Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [transformation]

Transformation has many meanings in mathematics. If using this tag, add another tag related to the object being transformed. If there is a tag for your specific kind of transformations, use that one instead: e.g., (laplace-transform), (fourier-analysis), (z-transform), (integral-transforms), (rigid-transformations).

0
votes
0answers
9 views

Proving DTFT pair as special case of another.

Consider these 2 basic discrete-time Fourier transform (DTFT) pairs... $$ \require{extpfeil}\Newextarrow{\xleftrightarrow}{15,15}{0x2194} \begin{array}{rcl} u[n] & \xleftrightarrow{\mathscr{F}} &...
0
votes
0answers
26 views

How to integrate over arbitrary quadrilateral

I need to integrate the product of two polynomial functions defined on an arbitrary (convex) planar quadrilateral defined by 4 points in $\mathbb{R}^3$. I was trying to firstly rotate the system of ...
1
vote
0answers
26 views

Kummer transform of the confluent hypergeometric function of second kind

I can see the kummer transformation of the confluent hypergeometric function of first kind throught the integral representation. However, I failed to see that for the second kind. More specificially, ...
1
vote
2answers
45 views

Transforming an arbitrary quadrilateral to a unit square

In this answer from Pedro Gimeno he proposed the following transformation to map the points of any arbitrary quadrilateral to the unit square $$\pmatrix{x'\\y'} = > \pmatrix{u_x&v_x&w_x\\...
0
votes
0answers
26 views

Extract param of $\sin$ from expression $y=2(\sin b-\sin a)/(\sin c-\sin a)$

$y=2\frac{\sin b-\sin a}{\sin c-\sin a}$, where $a=q(n+0)$ $b=q(n+1)$ $c=q(n+2)$ $q=\frac{2 \pi f}{s}$ Is it possible to extract $n$ from this formula? I already try this on WolframAlpha but I do ...
0
votes
0answers
24 views

Prove that for $x\neq 1, 0<y<\pi/2$ the system $u=\sin y/(x-1) ,v=x\tan y$ define a system of curvilinear coordinates.

Prove that for $x\neq 1, 0<y<\pi/2$ the system $u=\sin y/(x-1) ,v=x\tan y$ define a system of curvilinear coordinates. So this amounts to showing that the transformation is injective. ...
0
votes
0answers
14 views

What does it mean for a Wavelet transform to “commute” with a translation?

I'm referencing this paper here: https://arxiv.org/pdf/1203.1513.pdf Within this paper, it states that "A wavelet transform commutes with translations, and is therefore not translation invariant". ...
1
vote
2answers
64 views

If $T$ is an invertible linear transformation and $\vec{v}$ is an eigenvector of $T$, then $\vec{v}$ is an eigenvector of $T^{-1}$

I saw there is a proof for invertible matrices, but I don't know how to put this mathematically for a transformation. How do I prove an invertible linear transformation has the same eigenvectors as ...
0
votes
1answer
23 views

If $X$ is an exponentially distributed variable with mean $ \lambda$, $Y=−3\ln(X)$ has Gumbel distribution?

Let X be a random variable which follows an exponential distribution with parameter $\lambda$ ($\lambda>0$), find the distribution of the random variable $Y = −3\ln(X)$. So this is my answer for ...
0
votes
1answer
33 views

Rotation of an ellipse fixed at two points

I have a situation for which I have made a very crude drawing. Let's say we have an ellipse in $\mathbb{R}^2$ that is fixed at $x_0 = -a$ and $x_1 = a$ (as if it were resting on two poles). I am ...
0
votes
0answers
19 views

3D Affine Rotation Matrix from Orthogonal Vectors

How does one define an affine rotation matrix in order to rotate a 3D volume to align with a new coordinate system? The current coordinate system is $\mathbf{x}, \mathbf{y}, \mathbf{z}$ and I want to ...
0
votes
1answer
21 views

