All Questions
Tagged with projection geometry
148 questions
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Why are the projections of an area element onto the coordinate planes the components of its normal vector? [duplicate]
I understand that the components of the cross product $\bf{u}\times\bf{v}$ are the area of the parallelogram enclosed by $\bf{u}$ and $\bf{v}$, projected onto the respective coordinate planes (e.g. ...
- 916
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Different projection matrices, don't understand how to proceed
I'm stuck on the second part of an exercise:
In $\mathcal{E}^2$, identified with $\mathbb{R}^2$ through an orthonormal reference system $(i, j)$, be Pr: $\mathcal{E}^2 \to \mathcal{E}^2$, $v \mapsto v'...
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How to obtain the orthogonal basis of an ellipsoid section
Given an ellipsoid equation of the form \begin{equation} \textbf{v}^T \textbf{A} \textbf{v} \; = \; 1 \tag{1} \end{equation} where $ \textbf{A} \in \mathbb{R}^{n \times n} $ is positive definite and ...
- 372
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When is the product of two orthogonal projectors an orthogonal projector?
Let $V$ and $W$ be two vector subspaces of $\mathbb{R}^n$, let $p_V$ be the orthogonal projector onto $V$, and let $p_W$ be the orthogonal projector onto $W$. When is it the case that $p_Vp_W$ is the ...
- 197
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Isometric conversion from screen coordinates to an $N \times M$ grid when not at true origin?
Given a pre-rendered image and its associated 2d array of $N \times M$ size, is there a system of equations that can calculate from screen $(x,y)$ to $(\text{grid}x, \text{grid}y)$ in my array? The ...
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97
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How to project one circle onto another for the purposes of angles?
Two circles are concentric with the same radius. Circle 2 (blue) is tilted relative to the xy plane. They coincide at some nodal axis (dotted line) which is the x axis.
Now we advance to some point A ...
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Orthogonal projection of a double-sided cone onto a plane
I would like to solve the following problem:
a double cone with vertex at the origin of coordinates (0, 0, 0) is given (See fig. 1). The blue cone is symmetrical to the yellow one w.r.t. the origin;
...
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Orthographic Projection and Concentric Circles [closed]
Let $C$ be the bounded set of concentric circles centered at the origin.
Let $r_n = \sqrt{\frac{n}{\pi}}$ for any integer $n \in [1, k]$ be the radius of the circle $C_n \in C$.
Assume $C$ is a ...
- 115
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Determine if point can be projected on any segment in polyline
I am trying to implement a script that determines if a series of points can be projected on any segment of a polyline. The projection must be exactly on the segment, not in the rest of the line ...
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lengths of orthogonal projections of the standard basis on a subspace
Let $e_1,\ldots,e_n$ be the standard basis in $\mathbb{R}^n$. Suppose the scalars $\lambda_1,\ldots,\lambda_n$ satisfy $0< \lambda_1,\ldots,\lambda_n\leq1$ and $\lambda_1^2+\ldots+\lambda_n^2 = m$, ...
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How to calculate tangent vector with conical projection?
I'm currently reading some codes about conical projection.
Let the viewed target position point be $P$, and the observer point be $Q$, the local coordinate system's $x,y$ plane is perpendicular to $\...
- 465
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48
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Project a level set onto a plane
I have a level surface in $(x,y,z)$ space. For concreteness, let's say
$\frac{1}{r^2}x^2 + \left(\frac{\sin \theta}{r}\right)^2y^2 + \left(\frac{\cos \theta}{r}\right)^2z^2 + \frac{2 \cos \theta \sin \...
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Determining the relative dimensions of a right circular conical frustum from a perspective image
Suppose you have a right circular conical frustum with bottom base radius $r_1$ and top base radius $r_2$, and height $h$.
Now you take a (perspective) image of this frustum. The camera used to ...
user948761
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Determining the coordinates of the vertices of a triangle from two images taken by two cameras with known position/orientation
Suppose you have a triangle with unknown side lengths hanging in space with unknown position and orientation (i.e. nothing about the dimensions or position or orientation of the triangle is known).
...
user948761
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Determining the coordinates of the vertices of a triangle from its camera image
Suppose you have a pinhole camera (no lens) with known position and orientation with respect to the world.
Now with this camera, you take an image of a triangle that has known side lengths of ...
user948761
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Are the angles from the diagonal of a square to each side always equal? Even after rotation? And perspective?
