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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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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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Transforming inequalities [on hold]

Can someone explain to me how to transform it $\left( \frac{en}{s} \right)^{s} \le \left( \frac{en}{s+1} \right)^{s+1}(s+1)(n-s+1) $ to $\left( \frac{s+1}{s} \right)^{s} \le en(n-s+1) $
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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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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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2answers
38 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 $dx’=dx$ ...
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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 ...
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1answer
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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,...
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37 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
28 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, $$ \...
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1answer
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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 ...
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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 ...
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1answer
31 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 ...
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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 $\Bbb{R}$ .
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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 $...
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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 ...
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1answer
30 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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1answer
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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)) ...
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0answers
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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 ...
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1answer
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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{...
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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 ...
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2answers
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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}+\...
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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 ...
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1answer
29 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....
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1answer
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Question about group of automorphism of some $G$-structure.

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and I don't understand a thing at page 15. I don't understand why $U$ consists of transformation $a$ of $M$ that leave each ...
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1answer
41 views

Pdf of $\sin X$ when $X$ has the pdf $f(x)=\frac{2x}{\pi^2}1_{0<x<\pi}$

I would like to find the PDF of the random variable $Y=\sin(X)$ given the PDF of $X$: $$f(x) = \frac{2x}{\pi^2} \text{ for } 0<x<\pi \text{ and } 0 \text{ otherwise}$$ Following the tips in ...
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1answer
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Algebraic transformation — where is my mistake?

I tried to find the estimators of $\hat{\beta_1}$ and $\hat{\beta_0}$ via the least-squares method algebraically. Somehow I seem to have messed up. Can you tell me where? My Calculations.
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Continuous automorphism of a lie group in kobayashi's book

I'm reading Kobayashi's book Transformation Groups in Differential Geometry and i dont understand a thing at page 14. My question is why $A_\varphi$ is continuous? $G$ is a subgroup of ...
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48 views

Mapping $\Bbb N\to\Bbb Q$

I want to set up a map from $\Bbb N\to\Bbb Q.$ Take $\Phi_S(x)=e^{(S/\ln(1-x))}$ and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$ Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates. If $x$ happens ...
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1answer
42 views

How to solve this problem raven matrices problem?

I am doing this free test in http://test.mensa.no/ That, as far as I know, the only problem I can't solve. Basically we shift the first row to the right. From first to second is easy transformation. ...
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Matrix transformations.

I am currently learning matrix transformations and ran in to an exercise that I can't understand. We start with e1 = (1,0) e2 = (0,1) So the first transformation is a shear one where e2 becomes -2e1 + ...
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3D Vector Rotation Matrix with Radians

I have been working on a simple C++ vector library and needed 3D rotation so I found these 3D rotation matrices on Stack Overflow: ...
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1answer
41 views

Finding the pdf of $f(x,y) = e^{-x-y}$ where $Z = X+Y$

I am having trouble understanding how to find the pdf $f_Z(z)$ when $f_{X,Y}(x,y) = e^{-x-y}, x,y \space \epsilon(0,\infty)$ where $Z = X+Y$ My approach is that $$x = x, y = z-x$$ so using ...
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2answers
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How does squaring give you a monotonic transformation?

Consider the function: $$ f(x,y) = \sqrt {xy} $$ Is the function $$ f_1(x,y) = x^2 y^2 $$ a monotonic transformation of $ f $? I remember studying earlier that squaring does not give you a monotonic ...
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1answer
38 views

Show $f(n)= \frac{ (1-\alpha) a^{n+1} e^{-a}+\alpha b^{n+1} e^{-b} }{ (1-\alpha) a^{n} e^{-a}+\alpha b^{n} e^{-b} }$ is unique for $(\alpha,a,b)$

Suppose we have the following function \begin{align} f(n)= \frac{ (1-\alpha) a^{n+1} e^{-a}+\alpha b^{n+1} e^{-b} }{ (1-\alpha) a^{n} e^{-a}+\alpha b^{n} e^{-b} }, \text{ where }n=0,1,2,3,4, \ldots \...
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1answer
27 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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Pinch transform shapes