Help me understand how to use a Fourier Series to calculate an Σ sum

So, we're given a function $f(x) = \begin{cases} 2, &-\pi < x\le 0 \\ 6, &0 < x\le\pi \end{cases}$, while $f(x+2π) = f(x)$ for any $x\in\Bbb R$. Now, I've calculated the Fourier series ...
0
votes
1answer
15 views

Transformations' order

I get why you can move the curve for (a*(x-b))^2 where ever you want horizontally and then stretch as you want and it stays in the same place. Why does (ax-b)^2 behave any differently(and rather ...
0
votes
3answers
26 views

Image under billinear transformation

What is a image of $x+y>4$ under billinear transformation $B(z)=\frac{z-4-8i}{z-4}$? I got that $B(z)=1-\frac{8\sqrt{2}e^{i\frac{\pi}{4}}}{z+4}$, but I cannot conclude image correctly (it should ...
0
votes
1answer
14 views

$-\log(X)$ transformation of beta-distributed random variable $X$

Let $X \sim \text{Beta}_{(\theta, 1)} =: \mathbb{P}_\theta$ be a continuous random variable where $$\mathbb{f}_\theta(x) := \theta \cdot x^{\theta-1}\mathbb{1}_{[0,1]} = \cases{\theta \cdot x^{\theta-...
0
votes
0answers
30 views

Find joints positions in 3D robotic manipulator?

I have been trying to solve this 3D mechanics problem, but can't seem to be able to figure out what the best way to do it is. I have this $3D$ robot manipulator with $3$ rotary joints $(B, C, D)$. I ...
0
votes
2answers
22 views

Acos 90 degree matrix transformation.

I'm writing a program that transforms a matrix of points by 90°. In it, I have two vectors from which I am performing the rotation. Both vectors are normalized: ...
1
vote
1answer
15 views

How to interpret the vectors and design matrix in a linear model

In regression, linear models are of the form: $$y_i = \pmb z_i^T \pmb\beta_i + \epsilon_i$$ Or we can write this in a more general form with vectors and a design matrix: $$\pmb y = \pmb Z \pmb \beta ...
0
votes
0answers
24 views

Create a bijection with continuous function between finite subspace of $\mathbb R^3$ and predefined finite subspace of $\mathbb R^2$

Not sure if this is possible, but I want to make a continuous function that serves as a bijection between any finite subspace of $\mathbb R^3$, and a predefined subspace of $\mathbb R^2$. I'm looking ...
0
votes
0answers
33 views

How do I use the laplace transformations to solve this initial value problem?

So I have been given $\ddot{x} + 8\dot{x} +16x = e^t$ and $x(0) = 0, \dot{x} = 0$. How would I go about solving this initial value problem? As I am unsure of where to begin or what to do. Anything ...
1
vote
1answer
57 views

Using transformation to evaluate double integral

Given the transformation $T(x, y) = (x - y, x + y)$, evaluate the double integral $\iint_R (x^2+y^2) dA$, where $R$ is the rectangle in the $xy$-plane with vertices $A(1, 1)$, $B(2, 2)$, $C(-1, 5)$ ...
0
votes
0answers
24 views

solve for product of primes with difference of squares of primes

I had the following two algebras: $$C(p,q) = \frac{p^2-q^2}{4} \quad \text{and} \quad N(p,q) = pq$$ where p and q are primes greater than 10, moreover, the number of integers(i.e. the length of each ...
1
vote
1answer
13 views

How to adjust the diagonal so that a matrix is on the stability threshold?

