Tagged Questions
0
votes
0answers
10 views
Estimate for a rigid transform given a set of noisy measurements
I have a set of rigid transforms $\in \mathbb{R}^{4x4}$, where each transform is an approximation to some unknown, "correct" transform. I'm looking for an algorithm to estimate the correct transform ...
2
votes
1answer
32 views
Problem with finding the intersection point between a line and triangle
I have a mathematical problem that I'm trying to solve, but the equations I have derived don't give the correct output when utilised on concrete problems. However, I can't figure out what the problem ...
4
votes
0answers
48 views
Invariant of matrix under elementary transformations
$\DeclareMathOperator{\rank}{rank}$
Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix
$$
B = \begin{bmatrix} A & b \\ b^T & c ...
0
votes
1answer
36 views
Confirm: 2d transformation matrix cannot change rectangle into isosceles trapezoid?
This non-mathematician needs some help...
Given any 2D transformation matrix as typically used in game programming like this:
$$A=\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & ...
0
votes
0answers
27 views
Convert between View Matrix and Tuple of Camera Position, LookAt Vector, Up-Vector
given a View-Matrix $M$ that can transform world coordinates into camera space, how can I convert between this representation and a more human readable form of Position ($\vec p$), Look-at vector ...
1
vote
1answer
25 views
Decomposition of a single 4D rotation
I have a $4\times 4$ matrix $M$ which represents a general 4-dimensional rotation.
$$
M = \pmatrix{a_{11} &a_{12} &a_{13} &a_{14}\\a_{21} &a_{22} &a_{23} &a_{24}\\a_{31} ...
1
vote
1answer
40 views
Comparing a geometric definition of cross product to the “usual” one
Could you help me with my little problem?
Given this definition of cross product:
1) $a \times b$ is perpendicular to $a$ and $b$, whenever $ a,b$ are linearly independent
2) basis $a, \ b, \ a ...
2
votes
2answers
72 views
Finding a transformation matrix
Please can you help me with the following problem:
I have a rectangle with center $(x_1,y_1)$ and sides $a,b$ where side $a$ is parallel to axis $Ox$. I want to find a transformation matrix that:
a) ...
0
votes
3answers
107 views
Rotation matrix in 3-dimensional space with two angles.
I am trying to find a description of a rotation in a three-dimensional space with a matrix that uses only 2 angles. It is easy to find one which uses three angles, since I can always consider the ...
2
votes
0answers
31 views
volumes of balls under an affine transformation
Denote by $B_t(O,\rho) \subset \mathbb{R}^t$, the sphere centered at the origin with radius $\rho$, and $B_n(O,\delta) \subset \mathbb{R}^n$, the sphere centered at the origin with radius $\delta$.
...
1
vote
1answer
49 views
Relation between Hadamard product and scalar product
Is there a known relation/formula for
$$(A\circ B, C)$$
where $\circ$ is the Hadamard product and $(\cdot, \cdot)$ is the scalar (euclidean) product?
In particular, I have a vector $y$ and a two ...
1
vote
1answer
36 views
Finding an D-dimensional orthonormal change-of-basis matrix given a D-2 transformed points.
This is frustrating, I should be able to solve this but I'm having a mental fog.
I want to find an orthonormal change of basis: given a single point $(x_1,y_1)^T$ and its image $(x_2,y_2)^T$, find ...
-1
votes
1answer
78 views
How to change XYZ axes system into another 'xy' system
I have $3D$ point set lying on a vertical plane. This plane is not parallel to either $X$ or $Y$ axis but makes an angle (say, $\theta$) to $X$ axis. And also it has some ($+$ or $-$) intercept to the ...
0
votes
1answer
30 views
parabola in homogeneous coordinates
So if I have the parabola Y = X^2, how do I go about representing this homogeneously? I know I can parameterize it as F(t) = (t, t^2), but then what?
The reason I ask is because I have a 3*3 matrix ...
0
votes
1answer
91 views
2D Eculidian matrix to 2D cartesian graph/plan
Can anyone help ?
I am trying to convert a 2D matrix of distances to a 2D graph.
For instance, I would like to go from this :
...
1
vote
0answers
77 views
Sheared perspective projection matrix
I want to project a 3d scene onto a virtual screen.
I know how a perspective projection matrix would look like in the regualar case (transforming the red marked part of space into an unit cube). I ...
