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).
2,887
questions
0
votes
0
answers
25
views
Area-preserving continuous deformation of the graph of $r=a(1+\cos(2\theta))$ into a circle centered at the origin (graph included)
I want to know if the equation I provide actually does indeed model the transformation that I desire to model
Model the starting shape by
$$f\left(t\right)=\left(a\left(\cos\left(2t\right)+1\right)\...
- 399
2
votes
0
answers
21
views
Volume-preserving continuous deformation of an arbitrary ellipsoid centered at the origin into a sphere centered at the origin (graph included)
I want to know if the equation I provide actually does indeed accurately model the transformation that I desire to model
Model the ellipsoid by
$$f\left(u,v\right)=(A\cos\left(u\right)\sin\left(v\...
- 399
2
votes
0
answers
19
views
Modeling the continuous deformation of an arbitrary ellipse centered at origin into a circle of the same area centered at origin (graph included)
I want to know if my derivation is correct
Model the ellipse by
$$f\left(t\right)=\left(A\cos t,B\sin t\right)$$
Model the resultant circle by
$$g\left(t\right)=\left(\sqrt{AB}\cos t,\sqrt{AB}\sin t\...
- 399
1
vote
0
answers
81
views
Standard terminology for the operator $(a,b,c,d) \mapsto (b*c*d, a*c*d, a*b*d, a*b*c)$
Is there a standard name for the operator $s(a,b,c,d) := (bcd, acd, abd, abc)$ ? More generally, $s(x_1,x_2,...,x_n) = (z_1,\ldots,z_n)$, where $z_i = \prod_{j \ne i} x_j$.
- 9,230
0
votes
2
answers
53
views
Translation/Rotation properties for equations in the complex plane
Prove that complex numbers ($z_1,z_2,z_3$) that satisfy the relation below form an equilateral triangle in the complex plane.
$$z_1^2 + z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1$$
This answer first shows that ...
- 1,105
1
vote
1
answer
62
views
Can I see an example of transforming a tensor from polar to cartesian coordinates?
I have been learning about tensors, and I understand about creating the Jacobian matrix to obtain the coordinate transformation for infinitesimals, so that we have
$$d\bar{x}^j=\frac{\partial \bar{x}^...
- 13
0
votes
0
answers
31
views
Series expansion involving Kummer and Tricomi functions analogy
Good day to everyone. I've got in a pickle while toying around with some transformations.
It is well-known that the bivariate confluent hypergeometric function $\Phi_2(\cdot)$ can be expanded in the ...
- 51
-3
votes
0
answers
15
views
Find the bilinear transformation which maps z=(1,-i,2) respectively into w=(0,2,i) [closed]
Find the bilinear transformation which maps z=(1,-i,2)
respectively into w=(0,2,i)
- 1
0
votes
0
answers
50
views
Combine Rotation and Reflection matrices in one Matrix in 2D systems [closed]
Is it possible to combine the rotation and reflection matrix in one matrix?
$$Rot(\alpha)=\begin{pmatrix}\cos(\alpha) & -\sin(\alpha)\\\ -\sin(\alpha) & \cos(\alpha)\end{pmatrix}$$
$$Ref(\...
- 101
0
votes
0
answers
16
views
Equivalence with the Weierstrass transform
I have the following expression
$$\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}dx~ f(x-y) e^{-x^2/4 t} \tag{1},~~\forall ~y \in \mathbb{R}.$$.
I am trying to relate it with the generalized ...
5
votes
0
answers
132
views
How to $a^b \to b^a$ [duplicate]
Addition, multiplication and power are formed in a similar way. One follows from the other by repeating the preceding action several times. But while $a+b=b+a$, and $a \times b=b \times a$, in turn $a^...
- 51
2
votes
2
answers
57
views
Coordinate Transformation of Double Integral
Evaluate the integral by making an appropriate change of variables.
$$\iint_R\left[\cos \left(\frac{y-x}{y+x}\right)\right]^2 dA$$
where $R$ is the trapezoidal region with vertices $(2,0), (3,0), (0,3)...