I am working on the stability of food webs, which can be represented by a Jacobian matrix showing the interaction strengths between species. I know that a matrix is locally stable if all real parts of ...
0
votes
0answers
15 views

Finding a plane's RPY w.r.t. a global coordinate

The problem in short: I have the position of 3 points in a global coordinate frame - the points are co-planar. I would like to find the rotation of the plane stretched out by the points w.r.t. the ...
0
votes
0answers
17 views

From right-skewed to normal distribution

I have a variable that has this right-skewed (positive skew) distribution below: I aim to transform it in order to get a normal distribution. I have tried standard transformations (log10, natural ...
0
votes
0answers
58 views

Prove that $\sin^2(\pi x)$ is chaotic

My approach is based on the following from the book Chaos and Fractals: New Frontiers of Science, by Peitgen, Heinz-Otto, Jürgens, Hartmut, Saupe, Dietmar. To be more specific: "If $f$ is chaotic and ...
0
votes
0answers
41 views

Transforming $\left(\begin{smallmatrix} A^{T} \\ -A^{T} \end{smallmatrix}\right)^{T} x = -b$ after using Farkas Lemma

Let $A \in \mathbb{R}^{m\times n}$ and $u, b \in \mathbb R^{m}$. I am close to proving: $A x =b$ has a solution $\iff$ $b^{T}u \leq 0$ and $A^{T}u=0$. Using Farkas Lemma I get to the point where ...
0
votes
0answers
16 views

How to understand that the solution to least squares problem transformed with Box-Cox Transformation, is a generalized mean with $h(x)=x^\lambda$?

The least squares problem $\min_a \sum_i^n (x_i-a)^2$ is sometimes solved using transformed variables, that is, solving $\min_a \sum_i^n [h(x_i)-h(a)]^2$. The solution to this latter problem is $a=h^{-...
1
vote
0answers
20 views

Estimation / Calibration of Transformation of 2DOF laser pointing system in 3D space

Im creating a system where a Laser pointer should be able to point to various objects to direct a certain workflow. This laser pointer has two degrees of freedom, rotations about the local X and Y ...
0
votes
2answers
42 views

Nonlinear transformation of region from $\mathbb R^2\to\mathbb R^2$

If I have a given continuous nonlinear map $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$, and a region $D \subset \mathbb{R}^2$, is it necessarily true that $T(\partial D)=\partial T(D)$? That is, do ...
0
votes
0answers
13 views

Modeling the relationship between dilations and area?

On a rather basic high school level I want my students to understand the relationship between the scale factor of a dilation and the area of the pre-image and image. For instance if a rectangle has a ...
1
vote
2answers
40 views

Galilean transformation and differentiation

Given $x=x’-vt$ and $t=t’$, why is $\frac{\partial t}{\partial x’}=0$ instead of $1/v$? Maybe the answer has something to do with the fact that $dx’=dx$ in this Galilean transformation. Is $dx’=dx$ ...
3
votes
0answers
32 views

Matrix Transformation for 2D, how do I tell what this matrix does geometrically?

Given a 2x2 matrix, $$\begin{bmatrix}1&-1\\-1&\frac12\end{bmatrix}$$ What geometric effect does it have? So a way I did to solve this was to simply apply it to a unit square that I drew on a ...
0
votes
1answer
15 views

Does a Bijective Commutative transformation on a vector of angles exist?

I have a problem where I have two vectors a and b representing a list of angles. I need to find a transformation T where T(a,b) = T(b,a), where T has a distance metric to compare two transformations,...
0
votes
0answers
41 views

Changing rotation center

Things that we have: 2 dimensions, a object with it's coordinates (object P1), it's rotation center (pivot) C1. After that lets rotate it at pivot C1 by known angle A. Now let's move that pivot by ...
0
votes
1answer
29 views

Tensor product of vector with a tensor

I'm reading a paper describing transformation of gradient of a vector $\mathbf u$ (velocity vector) when I came across the following: $\nabla \mathbf u = \mathbf q$ after transformation is, $$ \...
1
vote
1answer
25 views

Transform $x^2u_{xx}-2xu_x+2u=\lambda x^2u$ into $w_{xx}=-\lambda w$ by choosing $M(x)$ where $u(x)=M(x)w(x)$