2
votes
0answers
104 views
Geometric interpretations of matrix inverses
Let $A$ be an invertible $n \times n$ matrix. Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through ...
1
vote
2answers
84 views
How to workout the determinant of the matrix $D_n(\alpha, \beta, \gamma)$.
I am going through an example in my lecture notes. This is it:
Let's introduce the matrix $D_n(\alpha, \beta, \gamma)$, which looks like this:
$$\pmatrix{\beta & \gamma & 0 & 0 ...
2
votes
3answers
85 views
Intuition about Hyperplane
I'm having a hard time understanding hyperplane ideas. So, can anyone explain to me how to easily understand what a Hyperplane is ?
2
votes
2answers
51 views
How does a normal vector change with respect to the points that define it?
$A$ is a nondegenerate $n \times n$ matrix with real-valued entries. If we interpret the rows of $A$ as points in $\mathbb{R}^n$, then $A$ defines a unique hyperplane that passes through each of ...
1
vote
0answers
71 views
transformation of symplectic structure by a matrix
Suppose that in canonical symplectic basis $e_1,e_2,f_1,f_2$ we have
$$\Omega=pf_1^*\wedge f_2^*+qe_1^*\wedge e_2^*+r(e_1^*\wedge f_2^*+e_2^*\wedge f_1^*)+s(e_1^*\wedge f_1^*-e_2^*\wedge f_2^*)$$
Let ...
1
vote
1answer
97 views
Rotating a line segment in 3D to a prescribed orientation
I have a general line segment with endpoints $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ referenced to a 3D Cartesian coordinate frame E. I wish to rotate this coordinate fram E to a new coordinate system F ...
9
votes
1answer
504 views
How to understand spectral decomposition geometrically
Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have
$$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$
and
$$A^{-1} = ...
5
votes
1answer
125 views
Geometric interpretation of the Kronecker product?
Is there any geometric interpretation of the Kronecker product of two $2\times 2$ matrices?
3
votes
1answer
122 views
How to prove $\left\|\ln\left(e^{iH_1}e^{iH_2}\right)\right\|\leq\left\|H_1\right\|+\left\|H_2\right\|$?
Let $H_1$ and $H_2$ denote arbitrary Hermitian operators (finite dimensional) and let $\left\|\ldots\right\|$ denote the usual operator norm. I conjecture that
$$
...
0
votes
0answers
41 views
Convert point coordinates
I have to create some transformations for a 3D application but I'm not very good at math.
I have 2 objects in space (let's call them ...
0
votes
2answers
110 views
3D transformation between two polylines problem
Say I have 2 separate objects.
One is a line defined by two points, the other is a polyline defined by three points.
Line 1 consists of the set of two points:
$a=(0,0,0)$ and b=$(0,0,1)$
Line 2 ...
1
vote
1answer
104 views
Normalizing 2D coordinates that are arbitrarily transformed.
This was originally posted as a programming question so I will try to keep it as abstract as possible.
We are dealing with two images. The first is the original and second is a distorted version of ...
0
votes
0answers
415 views
Explicit calculation of 3x3 rotation matrix from combining three angle-unit axis rotations?
I need to remove dependence on a programming library from a computer application I'm working on and instead hand code a geometric operation. Please can you show explicitly (for someone with little ...
1
vote
5answers
124 views
Independence of Rotation Matrix Definitions
I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix
$$
A = \left(
\begin{matrix}
a_{11}& a_{12}& a_{13}\\
a_{21}& ...
1
vote
3answers
460 views
3D to 2D rotation matrix
I have been trawling through this forum but am struggling to understand the maths a bit. Currently I have a 2D plane within a 3D space and I have the coordinates for them. I want to work on this 2D ...
0
votes
2answers
428 views
What is the rotation axis and rotation angle of the composition of two rotation matrix in $\mathbb{R}^{3}$
I was told in class that a rotation matrix is defined by a rotation
angle and rotation axis, if we call the rotation axis $v$ and take
a basis of $\mathbb{R}^{3}=\{v\}\bigoplus\{v\}^{\perp}$ then the
...
0
votes
1answer
219 views
A problem with definitions of rotation/reflection matrix/operator
I am a math undergraduate student taking a course called "Geometry
and symmetry" and I have something I don't understand with the definition
the lecture gave in class.
Definition: ...