- 51
0
votes
0
answers
51
views
Shape of a transform matrix that maps $\mathbb{R}^{a \times b}$ to $\mathbb{R}^{c \times d}$
I'm looking for the dimension of some tensor $T$ that transforms $a \times b$ matrices to $c \times d$ matrices. I initially thought it would have to be of a greater dimension, perhaps some shape like ...
- 33
0
votes
0
answers
38
views
Is the volume equivalent of every odd dimension conserved after deformations?
I don't know the best way to phrase this (english isn't my main language), so I'll do my best.
First we have the perimeter (one dimension, a line), which is conserved after a deformation. for example ...
0
votes
1
answer
46
views
Next Step After Finding Eigenvalues of Transformation Matrix T?
I've successfully determined the eigenvalues of the transformation matrix (T = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 5 & -10 \\ 1 & 0 & 2 & 0 \\ 1 & 0 & ...
- 335
0
votes
1
answer
28
views
Lie group of transformations in a plane.
I was trying to solve the following problem:
Check whether the transformations of the plane given below forms a Lie group.
$$x^{*}=x-\varepsilon y;\quad y^{*}=y+\varepsilon x$$
I tried to identify the ...
- 25
1
vote
0
answers
111
views
Proof-Check: Beta-distribution sample generation through variable transformation
Purpose of this thread: I want your feedback on my proof for the following problem and correct it where necessary.
Problem: For $\alpha > 0$ and $\beta > 0$, consider the following accept–reject ...
6
votes
1
answer
131
views
Is it possible to construct a coordinate system which gets "pulled in"
I really hope i can explain our issue adequately and keep this purely math bound, even if it is technically a graphics programming related question. But over on their exchange ill never get an answer. ...
- 201
0
votes
0
answers
18
views
Mixing conditional random variables by sampling
I am struggling to put my transformation of data into mathematical contexts. My goal is to define a mapping that transforms the original data into some awkwardly mixed data. In my simulation study, I ...
- 1
1
vote
1
answer
39
views
Finding a transformation that squares the components of a vector
I am looking for a transformation (of any type, as long as it can be written mathematically and not semantically) which takes a vector of infinite dimensions to another vector that has components ...
- 114
0
votes
0
answers
31
views
Exact Successor State Distribution for a Pendulum
I want to solve the following problem. Suppose we have a simple pendulum, which follows the differential equation
\begin{equation}
\dot{x} = f(x) = [x_2, -\sin(x_1)]^T, \text{with } x=[x_1, x_2]^T.
\...
- 1
0
votes
0
answers
24
views
Effects of horizontal scaling on area under curve?
If f(x) is a continuous function and we stretch it horizontally by a factor n (multiply x by 1/n in the formula) then my intuition and some examples i solved tell me that if we look at a segment of ...
- 87
0
votes
1
answer
74
views
Rotation of angle $2k\pi/p$ generates the group of all rotation
In the book Geometric Transformation of Razvan Gelca, there is an argument as follows:
I could understand most of the proof there, however is there any easier explanation for the yellow painted part, ...
- 343
0
votes
0
answers
17
views
Dual graph relation to star-mesh duality
I'm confused about the realtionship between dual graphs and the so called star-mesh transformation: https://en.wikipedia.org/wiki/Star-mesh_transform.
Take a simple triangle, its dual graph looks like ...
0
votes
2
answers
63
views
Intersection and union under arbitrary function
I have tried to grasp topology homeomorphism. I have heard that such transformations preserve intersections and unions. I have tried to prove it myself and it came out that actually any transformation ...
1
vote
1
answer
69
views
Is there a canonical form for rational expressions?
Polynomials have a canonical form $\sum a_n x^n$ which makes it easy to understand what a polynomial is. Yet rational expressions can often wear many disguises, where it's not obvious that two ...