I'm looking for an algorithm that can pinches a shape in a way to become pointy on both ends. Like this image, transforming the shape on the left to the right. The result is literally similar to ...
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41 views

How to transform basis functions

I know how to transform the basis if we have two sets of basis vectors. Now in my situation, I have two basis equation and I want to find out the transform between those basis functions. Specifically, ...
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1answer
46 views

How to find a 2D coordinate field's corners in a 3D Coordinate field if I have 3x 3D points with 3x2D Points?

In order to solve "this" problem, i have to transform my corner-points from a 2D Space to my 3D Space. But my two coordinate fields are only defined by their relation to each other. They have the ...
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Determining partial derivatives after coordinate transformation

I'm currently trying to figure out the partial deriviatives of a function, after there is a space transformation. $$ f(r_1, r_2, w) \rightarrow f(x, y, w)(2\sqrt{xy})^{-1} $$ Where $$ r_1 = x^2 \\ \\...
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Hankel transformation and inverse Hankel transformation of integer order.

Read the definition of Hankel transformation here... https://en.m.wikipedia.org/wiki/Hankel_transform Question: can we define Hankel transformation for any Integer order? Can we define inverse Hankel ...
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Calculate new X & Y coordinate based on compressed or enlarged rectangle

I have two Rectangles as Rect1 ___________ x'', y'' | |dy | | .x',y'| | | | | 0,0------------- Here the value of x''...
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1answer
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Logistic Regression and modelling probabiliy $\pi$

This is a General Linear Models topic but I believe it's just basic failure to remember some more basic math rules that's making it difficult for me. If the link function is $$g(\pi) = \log(\frac{\...
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1answer
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Is $\begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix}$ similar to $\begin{bmatrix} 0 & CAC^{-1} \\ C^{-1}BC & 0 \end{bmatrix}$ by some transformation?

Consider a matrix with two entries being some operator or matrix $$D=\begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix}.$$ I want to construct another $2\times2$ matrix $S$ such that $$SDS^{-1} = \...
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92 views

coordinate transformation in differential equation

I'm confused about coordinate transformations. What I understand is, that if we have $$ f(x(t))=g(x(t)), $$ that we can write for each (bijective) $h(x)$ $$ f\circ h(x(t))=g\circ h(x(t)). $$ If we can ...
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1answer
49 views

Transformation Theorem on n-dimensional Sphere

$ n \in \mathbb{N},b >0$ Define $S_n$ as the n-dimensional Sphere in $\mathbb{R}^n$. I cannot figure out the appropiate transformation to use the transformation theorem such that the following ...
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Problem with the basis of a linear transformation

I have this exercise Define and find the expression of the linear transformation Information given: $ T : \Bbb R^5\to \Bbb R^4$ Im T = ⟨(3, -1, 0, ...
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Mathematical expression as LP problem

I am seeking for a linear programming expression for the following expression: enter image description here B(x) is the neighborhod of f(y) (where delta is the size of the neighborhod). It can be an ...
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Affine rectification via vanishing line

I'm trying to understand how to rectify an image given some lines that should actually be parallel in the final image. For example: from the book Multiple View Geometry. I know that the idea is to ...
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3answers
40 views

Existence of a strictly increasing transformation between two functions [closed]

Assume $f$ and $g$ are two differentiable functions defined on a compact interval $X \subseteq \mathbb{R}$ mapping into $\mathbb{R}$ . I want to proof or disproof the following statement $ \forall x \...
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Expansion in power of $\frac{1}{Z}$ and $\frac{ln(Z)}{Z}$

When I read the paper I met the problem in the step expansion in power. We have \begin{align} s(\epsilon)=\frac{A\epsilon^{a}}{bB|\dot\epsilon|}e^{-Be^{b}} \left[1+\frac{a}{bB}\epsilon^{-b}+\frac{a(a-...