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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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How to calculate a double integral

I’ve got the following integral $$\int\int _D \frac{dxdy}{x+y}$$ D is the region bounded by $x+y = 1, x+y = 4, y=0, x=0$ and I have to use the transformation $x = u-uv, y=uv$ Anyone know what domain ...
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How to create a transformation matrix for a M22 → M22 transformation

I have a linear transformation, T, such that; T:${M_{22}}$→${M_{22}}$: T$\left(\begin{bmatrix}{x_{11}} & {x_{12}}\\{x_{21}} & {x_{22}}\end{bmatrix} \right)= \begin{bmatrix}{{x_{12}}-5{x_{21}...
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Image of a straight line under $w = Log(iz)$.

In detail, I have to find the image of the straight line parallel to the co-ordinate axes for the function $w = Log(iz)$. What I have tried is as follows (Since I am new to complex-analysis, I must be ...
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Is there an easy way to remove scale from a squared linear transformation matrix

Given a linear transformation matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}$, I know that one can use SVD or QR decomposition ...
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Trouble in mapping of möbius transformation

Question:- Show that the transformation $$ w = \frac{2z+3}{z-4}$$ maps the circle $x^2+y^2-4x=0$ onto the straight line $4u+3=0$ My attempt:- The circle $x^2+y^2-4x=0$ is $|z-2|=2$ . . .$(1)$ So ...
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Translate point after rotation relative to different origin

I have 200x380 input image and coordinates (63,146) where (0,0) is top-left: I rotate about the centre some amount of degrees and expand the "canvas" to avoid cropping resulting in larger output ...
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transformation in computer graphics [closed]

The figure $ABCD$ where $A=(-2,0)$, $B=(0,-2)$, $C=(-2,-4)$ and $D=(-4,-2)$ can be transformed into $A'B'C'D'$ where $A'=(1,-1)$, $B'=(3,3)$, $C'=(6,3)$ and $D'=(4,-1)$ by the composition of simple ...
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Find best transform to map points

I have a system with a robot picking parts based on feedback from a distance sensor. I have found that errors in alignment etc can cause problems over the working range. I want to create a correction ...
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Volume preservation Liouville's Theorem , explanation of proof

I am trying to understand the following proof of Liouville's Theorem , that states that trajectories generated by Hamiltonian equations are volume preserving.The proof can be found in the following ...
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Calculating Ego-Motion from planar keypoints

I'm working on a module to estimate the ego motion from a robot with the help of a camera. In order to simplify that system I decided to track keypoints on the floor below the robot instead of solving ...
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Joint distribution of $Y_1=X_1/X_2, \quad Y_2=X_2$ when $h(x_1,x_2) = 8x_1x_2$

I am having trouble finding the joint distribution of the following. Joint distribution of $Y_1=X_1/X_2, \quad Y_2=X_2$ when $h(x_1,x_2) = 8x_1x_2$ when $0 < X_1 < X_2 < 1$. I think I am ...
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Find transformation matrix $[T]_{B,B}$ representing $T$ in the basis $B$

Info provided: $T:P_2→P_2$ given by $T(p(x)) = p(kx)$ where $k>0$ Find matrix $[T]_{B,B}$ representing $T$ in the basis $B$ Attempted Solution: Using the standard basis for $P_2$, {$1,x,x^2$}, ...
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If $\Sigma^{-1}=(A^{-1})^TA^{-1}$, then why does $|A^{-1}|=|\Sigma|^{-1/2}$?

In the derivation of the joint pdf of $f_\textbf{X}(\pmb{x})$, where $\textbf{X}=\pmb\mu+A\pmb Z$ and $\textbf{X}\sim~N_n(\pmb\mu,\Sigma)$, there is a step I do not understand. In particular, it is ...
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Confusion over order of transformations of graphs

I did a search on the order of transformations applied to graphs, and mostly found the following, e.g. in this post. Given a function $f$ always perform transformations $$Af(Bx+C)+D$$ in the order $...
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Matrix transformation for finding eigenvalues

For each $m\in\{1, 2, ..., n\}$, is there a transformation $\phi_m$ that I can apply to a matrix $M\in \mathbb{R}^{n\times n}$ such that the $m^{th}$ largest eigenvalue $\lambda_m$ of $M$ is the ...
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Logarithm transformation for a function in Machine Learning

I was looking for a clarification on why the function (the one raised to the power of M) was transformed to a logarithm function. I know it will be used to find p when L is maximised ( derivative of ...
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Transformation matrix in affine matrices

I'm studying this for my computer graphics course under the topic of affine transformations but I've searched everywhere and I don't get how to do this Given the transformation matrix: $$ M = \begin{...
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Pure pursuit algorithm - Transform Global to vehicle coordinates?

project: autonomous driving car I have a small RC car, a Raspberry Pi and a camera which are fixed on the car. Now I want to let the car drive autonomous with image processing. So far I can detect ...
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How is function transformation related to geometric transformation?

I know how $y = \mathrm{a} f(\mathrm{b} x + \mathrm{c}) + \mathrm{d}$ transforms $y = f(x)$. I wonder how function transformation is related to geometric transformation. The MathWorld page about ...
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How can I get local scale from a rotation matrix and a world scale?

