ASCII case-insensitive. Thus, calc(InFiNiTy) is perfectly valid. However, NaN must be serialized with this canonical casing.

Note: While not technically numbers, these keywords act as numeric values, similar to e and pi. Thus to get an infinite length, for example, requires an expression like calc(infinity * 1px).

Note: These constants are defined mostly to make serialization of infinite/NaN values simpler and more obvious, but can be used to indicate a "largest possible value", since an infinite value gets clamped to the allowed range. It’s rare for this to be reasonable, but when it is, using infinity is clearer in its intent than just putting an enormous number in one’s stylesheet.

11.8. Syntax

The syntax of a math function is:

<calc()>  = calc( <calc-sum> )
<min()>   = min( <calc-sum># )
<max()>   = max( <calc-sum># )
<clamp()> = clamp( <calc-sum>#{3} )
<round()> = round( <rounding-strategy>?, <calc-sum>, <calc-sum> )
<mod()>   = mod( <calc-sum>, <calc-sum> )
<rem()>   = rem( <calc-sum>, <calc-sum> )
<sin()>   = sin( <calc-sum> )
<cos()>   = cos( <calc-sum> )
<tan()>   = tan( <calc-sum> )
<asin()>  = asin( <calc-sum> )
<acos()>  = acos( <calc-sum> )
<atan()>  = atan( <calc-sum> )
<atan2()> = atan2( <calc-sum>, <calc-sum> )
<pow()>   = pow( <calc-sum>, <calc-sum> )
<sqrt()>  = sqrt( <calc-sum> )
<hypot()> = hypot( <calc-sum># )
<log()>   = log( <calc-sum>, <calc-sum>? )
<exp()>   = exp( <calc-sum> )
<abs()>   = abs( <calc-sum> )
<sign()>  = sign( <calc-sum> )
<calc-sum> = <calc-product> [ [ '+' | '-' ] <calc-product> ]*
<calc-product> = <calc-value> [ [ '*' | '/' ] <calc-value> ]*
<calc-value> = <number> | <dimension> | <percentage> |
               <calc-constant> | ( <calc-sum> )
<calc-constant> = e | pi | infinity | -infinity | NaN

In addition, whitespace is required on both sides of the + and - operators. (The * and / operators can be used without white space around them.)

Several of the math functions above have additional constraints on what their <calc-sum> arguments can contain. Check the definitions of the individual functions for details.

UAs must support calculations of at least 20 <calc-value> terms. If a calculation contains more than the supported number of terms, it must be treated as if it were invalid.

11.9. Type Checking

A math function can be many possible types, such as <length>, <number>, etc., depending on the calculations it contains, as defined below. It can be used anywhere a value of that type is allowed.

Additionally, math functions that resolve to <number> can be used in any place that only accepts <integer>. (It gets rounded to the nearest integer, as specified in § 11.12 Range Checking.)

Operators form sub-expressions, which gain types based on their arguments.

Note: In previous versions of this specification, multiplication and division were limited in what arguments they could take, to avoid producing more complex intermediate results (such as 1px * 1em, which is <length>²) and to make division-by-zero detectable at parse time. This version now relaxes those restrictions.

Math functions themselves have types, according to their contained calculations:

• The type of a calc() or abs() expression is the type of its contained calculation.

• The type of a min(), max(), or clamp() expression is the result of adding the types of its comma-separated calculations.

• The type of a sign() expression is «[ "number" → 1 ]».

• The type of a sin(), cos(), or tan() expression is «[ "number" → 1 ]».

• The type of an asin(), acos(), atan(), or atan2() expression is «[ "angle" → 1 ]».

• The type of a pow(), sqrt(), log(), or exp() expression is «[ "number" → 1 ]».

• The type of a hypot(), round(), mod(), or rem() expression is the result of adding the types of its comma-separated calculations.

For each of the above, if the type is failure, the math function is invalid.

A math function resolves to <number>, <length>, <angle>, <time>, <frequency>, <resolution>, <flex>, or <percentage> according to which of those productions its type matches. (These categories are mutually exclusive.) If it can’t match any of these, the math function is invalid.

Division by zero is possible, which introduces certain complications. Math functions follow IEEE-754 semantics for these operations:

• Dividing a positive value by zero produces +∞.

• Dividing a negative value by zero produces −∞.

• Adding or subtracting ±∞ to anything produces the appropriate infinity, unless a following rule would define it as producing NaN.

• Multiplying any value by ±∞ produces the appropriate infinity, unless a following rule would define it as producing NaN.

• Dividing any value by ±∞ produces zero, unless a following rule would define it as producing NaN.

• Dividing zero by zero, dividing ±∞ by ±∞, multiplying 0 by ±∞, adding +∞ to −∞ (or the equivalent subtractions) produces NaN.