- 2,722
1
vote
0
answers
19
views
How would I scale up a set of values while preserving their original increasing order using logarithmic functions?
I've got a set of data ranging between 0 and 1. Most of the values in this set are extremely small (like, between 0.003 and 10^-7). I want to scale up these values, so they are less small but still ...
- 143
0
votes
1
answer
39
views
Expected value of transformation of exponential PDFs
I have the following: $X$ and $Y$ are exponentially distributed with parameter $\lambda$.
They are independent of each other. I also have $U = X+Y$ and $V = X-2Y$.
I am asked to find $E(V|U=1)$.
What ...
0
votes
0
answers
23
views
General exponential grid
A known function that maps the interval $[-1,1]$ onto itself that is used to modify a linear gridding into an exponential one is given by,
\begin{align*}
f \colon [-1,1] &\to [-1,1]\\
x &\...
- 194
0
votes
0
answers
18
views
Transform ratios with negative numerators
I am working on a regression model that takes in features that are in the form of ratios.
For example, one of the ratios is:
...
- 101
1
vote
0
answers
66
views
What is Meant by "Change of Origin" in Coordinate Geometry?
I don't think I understand what is meant by "to shift the origin of coordinates to the point $(h,k)$ in coordinate geometry. I've read Loney's book on coordinate geometry in which he says that to ...
- 1,377
2
votes
1
answer
41
views
Problem With Transformation of Coordinates
In S.L. Loney's book on Coordinate Geometry, he introduces the reader a way to switch from one origin of coordinates to another. The method he shows is: to change the origin to the point $(h,k)$ you ...
- 1,377
4
votes
0
answers
56
views
Removing four terms from the decic using a Tschirnhausen transformation?
I. Transformation
In this 2021 paper, one can remove four terms from an equation of degree $n$ using a Tschirnhausen transformation of degree $n-1$ (and with radical coefficients), but only if $n\...
- 52.2k
2
votes
1
answer
87
views
Generalization of Gauss multiplication formula for $\Gamma(jm+kn+a);j,k\in\Bbb N$?
A hypergeometric single sum, like a Mittag Leffler function uses the Pochhammer symbol $(a)_n$ multiplication formula to easily have a univariate hypergeometric function $_p\text F_q$ closed form:
$$\...
- 9,987
1
vote
0
answers
69
views
Reducing the general sextic to "cubic" form and other Tschirnhausen transformations of higher degrees
I. Methods
Using a quartic Tschirnhaus transformation, one can eliminate three terms $x^{n-1}, x^{n-2}, x^{n-3}$ without solving a $1\times2\times3 = 6$-deg equation, a simple explanation of which is ...
- 52.2k
0
votes
0
answers
17
views
Formula for Transformation of Polynomial Coefficients under Rotation
I have a function represented in a basis of 2D Legendre polynomials,
$$
f(x,y) = \sum_{n=0}^N c_n P_n(x,y)
$$
where $P_n(x,y)$ is a 2D Legendre polynomial given by
$$
P_n(x,y) = P_l(x)P_m(y)
$$
where $...
- 266
0
votes
0
answers
8
views
Rectifying the isophote to second order by homothetic transofrmation
I'm writing a Python script following a research paper's algorithm (https://articles.adsabs.harvard.edu/pdf/1990A%26A...233...82C). I've got my isophote points and the points of the closest-fitted ...
- 101
0
votes
1
answer
53
views
Find $\sum^{30}_{k=7}(\sqrt{k - 4} - \sqrt{k - 3})$ without explicitly calculating the 24 complements
Is it possible to find $\sum^{30}_{k=7}(\sqrt{k - 4} - \sqrt{k - 3})$ without explicitly calculating the 24 complements ?
I have this task in a text book and it doesn't provide any equivalent example, ...