I'm working on a tool for game modding, intended to help users create mesh morphs. This part of the tool concerns joint-based morphs. My problem is this: I want users to be able to enter delta ...
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Showing that a transformation is a variational symmetry

I'm trying to solve problem 9.2.1 in the book 'The calculus of variations' by Bruce Van Brunt. I was given the functional $$J(y)=\int_{x_0}^{x_1}xy'^2 dx$$ Now I'm supposed to show that the ...
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How to make a transformation to the floor function to the right or left?

Assume a function called $f(x)$ , Then all of us know that of we draw $f(x+a)$ it will be a transformation to the left or right and $f(x)+b$ to up or down. But when I drew floor function on Desmos ...
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Is there a way to encode the prime counting function in the unit square?

Here's what I tried: Let $$\Psi(x)=\int_0^x e^{\frac{1}{\ln(t)}} dt $$ and consider a map that associates the logarithmic integral with $\Psi(x)$: $f: Li(x) \mapsto \Psi(x).$ See this question ...
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transformation of the Gaussian function

consider a gaussian function: $$ \frac{1}{\sqrt{\pi}} \exp \left\{-\frac{1}{2}\left(x^{2}+y^{2}\right)\right\} $$ And I have to prove that transformation of this function: $$ \exp \left\{-\eta\left(...
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General Transformation Matrices for Images

for a vector (x, y) the matrix (a, b; c, d) can represent arbitrary linear transformations, using the vector (x, y, 1) and a 3x3 matrix arbitrary transformations (projective?) Now I want to transform ...
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Problem with Graphing Transformation of Cartesian coordinate into Polar coordinate.

I was trying to map Rectangle from cartesian to polar coordinates. I started by making a rectangle in a Cartesian Plane. From the x and y coordinates of the rectangle, I calculate the radius and angle ...
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Obtaining the equations after scaling and shifting?

I am thinking about this coordinate shift. Suppose we have the two equations as below, where $x_0,y_1, x,y,\bar x,\bar y$ are variables while others are constants: $$\begin{aligned} \bar{x_0} &= ...
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1answer
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Find the CDF and PDF of $W$ where $W =.7 - b(.7 - y)^2, 0<b<1$, with $Y \sim U[0,1]$

A random variable $Y \sim U[0,1]$. Let $W = \frac7{10} - b\cdot(\frac7{10} - y)^2, 0<b<1$.Completely specify the CDF and PDF of $W$.Also show that the PDF of W integrates to $1$. So I have ...
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Normal distribution non linear transformation

I have the following problem : Given $X \sim N(\mu,\sigma^2)$ and $X' = h(X) = (\frac{x-\mu}{\sigma})^2$ Find $E[X']$ and $V[X']$. My reasoning is as follow : Since $X' \sim (\frac{x-\mu}{\...
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Orbit of a point and flow

Let $\varphi:X\times \mathbb{R}\to X$ be a continuous flow on compact metric space $X$ without singularity. For $\delta>0$ and $x\in X$, take \begin{equation} \Gamma_\delta(x)=\bigcup_{h\in\...
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Given that x~U[0,1] and y~U[0,1], derive the conditional CDF of W=x-b*(x-y)^2 where 0<b<1? Condition on x (i.e. treat x as a constant).

Given that $x\sim U[0,1]$ and $y\sim U[0,1]$, derive the conditional CDF of $W=x-b\cdot(x-y)^2$ where $0<b<1$? Condition on $x$ (i.e. treat $x$ as a constant). I am running into difficulties ...
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How do I rotate a bitmap image?

I am trying to write an algorithm to rotate a bitmap image of $n$ by $n$ size by an angle $\alpha$. I know that I have to find a rotation matrix, then perform matrix multiplication of the rotation ...
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A question on transformation

I am doing a transformation problem of getting the graph of $\sin (2x – \pi/6)$ by applying transformations to $F(x) = \sin x$ In the process, I let $f(x) = F(2x) = \sin 2x$. Next, I then let $g(x) ...
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Connection between rotate / translate / scale as operations in 3D space

This feels like a silly question, what am I missing: I'm a 3D artist working on videogames. Sometimes I make 3D artworks like the ones here: http://samuelthomson.org/blog/2012/06/07/topologic-...
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Why heat kernel is a dirac sequence?

I want to show that the heat kernel $\displaystyle\gamma_t(x) := \frac{1}{(4\pi t)^{d/2}} \exp\left( - \frac{|x|^2}{4t} \right), \quad x \in \mathbb{R}^d, \ t > 0$ is a Dirac sequence. $\gamma_t \...
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Is there an appropriate transformation of weights?

Background Sometime back I managed to conjecture an interesting formula: $$ \lim_{k \to \infty} \lim_{n \to \infty}\ \sum_{r=1}^n \lambda_r \left( f(\frac{k}{n}r)\frac{k}{n} \right) = \lim_{s \to ...
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Back transforming the intersect of 2 log normal curves

Lets say i have a dataset that follows a double log normal distribution. these 2 functions intersect at $(x,y)$ when $y = p(\frac{1}{\sigma_1\sqrt{2\pi}}exp(-\frac{(x-\mu_1)^2}{2\sigma_1^2})) = (...
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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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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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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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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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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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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
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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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1answer
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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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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
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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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1answer
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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 ...