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), ...

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Why does “up to scale” make homograph matrix lose one freedom?

Can anyone explain "if H is up to scale, then dof(H)=8" in the following discussion? degree of freedom of Homography matrix Thank you!!!
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12 views

How to go from Affine to a Non-linear transformation

If you were able to take two sets of matrices and transform one to attempt to match the other using an affine method such as Least Squares, how could you replicate this process better using a ...
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1answer
20 views

State transformation for non-holonomic differential equation.

Given a non-holonomic dynamical system, \begin{align*} \dot x = v\cos\theta \\ \dot y = v\sin\theta \\ \dot \theta = \omega \end{align*} with constraints $|v| < v_{max}, |\omega| < ...
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14 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
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11 views

Transforming UV Region to XY Bounded By Hyperbolas and Lines

Suppose I have a region in the x-y plane bounded by: $y=\frac{1}{x}, y=\frac{4}{x}, y=x, y=4x$ We see that: $1\leq yx \leq 4$, and $1\leq \frac{y}{x} \leq 4$ If I let $u=yx$ and $v = ...
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3answers
33 views

How can you enlarge a shape about a point other than (0,0), using matrices?

If I want to enlarge a shape, $A$, by scale factor $k$ about $\left(0,0\right) $ I multiply each point (in the form $\begin{bmatrix}x\\y\end{bmatrix}$) by $kI$. However, I can't work out a general ...
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1answer
18 views

Inverse Z-transform of $\frac{10}{z-5}-\frac{2}{z-1}$

I must calculate the inverse z-transform of $\frac{10}{z-5}-\frac{2}{z-1}$. I decided to use the known formula $H(n-1)a^n\rightarrow \frac{a}{z-a}$, where $H(n)$ is the heaviside signal. I finally get ...
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2answers
34 views

How to determine that the 3 points given in homogeneous coordinates are collinear? [closed]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
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1answer
18 views

Translate a Rectangle Position from 1 Image to another [closed]

I have a Large Size Image.Since its too large for processing within a small time, i need to resize it.I have the coordinates of a rectangle in the resized image.Is there a way i can translate this ...
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22 views

Finding transform matrix from resulting multiplypoint function

Two matrix transformation functions exist within the Unity3D API: 1) MultiplyPoint 2)MultiplyPoint3X4 3X4 matrix (2) preforms a standard transform against a vector (And ofc is easily replicated ...
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23 views

Fourier transform of $H(-t)e^{5t}$

i have to calculate the Fourier transform in the title. My professor says the result is $\frac{1}{5-2\pi i f}$. I start from $H(t)e^{\alpha t}$, and i calculate the transform $H(t)e^{5t}\rightarrow ...
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1answer
16 views

Can you kindly explain me in detail this Fourier transform?

I've this function to transform not using the general formula, but just substituting the known transform (i.e. $\text{rect}(t)\rightarrow \text{sinc}(f)$): $\frac{\sin(6\pi t)}{t}$ I know the ...
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0answers
17 views

Transforming parts of functions

I have a function in the form: $$ \mathrm{e}^{-t\lambda} \cdot \left[t\lambda - {(t\lambda)^2 \over 2}\right] $$ If one were to plot this for say $\lambda = \frac{2}{3}$ and $t$ from $0$ to $20$, ...
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22 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
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12 views

Perspective correction from 3 points and foreshortening factor

I'm working on creating a homography 3x3 matrix to do a perspective correction of a photograph 2D piece of paper. The paper contains 3 markers (like the 3 corner markers of a QR code) in its corners, ...
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3answers
46 views

Image of a family of circles under $w = 1/z$

Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of ...
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1answer
43 views

Fourier transform of $\frac{\sin(6\pi t)}{t}$

I have to calculate the fourier transform of this function in time domain: $\frac{\sin(6\pi t)}{t}$. First I tough to use the definition of $\operatorname{sinc}$ function as ...
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1answer
13 views

Prove a transformation is injective if its restrictions are injective.

Suppose that $V$ is a vector space, and let $V → W$ be a linear map. $$V_0 ⊆ V_1 ⊆ · · · ⊆ V_i ⊆ V_{i+1} ⊆ · · · ⊆ V$$ are subspaces of $V$ (one for each $i = 0, 1, 2, \ldots$) and inclusions ...
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1answer
40 views

$Z$ coordinates disappear in the general rotation transformation matrix.