- 885
0
votes
1
answer
52
views
Rotation of functions without using matrices
I was reading a Linear Algebra book when I stumbled upon this relation
$$
\begin{bmatrix}
x'\\
y'
\end{bmatrix}
=
\begin{bmatrix}
\cos \alpha & -\sin \alpha\\
\sin \alpha & \cos \alpha
\end{...
- 169
2
votes
1
answer
43
views
Finding a transformed random variable's distribution
Let $Y\sim \operatorname{Geo}(p)$ with $P(Y=k)=p(1-p)^k$. Furthermore, $\hat{Y}\sim \operatorname{NegBin}(2,p)$, i. e. $P(\hat{Y}=k)=kp^2(1-p)^{k-1}$.
I want to find $P(\lfloor U\widehat{Y} \rfloor=k)$...
- 2,388
1
vote
1
answer
53
views
How to prove $\sum_{n\ge 0} a_n x^n = \sum_{n\ge 0} \left(\sum_{k=0}^{n}(-1)^{k+1} \binom{n}{k}a_k\right) \frac{x^n}{(x-1)^{n+1}}$?
While reading the Wikipedia article on the Binomial transform, which is defined for a given sequence $\{a_n\}$ as
$$
s_n = \sum_{k=0}^{n}(-1)^k \binom{n}{k} a_k
$$
I found the following relation ...
- 6,595
0
votes
0
answers
32
views
How does a constant scaling of the metric affect the geodesics? [duplicate]
Let $(M,g)$ be a Riemannian manifold, and let $\lambda$ be a positive number. Then we know $(M, \lambda g)$ is also a Riemannian manifold.
Also, if we have a point $p$ in $M$ and a vector $V$ in $T_pM$...
0
votes
0
answers
15
views
Question regarding transformation of stiffness matrix
I am currently doing my bsc thesis and I need to find the systems stiffness matrix of a beam structure in 3d. I have the element matrix of all the beams but I need to rotate them to the global system. ...
1
vote
1
answer
53
views
A Problem on Compositions of Rotations [closed]
Let $A$, $B$, and $C$ be three different points such that compositions of the following three rotations
$$
R(C, 256^\circ) \circ R(B, 244^\circ) \circ R(A, 220\circ)
$$
has a fixed point. Determine ...
- 297
2
votes
1
answer
63
views
Is there a way to approximate a function as sum of Gaussian?
I'm a chemist, and my friend give me a spectroscopic data that was supposed to be sum of several Gaussian function $(G(x)=Ae^{-B(x-C)^2})$.
Let's say the data is given as function $f(x)$, is there a ...
- 147
0
votes
1
answer
38
views
Graph Transformation: $f(x)=(\sin x)^2$ and $a=x^{−1/2}$
My understanding is that the transformation $f(ax)$ to a transformation $f(x)$ will increase the frequency of $f(x)$ by scale factor $a$.
Furthermore, $a$ can itself be a function of $x$.
Thus if we ...
- 139
0
votes
0
answers
42
views
Transformation matrix 3d beam element
I am currently doing my bsc thesis and I need to find the systems stiffness matrix of a beam structure in 3d. I have the element matrix of all the beams but I need to rotate them to the global system. ...
0
votes
0
answers
19
views
Shouldn't an object orbit and not spin when it first translated and then rotated?
I understand that the transform order matters, but I don't understand why the object in this tutorial on minute 7:00 spins and not orbits.
Line 156 applies rotation and then translation on Model ...
- 113
0
votes
0
answers
34
views
Prove every parallelogram preserving transformation can be constructed from an isometry and two stretches
Prove that any transformation of the Euclidean plane which preserves parallelograms can be constructed from the composition of a rigid transformation (an isometry) and two stretches, using synthetic ...
- 2,722
0
votes
0
answers
19
views
Radon Transform over a finite domain
Im having trouble understanding how to apply the Radon Transform.
Algorithmically, it is simply applying the line integral of the form:
$$\mathbb{R}[f(x,y)] = \int_L f(x(s), y(s))ds$$
where $x(s) = r\...
- 47