• Any operation with at least one NaN argument produces NaN.

Additionally, IEEE-754 introduces the concept of "negative zero", which must be tracked within a calculation and between nested calculations:

• Negative zero (0⁻) can be produced by a multiplication or division that produces zero with exactly one negative argument (such as -5 * 0 or 1 / (-infinity)), or by certain argument combinations in the other math functions.

  Note: Note that negative zeros don’t escape a math function; as detailed below, they’re "censored" away into an "unsigned" zero.

• 0⁻ + 0⁻ or 0⁻ - 0 produces 0⁻. All other additions or subtractions that would produce a zero produce 0⁺.

• Multiplying or dividing 0⁻ with a positive number (including 0⁺) produces a negative result (either 0⁻ or −∞), while multiplying or dividing 0⁻ with a negative number produces a positive result.

  (In other words, multiplying or dividing with 0⁻ follows standard sign rules.)

• When comparing 0⁺ and 0⁻, 0⁻ is less than 0⁺. For example, min(0⁺, 0⁻) must produce 0⁻, max(0⁺, 0⁻) must produce 0⁺, and clamp(0⁺, 0⁻, 1) must produce 0⁺.

If a top-level calculation (a math function not nested inside of another math function) would produce a value whose numeric part is NaN, it instead act as though the numeric part is +∞. If a top-level calculation would produce a value whose numeric part is 0⁻, it instead acts as though the numeric part is the standard "unsigned" zero.

Note: Algebraic simplifications do not affect the validity of a math function or its resolved type. For example, calc(5px - 5px + 10s) and calc(0 * 5px + 10s) are both invalid due to the attempt to add a length and a time.

Note: Note that <percentage>s relative to <number>s, such as in opacity, are not combinable with those numbers—opacity: calc(.25 + 25%) is invalid. Allowing this causes significant problems with "unit algebra" (allowing multiplication/division of <dimension>s), and in every case so far, doesn’t provide any new functionality. (For example, opacity: 25% is identical to opacity: .25; it’s just a trivial syntax transform.) You can still perform other operations with them, such as opacity: calc(100% / 3);, which is valid.

Note: Because <number-token>s are always interpreted as <number>s or <integer>s, "unitless 0" <length>s aren’t supported in math functions. That is, width: calc(0 + 5px); is invalid, because it’s trying to add a <number> to a <length>, even though both width: 0; and width: 5px; are valid.

Note: Altho there are a few properties in which a bare <number> becomes a <length> at used-value time (specifically, line-height and tab-size), <number>s never become "length-like" in calc(). They always stay as <number>s.

Note: In Quirks Mode [quirks], some properties that would normally only accept <length>s are defined to also accept <number>s, interpreting them as px lengths. Like unitless zeroes, this has no effect on the parsing or behavior of math functions, tho a math function that resolves to a <number> value might become valid in Quirks Mode (and have its result interpreted as a px length).

11.10. Internal Representation

The internal representation of a math function is a calculation tree: a tree where the branch nodes are operator nodes corresponding either to math functions (such as Min, Cos, Sqrt, etc) or to operators in a calculation (Sum, Product, Negate, and Invert, the calc-operator nodes), and the leaf nodes are either numeric values (such as numbers, dimensions, and percentages) or non-math functions that resolve to a numeric type.

Math functions are turned into calculation trees depending on the function:

calc()

The internal representation of a calc() function is the result of parsing a calculation from its argument.

any other math function

The internal representation is an operator node with the same name as the function, whose children are the result of parsing a calculation from each of the function’s arguments, in the order they appear.

11.10.1. Simplification

Internal representations of math functions are eagerly simplified to the extent possible, using standard algebraic simplifications (distributing multiplication over sums, combining similar units, etc.).

11.11. Computed Value

The computed value of a math function is its calculation tree simplified, using all the information available at computed value time. (Such as the em to px ratio, how to resolve percentages in some properties, etc.)

Where percentages are not resolved at computed-value time, they are not resolved in math functions, e.g. calc(100% - 100% + 1px) resolves to calc(0% + 1px), not to 1px. If there are special rules for computing percentages in a value (e.g. the height property), they apply whenever a math function contains percentages.

The calculation tree is again simplified at used value time; with used value time information, a math function always simplifies down to a single numeric value.

Given the complexities of width and height calculations on table cells and table elements, math expressions mixing both percentages and non-zero lengths for widths and heights on table columns, table column groups, table rows, table row groups, and table cells in both auto and fixed layout tables MUST be treated as if auto had been specified.

11.12. Range Checking

Parse-time range-checking of values is not performed within math functions, and therefore out-of-range values do not cause the declaration to become invalid. However, the value resulting from an expression must be clamped to the range allowed in the target context. Clamping is performed on computed values to the extent possible, and also on used values if computation was unable to sufficiently simplify the expression to allow range-checking. (Clamping is not performed on specified values.)

