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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).

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Linear Algebra Question ( rank of matrix )

Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively prove $\operatorname{rank}(\mathbf{PA}) = \operatorname{rank}(\bf{A})$...
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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}} &...
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1answer
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Transforming NED Acceleration Profile to Body Frame through Quarternions

I have an acceleration profile which is in the North-East-Down coordinate system. The moving object in question is 6 DOF, however, and frequently approaches 90 degrees in roll, pitch, and yaw, making ...
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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 ...
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0answers
27 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, ...
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2answers
35 views

Affine Transformation as Rotation

Im trying to do this textbook question which asks me to "express" a motion T(x) = Ax + b in the form T = Rot(P, $\theta$) (A is the rotation matrix) I know that if I draw the transformation, the ...
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2answers
4k views

How do I map the torus to a plane?

Please see my answer on Perlin noise first. A bit of background. Imagine a solid texture, like an actual block of sky and cloud. If you "cut a sheet" of sky and display it as an image, you'd get ...
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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\\...
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1answer
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Scaling - Rigid or Non-Rigid Transformation

I am trying to look for a precise definition of what rigid and non-rigid transformation is, and to which categories does 'scaling' belong. This is connected to a Point-Set registration problem that I ...
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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 ...
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3answers
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Transforming a square into a parallelogram

as an exercise I wanted to calculate the transformation matrix in order to make the square $ABCD$ into the parallelogram $A'B'C'D'$. I am able to get the matrix so that the square is first at the ...
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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 ...
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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. ...
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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". ...
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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 ...
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1answer
5k views

Negative & Positive Shear Factor

My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant ...
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2answers
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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 ...
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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 ...
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2answers
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Getting a transformation matrix from a normal vector

I'm trying to randomly generate coordinate transformations for a fitting routine I'm writing in python. I want to rotate my data (a bunch of $(x,y,z)$ coordinates) about the origin, ideally using a ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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Can non-linear transformations be represented as Transformation Matrices?

I just came back from an intense linear algebra lecture which showed that linear transformations could be represented by transformation matrices; with more generalization, it was later shown that ...
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1answer
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Linear Algebra - Preservation of inner product

Consider the vector space $\mathbb{R}^2$ with the standard inner product given by $ \langle(a, b), (c, d)\rangle = ac + bd$. (This is just the dot product.) (a) Let $\theta \in [0,2\pi)$ and let T : $...
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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 ...
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1answer
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$-\log(X)$ 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-...
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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 ...
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2answers
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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: ...
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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 $A(1, 1)$, $B(2, 2)$, $C(-1, 5)$ ...
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1answer
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Geometric transformation on circle equation

Suppose that I have variables $x_1,x_2$ and following circle equation: $x_1^2+x_2^2=1$. Now I have a matrix $A$ which will be used to transform my circle equation. $Z=AX$ where $X$ is a vector with $...
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0answers
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
763 views

Under the transformation $w=z^2,$ find the images of $\arg z=\theta$

please help with this exercise. Under the transformation $w=z^2,$ find the images of straight line $y=x$ "rayo in spanish" $\arg z=\theta$ I try 1- $$K=\{z=x+iy:y=x\}$$ $$w=(x+iy)^2=x^2-y^2+2ixy=...
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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 ...
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1answer
43 views

Sum of a rectangular and a triangular distribution

It is given that $X \sim R(0,1)$ and the density of $Y$ is given as: $f(y)= \begin{array}{cc} \Bigg\{ & \begin{array}{cc} y & 0<y<1 \\ 2-y & 1<y<2 \\ ...
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1answer
113 views

Monotonic transformation to smooth the probabilities

I am studying some event for a set of objects that can be plotted on a square $[0, 100] ^ 2$. I have used logistic regression to calculate probabilities that event occur for different objects and the ...
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1answer
699 views

Calculate transform horizontal/vertical skew and scale from 2d coordinate

I'm currently working on a javascript which allows to create a 3D rotating cube. I successed to create the 3D cube thanks to 8 points coordinates. However, I need to add an image on one the cube face. ...
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2answers
797 views

Show that if $T$ is surjective and spans $V$, then $T(S)$ spans $W$.

Given that $T: V \to W$ is a linear transformation from $V$ to $W$. Show that if $T$ is surjective and $S\subset V$ spans $V$, then $T(S)$ spans $W$. I think the main thing stumping me right now is ...
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0answers
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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 ...
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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 ...
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0answers
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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^{-...
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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 ...
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0answers
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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 ...
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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 ...
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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| ...
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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 ...
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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)) ...