I wanted to generate the general rotation transformation matrix ($3D$). But when I did the multiplication the result didn't include the original $Z$ coordinates,I don't know why the $Z$ disappeared. ...
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14 views

How to determine changing scale factors when performing coordinate transfomations?

To explain: I have two coordinate systems. One (x,y) and the other (x',y') as seen in this photo. Coordinate systems I am trying to convert the coordinate in the (x,y) system to the rotated red ...
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1answer
12 views

Can you stretch a function with a zero or undefined gradient?

If $y=f(x)$ is either $y=3$ (zero gradient) or $x=2$ (undefined gradient), is it possible to stretch $y=f(x)$ by graphing $y=af(x)$ or $y=f(ax)$? If it is possible to stretch them, can you only ...
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0answers
20 views

Homography with line correspondences

When calculating a homography with line instead of point correspondences, what is the derivation of the formula: $$ l_i = H^T\cdot l^{'}_i $$ I know that: $$ l^T\cdot x = 0 \quad\text{and}\quad ...
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1answer
44 views

Find the density of $Z=\frac{X}{Y}$ for an exponential distribution?

We have the iid random variables $(X,Y)$ where $f_x(x)=\lambda e^{-\lambda x}$, $x>0$. We are given $Z=\frac{X}{Y}$ and asked to find the cdf and the density function. Here's my attempt. ...
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18 views

affecting the final result of function depending on external factor

Suppose I have a function $f(x) = \frac{x}{x+y}$ whose range is in the interval $[0,1]$ and there is an external factor say $a$, such that $a$ is in the interval $[0,1]$, moreover, a predefined ...
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9 views

Solving a transformation problem in complex (Argand) planes

Below is an example in a text-book section on using complex-number arithmetic to represent transformations in Argand planes Example A transformation $T$ of the $z$-plane to the $w$-plane is ...
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1answer
22 views

What's the formulation of N-point radix-N for NTT

We can write the formulation for the buttlerfly function applied in FFT as \begin{align*}y_0 &= x_0 + x_1,\\ y_1 &= x_0 - x_1. \end{align*} As seen here. For FFT (Fast Fourier Transform) we ...
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0answers
8 views

Characteristik values of linear transformation?

Consider this inner product space on C^2. T be a linear operatör from C^2 to C^2 s.t every (x,y) element of C^2 T(x,y)=(x+iy,y+ix). Find all characteristic values of T. And T is self adjoint?
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1answer
15 views

How to determine if an affine transformation would cause reflection?

I have a list of affine transformation matrices and I want to write a code to delete the transformation matrices that applying them on an image would cause reflection. after seeing this image in ...
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20 views

Distribution of discrete function of continuous random variable?

It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the ...
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1answer
19 views

Invariance properties of transformations

In Gentle's Matrix Algebra (2007, p. 175), he presents a table of what features of vectors various transformations preserve. What does it mean to say a transformation T preserves some property of a ...
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1answer
54 views

Bring system in normal form up to the second order

Bring the system $$ x'=y+xz,\quad y'=x^2+y^2+z^2,\quad z'=-2z+xy $$ to a normal form up to the second order (kill all non-resonant quadratic terms). The equilibrium is $(0,0,0)$ ...
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1answer
25 views

How to approximate linear relationship between two timeseries? [closed]

I have two time series A and B I would like to solve for the equation in the form $$y_t = m x_t + b$$ that transforms a point $y$ at time $t$ from series $B$ to the corresponding point $x$ at time ...
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101 views

PDFs of Piecewise Transformations: why doesn't it apply in this case?

This is from Casella and Berger's Statistical Inference, although it is more of a probability question than a stats question. Theorem 2.1.8 Let $X$ have pdf $f_{X}$, let $Y = g(X)$. Define the ...
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1answer
28 views

rational integral with a quartic function in the denominator

Let's say I've got the integral: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ$ This integral evaluates to: $\int [ 2 Q^4 - 5 Q^2 + 3 ]^{-1} dQ = \tanh^{-1}\left( Q \right) + \sqrt{ \frac{2}{3} } ...
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1answer
22 views

The correct order for applying transformations?