If you draw a square on a sheet of paper, and draw the diagonal, are the apparent angles from the diagonal to each side equal, even if you rotate the sheet in 3D space?
Visual Example
Transcript of ...
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How do I project a point $P$ onto a sphere from a projective point $C$ that is NOT located at the sphere's center or surface?
I have a sphere of radius $1$ centered on the point $(0, 0, 1)$.
I also have a point $C = (0, 0, H)$ where $0 < H < 1$ from which to project objects onto the sphere. It is located inside the ...
- 1,871
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Why can the area projected onto a plane be calculated with the dot product of the vector area and the unit vector of the plane?
I was trying to find the projection of a flat surface onto an arbitrary plane and I came across this Wikipedia article,
The projected area onto a plane is given by the dot product of the vector area $...
- 801
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Can one compute the location of the unseen point?
My question is quite simple. I have two images, on the first one I know the location of points $P1, P2, P3$, and $P4$. In the second image, I know the location of $P2'$, $P3'$, $P4'$, and point $Q'$. ...
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How to algebraically prove the sum of the projections of two sides of a triangle is equal to the projection of the third side of the triangle?
Look at the picture above, graphically/geometrically, the green line is equal to the sum of the blue and red lines. But I am trying to think of an algebraical proof. Let's say the lengths of the side ...
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Formula to project a 3D point onto a sphere, when centre of projection is not necessarily at the centre of the sphere
People sometimes talk about "projecting onto a sphere" analogously to projection onto a plane, but how is this defined (a formula would be nice)? It's not obvious to me how we should handle ...
- 1,660
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Area of projection of ellipsoid onto the $xy$ plane
Given the ellipsoid
$$ (\mathbf{r} - \mathbf{r_0} )^T Q (\mathbf{r} - \mathbf{r_0} ) = 1 $$
with $Q, \mathbf{r_0}$ known.
Question: Find the area of its projection onto the $xy$ plane.
If $a,b,c$ are ...
user948761
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Area of a cut column
An infinite column with its axis along the $z$ axis, and with square cross section of side length $10$, is cut by the plane $4x - 7y + 4z = 25$. The cut is in the shape of a parallelogram. Find the ...
user948761
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Projections of zonohedra
I noticed that any generic orthogonal projection of a cube has exactly two crossings.
This led me to wonder about generalizations. A friend suggested I look at zonohedra.
A zonohedron is the ...
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Reversing perspective projection of cricles
I have a set of real-world images containing circles, but due to the images not being taken parallel to the circle plane, the circles appear as ellipses in the images. I need to crop out the inside of ...
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Understanding a claim of Makai and Martini: why is an ellipsoid's cross-section body the same as its projection body?
In The Cross-Section Body, Plane Sections of Convex
Bodies and Approximation of Convex Bodies, I (Makai and Martini, 1996), the authors define for a convex body $K$ its cross-section body $CK$ and its ...
- 18k
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Finding the Center of Many Intersecting Lines
I have the following matrix of linear equations which gives rise to the following plot. (All of my examples and code use R.)
...
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Is there anything to stop an image being projected onto the side walls in a pinhole camera/camera obscura?
I always see diagrams of how a camera obscura works where the projected image neatly stops before or at the edges of the wall opposite the pinhole.
But when I look at pictures and videos of rooms ...
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Extending a vector so its projection match another vector's length, there's a name for that?
My Problem
I'm a game developer, and I had a problem yesterday in which I needed to extend a vector $\vec{v}$ so its projection into $\vec{w}$ was equal to $||\vec{w}||$.
(Illustrating the Problem)
...
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"Clamp" barycentric coordinates?
Let's begin with something we know how to do. Given a point in 2D, one can project it to a segment by projecting the point onto the line containing the segment.
You do this by doing $s = (p - o)\cdot ...
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Projection of a Pentagonally-tiled Sphere
I know that a regular pentagonal tiling does not work in Euclidian space, but does work on a sphere. But this got me wondering something that I hope people can help with here, because I can't find any ...
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Perspective image of a cuboid
You are given a perspective image $A'B'C'D'E'F'G'H'$ of a cuboid (rectangular prism) $ABCDEFGH$. The image is such that in the image, in any face, opposite sides are not parallel to each other. In ...
user948761
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Orthogonal projection of a conic onto a plane
It is well known that conic sections are the intersections of a plane with a cone. Let the cone be $z^2 = x^2 + y^2$ and the plane be $z = ex + b$. Project orthogonaly the resultant conic onto the $...