Note: This requires all contexts accepting calc() to define their allowable values as a closed (not open) interval.

Note: By definition, ±∞ are outside the allowed range for any property, and will clamp to the minimum/maximum value allowed. Even for properties that explicitly allow infinity as a keyword value, such as animation-iteration-count, will end up clamping ±∞, as math functions can’t resolve to keyword values; the numeric part of the property’s syntax still has a minimum/maximum value.

Additionally, if a math function that resolves to <number> is used somewhere that only accepts <integer>, the computed value and used value are rounded to the nearest integer, in the same manner as clamping, above. The rounding method must be the same as is used for animations of integer values.

width: calc(5px - 10px);
width: calc(-5px);
width: 0px;

11.13. Serialization

This section is still under discussion.

See [CSSOM] for further information on serialization.

11.14. Combination of Math Functions

Interpolation of math functions, with each other or with numeric values and other numeric-valued functions, is defined as V_result = calc((1 - p) * V_a + p * V_b). (Simplification of the value might then reduce the expression to a smaller, simpler form.)

Addition of math functions, with each other or with numeric values and other numeric-valued functions, is defined as V_result = calc(V_a + V_b). (Simplification of the value might then reduce the expression to a smaller, simpler form.)

Appendix A: IANA Considerations

Registration for the about:invalid URL scheme

This sections defines and registers the about:invalid URL, in accordance with the registration procedure defined in [RFC6694].

The official record of this registration can be found at http://www.iana.org/assignments/about-uri-tokens/about-uri-tokens.xhtml.

Acknowledgments

Firstly, the editors would like to thank all of the contributors to the previous level of this module.

Secondly, we would like to acknowledge Anthony Frehner, Koji Ishii, and Xidorn Quan for their comments and suggestions, which have improved Level 4.

Changes

Recent Changes

Changes since 7 July 2021 WD (this is a subset of Additions Since Level 3):

• Added mix() notation for representing interpolated values.
• Defined generically the computation of <integer>, <number>, <percentage>, and <length>.
• Clarified that only non-zero lengths create a percentage+length mix that switches table cells to auto sizing.

Changes since 11 November 2020 WD (this is a subset of Additions Since Level 3):

• Updated interpolation of colors to reference [CSS-COLOR-4] instead of [CSS-COLOR-3].
• Added the svh, svw, svi, svb, svmin, and svmax small viewport-percentage units; lvh, lvw, lvi, lvb, lvmin, and lvmax large viewport-percentage units; and dvh, dvw, dvi, dvb, dvmin, and dvmax dynamic viewport-percentage units. (Issue 4329 and Issue 6113)
• Clamped excessively large <angle> values to multiples of 360deg. (Issue 6105)
• Added back rules on range-checking combined values lost during move from the CSS Transitions specification. (Issue 6097)
• Specified that UA-imposed minimum font sizes apply to the used font-size and not to resolution of font-relative lengths. (Issue 5858)
• Clarified how min() and max() percentages can partially simplify. (Issue 6293)

Additions Since Level 3

Changes since CSS Values and Units Level 3:

• Explicitly undefined numeric precision/range.
• Added rules for interpolation per value type, and their clarified computed values.
• Updated interpolation of colors to reference [CSS-COLOR-4].

Additions since CSS Values and Units Level 3:

• Added the mix() notation for interpolation.
• Defined the <dashed-ident> type.
• Defined the <ratio> type.
• Added src() to the <url> type.
• Added the vi, vb, ic, cap, lh and rlh length units.
• Added the svh, svw, svi, svb, svmin, and svmax small viewport-percentage units and dvh, dvw, dvi, dvb, dvmin, and dvmax dynamic viewport-percentage units.
• Added the x alias to dppx.
• Added min(), max(), and clamp() comparison functions.
• Added round(), mod(), rem(), sin(), cos(), tan(), asin(), acos(), atan(), atan2(), pow(), sqrt(), hypot(), log(), exp(), abs(), sign() math functions.
• Added e, pi, infinity, -infinity, NaN constants for use in calc().
• Added unit algebra to calc(), allowing multiplication and division of dimensions.
• Added back toggle() (punted from level 3 originally).
• A non-integer in a calc() automatically rounds to the nearest integer when used where an <integer> is required.
• Defined serialization of math functions.

Security and Privacy Considerations

This specification mostly just defines units that are common to CSS specifications, and which present no security concerns.

Note: Does URL handling have a security concern? Probably.

This specification defines units that expose the user’s screen size and default font size, but both are trivially observable from JS, so they do not constitute a new privacy risk.