I had a line, y=-1/2 x I wanted to reflect everything onto the line so as you can see by my steps in the picture below(sorry, I dont know the coding) I first rotated the line theta degrees then ...
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1answer
31 views

Transform PDE to ODE (3 variable case) with given boundaries

How can I transform the following PDE into an ODE? I tried using three different functions $H(x),G(y)$ and $F(t)$ but that didn't help hence I did not post it here. I really hope someone can help me ...
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1answer
10 views

Three Variable Transformation and Independence

If $X_1$, $X_2$, and $X_3$ are independent, identically distributed random variables and $Y_1$, $Y_2$, and $Y_3$ are functions of them, how do I show that the $Y$ variables are mutually independent? ...
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1answer
20 views

Integral with more-dimensional substitution variables

Good day, In the lecture of partial differential equations we had the following transformation: $$\int_{||\nu||=1} h(x+\nu c t, \tau) d\nu = \frac{1}{c^2 t^2} \int_{||y-x||=ct} h(y,\tau) dy$$ for ...
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33 views

Is there a product integral that preserves zeroes?

The integral essentially takes the arithmetic mean of the range of a function multiplied by the domain, adding together each possible output weighted by the amount of the domain accounted for by that ...
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30 views

(Answered) Transformation between AutoCAD OCS and WCS coordinate systems

This question originates in a problem I am having with transforming between two coordinate systems, that come from AutoCAD. This question is purely about the math of the problem, I just include the ...
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0answers
22 views

Fourier Transform of triangle function

I have a question regarding the FT of the triangular function: How does $e^{-j\omega t}$ becomes the cosine function in the first line? What happened to the sine when you go from $e^{j \omega t}$ ...
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3answers
32 views

Transformation of exponentials

Find the transformation that takes $y=3^x$ to $y=\textit{e}^x$. I have tried: Let $y=3^x$ to $y=e^{x'}$ $$\log_{3}(y)=x\quad\text{hence}\quad\log_{3}(y)=\frac{\log_{e}(y)}{\log_{e}(3)}$$ ...
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26 views

Graph Transformation

This question seems trivial to me but I still cannot figure out how to approach it. For any $x \gt 0$ and $f(x) \le 2^x$, how do I prove the following? $$ f(2x) \le f(x)^2 $$
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1answer
51 views

How can I find the Fourier transform of constant value like $1$.

The textbook told me that $\mathbb F[1] = \delta(f)$ and $\mathbb F[\delta(t)]=1$. It is easy to prove that $\mathbb F[\delta(t)] = 1$. $$ \mathbb F[\delta(t)] = \int_{-\infty}^\infty ...
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0answers
36 views

pdf of transformed random variable g(X) as integral over X?

I am not a mathematician, so I am sorry if this question is too easy or some notational detail is not correct. I am trying my best! I have got a random Variable $X$ in $\mathbb{R}^N$ with pdf $p(X)$ ...
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0answers
10 views

Name for affine transform which consists of scale and translate

Is there a concise name for this type of transform? Viewed as an affine transform, the matrix would have the form: k 0 dx 0 k dy 0 0 1 The best I can come ...
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21 views

Is there a similarity solution for this PDE? (with discussion, kindly check)

I have a PDE for $h(x,t)$ of this form $$h_t+Ah^{-1}+(h^3h_x)_x+Bh_{xx}+(h^3h_{xxx})_x=0,$$ where the subscripts denote the partial derivatives, and $A$ and $B$ are all constants. I'm wondering ...
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1answer
14 views

Basic inverse $z$ transform

I have trouble finding a (probably) pretty easy inverse of a $z$ transform. $$H(z) = \frac{z-0,5}{z+0,5}$$ I used the polynomial division on it to get a proper fraction and got $$H(z) = 1 - ...
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2answers
22 views

Range of continuous transformation on closes set

let $f$ be a continuous transformation and $F$ closed set. Prove that the range $f(F)$ does not have to be closed.
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3answers
188 views

Find the distribution of the series $Z = X_1+X_2+…+X_N$

"Let $0<p=1-q<1$. Suppose that $X_1,X_2,...$ are independent Ge(q)-distributed R.V.'s and that $N \in Ge(p)$ is independent of $X_1,X_2,...$. Find the distribution of $Z=X_1+X_2+...+X_N$." I ...