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Orthographic projection of a rectangle
Rectangle $ABCD$ is projected using orthographic projection onto a plane that makes a known angle with the plane of the rectangle, as shown in the figure above. Can the lengths of the sides of the ...
user948761
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Point on a line closest to the origin, general formula using linear algebra only
Given a line $\ell$ in $\mathbb R^2$ containing points $p, q$, find point $r$ on $\ell$ closest to the origin, using linear algebra only (no calculus).
My answer is below, but I seem to have made a ...
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How could one convert 3d coordinates which are on a plane to coordinates relative to said plane?
I have a plane, defined by ax1+bx2+cx3=d, and a point which I know is on said plane. How could I convert the coordinates of the point to coordinates relative to the plane? I have attempted to find a ...
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Proof of method to determine whether or not equally-sized axis-aligned disjoint cubes overlap each other when rendered.
Consider two opaque equally-sized axis-aligned disjoint cubes somewhere in three dimensional Euclidean space. Additionally there is a point in that space outside of the cubes that represents a camera ...
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The transformation matrix of an dimetric projection onto the plane Z=0
I need to make a dimetric projection of the point onto the Z=0 plane. I found the transformation matrix of an isometric projection onto the Z=0 plane and it looks like this:
\begin{align*}
M &= \...
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how to draw a circumference around the centroid of the geometry?
I am working on a task in which I have to project nodes (X and Y coordinates of nodes) of a point cloud to the circumference located exactly on the centre of gravity of the mesh (nodal Geometry). My ...
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Is a projection from a conic to itself through a point a projective transformation?
Given a conic $C$ is the map For $A\in C,$ $A\to AX\cap C$ Where we take the point which isn't $A$.(except when $AX$ is a tangent) a projective transformation from the conic to itself? where $X$ is a ...
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Projection of a cylinder from XYZ 3D space on to XY plane. [closed]
A cylindrical body of length $ \ L \ $ lies in the XYZ 3D space. The base of the cylinder makes an angle $\psi$ with the Z axis, as shown in image link. I would like to know the projection of that ...
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Question about oblique and orthographic projections
The question in case of TL;DR: Given an extremely long distance (or focal length), would the image generated of a given building or buildings (or indeed of any real world object) always specifically ...
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Why matrix dot product (in linear algebra) is equal to vector dot product (in geometry)? A link between linear transformation and vector projection? [closed]
The question:
[Matrix dot product][1]
[Projection in dot product][2]
Why are these two equal?
My work till now. It may help:
I have attached images because pdf doesn't work, but still I can not poste ...
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Why does eyes initially on the horizon line always remain on the horizon line when moving on a ground without elevation?
In this video by Youtuber Love Life Drawing , it's shown objects having their eyes along the horizon line initially still have their eyes on the horizon line when they get closer to the camera. Photos:...
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Find location and orientation of a pinhole camera from a given image of a triangle with known sides contained in a known plane
So, I am playing with problems related to perspective images produces by a simple pinhole camera.
I came up with the following problem. Suppose you have a triangle of known side lengths, that is ...
user948761
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59
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Calibrating a pinhole camera (finding $z_0$)
A pinhole camera is a very simple theoretical device for generating perspective images on a plane that a distance $z_0$ from the pinhole (a point) and whose normal vector is the direction vector at ...
user948761
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58
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Locating vertices of a known triangle in $3D$ from a single image
Suppose you have a labelled triangle with known side lengths, and you take one image of this triangle using a known pinhole camera (i.e. the focal length is known), from a point with known coordinates,...
user948761
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generalization oh lhuillier´s thorem
lhuillier´s thorem states thatEvery triangle can be considered
as the normal projection of a triangle of given form (an equilateral triangle, for indtance). see the discussion here : Projection of an ...
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Projection of $\lambda y$ onto a polytope is fixed for sufficiently large $\lambda$?
Le $y\in \mathbb R^n\setminus\{0_n\}$.
Let $X\subset \mathbb R^n$ be a compact polytope (intersection of finite half-spaces).
For a sufficiently large $\lambda\in \mathbb R_+$, I have the impression ...
- 1,354
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Relationship Between Hyperbolas and Hyperbolic Spaces
I am trying to understand the difference between Hyperbolic Functions and Hyperbolic Spaces.
In my very limited knowledge of mathematics, I have only come across:
Hyperbolas : https://i.ytimg.com/vi/...
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