1
vote
1answer
683 views
2D transformation matrix to make a trapezoid out of a rectangle
In most vector graphic software libraries I can use (3x3) matrices to transform 2D geometry (e.g. scale, rotate, skew). How does a matrix need to look like to transform a 2D rectangle to a symmetrical ...
0
votes
0answers
22 views
Recovering relative positions of points in d-dimensions
On page 5, right column of
http://www.dimitris-agrafiotis.com/Papers/jcc20078.pdf
I found that "that the
relative configuration of N points in a D-dimensional space can
be fully described using only ...
1
vote
2answers
305 views
How to find a 2D basis within a 3D plane - direct matrix method?
I have a plane equation in 3D, in the form $Ax+By+Cz+D=0$ (or equivalently, $\textbf{x}\cdot\textbf{n} = \textbf{a}\cdot\textbf{n}$), where $\textbf{n}=\left[A\:B\:C\right]^T$ is the plane normal, and ...
2
votes
2answers
104 views
Orientation matrix for three points in the plane
In this Wikipedia entry for determining the orientation of a simple polygon, the following explanation is given for one of the steps:
In computations, the sign of the smaller angle formed by a ...
2
votes
3answers
354 views
Forming equation of a plane by solving linear equation set
Given three points on the plane: $ A(x_1, y_1, z_1) $, $ B(x_2, y_2, z_2) $ and $ C(x_3, y_3, z_3) $.
I'm trying to obtain the equation of the plane in this format:
$ ax + by + cz + d = 0 $
I ...
1
vote
1answer
102 views
Intuitive interpretation of the 3D to 2D mapping
Suppose a 3D configuration of points is given, $X\in\mathbb{R}^{n\times 3}$, and a non-zero matrix $Q\in\mathbb{3\times 2}$, with orthonormal columns. Now, suppose a mapping to 2D is obtained as
...
1
vote
0answers
138 views
2D Cartesian Matrix / coordinate transformation.
I has initially asked this question in the programming site but did not get an answer that worked. This is my first question on this site so please bear with me.
Consider a page with three distinct ...
2
votes
1answer
941 views
Matrix for rotation around a vector
I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
3
votes
1answer
379 views
What is the geometrical action of a skew-symmetric matrix on an arbitrary vector?
What is the geometrical action of a skew-symmetric matrix on an arbitrary vector?
The rotation matrix is a skew-symmetric matrix when $\theta$ is some multiple of $\frac{\pi}{2}$. But it cannot be ...
11
votes
2answers
377 views
Does this expression have a (“better”) determinant form?
[Edit I've found a determinant that satisfies the letter of the previous version of this question, but not its "Cayley-Menger" spirit.]
A tetrahedron with face areas $w$, $x$, $y$, $z$ and ...
0
votes
0answers
43 views
Maintaining the line with the 2D iterands
Suppose a linear system is given
$$AX=B,$$
where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
1
vote
1answer
351 views
Find 3D rotation vector and angle to transform a rectangle into a given quadrilateral
I have a given rectangle that I need to transform into a given quadrilateral shape that resulted from a rotation and translation in 3D space, and subsequent projection.
...
0
votes
1answer
99 views
Chain Rule and Homogenous Coordinates
I have a vector $\tilde{p} = (x,y,z)$ (homogenous coordinates). The corresponding non-homogenous vector is $p = (x/z, y/z)$.
Now the $\tilde{p}$ is a result of some linear transform $R(\theta)$ of ...
1
vote
1answer
679 views
(Graphics Gems IV, Shoemake) From matrix to euler angles explanation
I am trying to understand matrix to Euler angles conversion. So I read Graphics Gems IV, page 222 from Ken Shoemake. It states:
"Suppose we have code to convert a rotation matrix to XEDS angles, $R = ...
1
vote
1answer
78 views
Need some help to understanding the formula
This is pinhole camera model
(I don't get, is there [R t], or (R, t))
This formula is used to model the projection from a space point M to an image point m.
Projection drawing
Tilde over vector, ...
2
votes
2answers
1k views
How to multiply vector 3 with 4by4 matrix, more precisely position * transformation matrix
All geometry in computer graphics are transformed by position * transform matrix; The issue is the fact that position is a vector with 3 components (x,y,z); And transform matrix is a 4 by 4 with one ...
2
votes
1answer
88 views
Computing the point which is closest to many Planar surfaces
Suppose, i have been given different planes which orients to different direction (i.e. i know only the plane parameter of those planes). If i am able to find out planes (probably more than 3 planes) ...