Consider the eigenvalue problem, $$x^2u_{xx}-2xu_x+2u=\lambda x^2u$$ for $0<x<1$, with boundary conditions $u_x(0)=0$ and $u(1)=u_x(1)$. Determine a function $M(x)$ so that, under the change of ...
0
votes
0answers
16 views

Centering a polygon on the origin by affine translation

Given a set of tuples, each representing the vertices of a polygon, I would like to center it on the origin. Having calculated the centermost point of the polygon (not the arithmetic centroid) and the ...
1
vote
1answer
33 views

Duffing equation transformation $(t,x) \rightarrow (-t,-x)$

We have the Duffing equation, $\ddot{x}+ λ\dot{x}=x-x^3$, which can also be written as $\dot{x}=y$ $\dot{y}=-U'(x)- λ y=x-x^3-\lambda y $ Show that the transformation $(t,x) \rightarrow (-t,-x)$ of ...
2
votes
2answers
32 views

Can 3D co-ordinates be transferred into 2D co-ordinates?

Is it possible to transform co-ordinates $(a,b,c)$ into $(x,y) $ such that $(x,y)$ is unique for each $(a,b,c)$ ? $a, b, c, x, y$ are in $\Bbb{R}$ .
0
votes
0answers
15 views

3D to 3D correspondence norm derivation

I've been going through a set of slides about a modified version of the Procrustes problem. The whole problem is described by trying to find a transformation that satisfies $$A_i = sRB_i + T$$ where $...
0
votes
0answers
22 views

Calculate contour area of an object on image plane when the tilt angle of a camera have changed

The camera is always oriented in a way that lower border of HFOV (horizontal field of view) is aligned with the bottom side of the object. The dimension of the object for a reference camera tilt ...
1
vote
1answer
34 views

Adding nodes to graphs while preserving harmonic solution

I have a graph and I'm interested in adding node to my graph such that it preserve the harmonic solution (page 2). Concretely, given a graph $G = (V,E)$ with $|V| ...
0
votes
1answer
57 views

What is image of $f(z)=\tan(z)$ where $\Im(z)=cst$?

Can you help me figure out what is the image of line segments ${z =x+iy: -π/2<x<π/2, y=const}$ under $f(z)=\tan(z)$. I've got $tan(x+iy) = sin(2x)/(ch(2y)+cos(2x)) + i sh(2y)/(ch(2y)+cos(2x)) ...
1
vote
0answers
17 views

Map one point cloud to another

I have two set of M 2D points (A, B) that are scaled to <0,1> intervals in both dimensions. How can I create a mapping function, that will map the set A to the set B if I select manually few points ...
0
votes
1answer
15 views

reflect a point over another point using matrix transformation

We know that if we want to reflect any point over an origin, i.e. $ O\left(0, 0\right) $, we can use matrix transformation like this $$ \left(\begin{matrix}x' \\ y'\end{matrix}\right) = \left(\begin{...
0
votes
0answers
13 views

Transform an Inertia Tensor

I am trying to provide colleagues with a spreadsheet method of transforming the inertia properties of a complex shaped body to a different coordinate system, involving only rotation. I've read that ...
1
vote
2answers
33 views

Prove that General Affine Transformations preserve ratios of lengths

Let $A$ be a matrix with determinant 1. Then we call a general affine transformation, a transformation of the form \begin{align*} \begin{bmatrix}x'\\y'\end{bmatrix}=A\begin{bmatrix}x\\y\end{bmatrix}+\...
1
vote
0answers
26 views

Mathematical Source for an algorithm that turns vector to euler angles

Am using the algorithm described here (https://stackoverflow.com/questions/21622956/how-to-convert-direction-vector-to-euler-angles) in the first answer for a thesis in software development. I do ...
1
vote
1answer
31 views

Transformation of RV: Finding PDF

I am trying to work through this example problem in my textbook but I keep getting the wrong final answer. My Notation: PDF X : pX(x) CDF X : FX(x) Question: Consider the transform Y=X2 if pX(x) = o....