# Tagged Questions

In mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic approach to the study of geometry by focusing on groups of geometric transformations, and the properties of figures that are invariant under them. (Def: http://en.m.wikipedia.org/wiki/...

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### Is there a name for this mapping?

I wish to take a set of points described using Cartesian coordinates ($x, y$) and map them to a set of polar coordinates ($r, \theta$), such that $r = y$ and $\theta = x$. Thus horizontal lines become ...
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### What is special about glide reflection?

In wikipedia article https://en.wikipedia.org/wiki/Translation_(geometry) it is written: A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and ...
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### Is there any ongoing research in transformation geometry?

I'm a 3rd year undergrad Math student and totally confused right now on what field I'm gonna choose in my thesis next semester. We we're introduced in Transformation Geometry (Introduction to ...
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### Projecting from one $2D$ plane onto another $2D$ plane

I would like to project from one $2D$ plane onto another. Imagine that I have a picture taken with a camera that was looking onto a plane. Given camera's extrinsic and intrinsic parameters I want to ...
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### is the followng function $f$ surjective?

$f$ is a function mapping $x$ axis to plane $V$ defined by if $P(x,0)$ then $f(P) = (x,x^2)$ ? I am not so sure for my following answer : to show that $f$ is surjective, for every $A(x,x^2)$ there ...
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### Finding the glide reflection using a compass or straightedge

Given two congruent triangles that are not a rotation, translation or reflection of each other; how can I find the glide reflection (the last remaining option) using only compass and straightedge. ...
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### How to proof that Möbius transformation is isometric?

How to proof that the following Möbius transformation is isometric with respect to the spherical metric? $\varphi (z)=\frac{az+b}{cz+d}$ The answer is : $a=\bar{d},b=-\bar{c}$ However, I do not ...
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### vector analysis and co-ordinate transformation

Suppose one try to introduce a new product of two vectors as $C = A\operatorname{XX}B$, where $A,B,C$ are all vectors. Now it is defined as \begin{align} C_x &=A_y B_z + A_z B_y\\ C_y &= ...
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### Transforming a trapezium to a rectangle

I need to transform a certain area of an image to a rectangle. I know the 4 corner coordinates of the trapezium area. For example, this is the original image: The x and y coordinates of points A, ...
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### Geometric transformations between dimensions

I have a ratio I want to illustrate by geometry. That ratio is $27/25 = (\frac{3\sqrt3}{5})^2$ = 1.08 So I have made this illustration but I can't figure out, how to make a transformation between ...
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### Write the equation that results in the desired transformation [closed]

The cubing function, vertically shrunk by a factor of $0.2$, and reflected the $x$-axis then $y=$?
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### Generalising Plane Isometries to $\mathbb{R}^3$

Firstly, I DO NOT WANT PROOFS OF ANY OF THESE THEOREMS, as I wish to prove them myself. However, I would like to know the correct generalizations to $\mathbb{R}^3$ of the following theorems: An ...
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### Find the coordinates of center for the composition of two rotations

The combination of a clockwise rotation about $(0, 0)$ by $120◦$ followed by a clockwise rotation about $(4, 0)$ by $60◦$ is a rotation. Find the coordinates of its center and its angle of rotation. ...
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### The Composition of Two rotations

So far I rewrote the halfturns of d,c,b,a to halfturn (p,n)(m,l) where n=m because lines c and d are parallel so I can make ambiguous lines n and p parallel too. I also know that lines c,d can be ...
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### Definition of some terms of transformation geometry

Recently I was studying transformation geometry from problem solving strategies . I liked the subject but could not understand some terms. Please anyone help me--- What is isometry? What is ...
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I have used ...
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### Images of Lines

I'm studying for this exam and one of the questions I am stuck on is: Find the image of the line $$3x-y+1 = 0$$ under the transformation $$z \mapsto \frac{2}{z+1}$$ So I know I have to convert the ...
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### A question of H.G. Wells' mathematics

H.G Wells' short story The Plattner Story is about a man who somehow ends up being "inverted" from left to right. So his heart has moved from left to right, his brain, and any other asymmetries ...
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### Composition of isometries

I'm having trouble wrapping my head around the composition of isometries. For example, we've learnt that that the composition of three reflections is a glide reflection if they are not all parallel ...
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### Calculate the slope of a line on an auto-scaled chart?

Let say, I have a line with two points A=(1, 10.09) and B=(3, 10.42) on an auto-scaled chart like this. I would like to ...
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### geometry: linear transformation

I know I do it wrong but where is the mistake??? In E3* are given the points $A(1,0,0,0)$, $B(0,1,0,0)$, $D(0,0,1,0)$, $O(0,0,0,1)$ and $E(1,1,1,2)$. The linear transformation $\Phi$ operates ...
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### Set with plane having two axes of symmetry with angle of intersection

Proof that a closed and compact subset of plane having two axes of symmetry with angle of intersection not a rational multiple of $\pi$ is either a disc or a whole plane. Proof The composite of two ...
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### transformation between square and a polygon?

I have a square and a polygon. I want to transform all the points inside this square such that they are mapped inside the polygon. I was trying using scale and rotate matrices but I am not able to ...
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### 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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### What are spatial Transformations?

What are spatial Transformations? Are Affine transformations also part of spatial transformations?
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### Transforming a point from one coordinate system to another

I am trying to map the position of an object in one image to another. I have four points in one image with corresponding points in another image so as to bound an area say A. Now, if I have points in ...
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### Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a$if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook
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### How to get around non-commutativity of matrix multiplication?

I have a problem with a matrix equation/transformation problem which I need solving. I have two transformations $A_1$ and $A_2$, both of which can be expressed as $A_i = R_i \times B_i$, $R_i$ ...
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### Hyperbolic Uniformization Metrics

I have been working Euclidean Ricci Flow but have been having considerable trouble trying to implement the same discrete gradient descent functionality in hyperbolic space. I am following the ...
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### Hyperbolic Universal Covering Space

I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic meshes....
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### Isometries of $\mathbb{R}^3$

So I'm attempting a proof that isometries of $\mathbb{R}^3$ are the product of at most 4 reflections. Preliminarily, I needed to prove that any point in $\mathbb{R}^3$ is uniquely determined by its ...
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### Finding Transformation matrix between two 2D coordinate frames [Pixel Plane to World Coordinate Plane]

The question I'm trying to figure out states that I have N points (Pa1x,Pa1y) , (Pa2x,Pa2y)...(PaNx,PaNx) which correspond to a Pixel plane xy of a camera, and other N points (Pb1w,Pb1z) , (Pb2w,Pb2z)....
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### If a rectangular grid, where left<->right and top<->bottom wrap, can be mapped onto the surface of a torus, what does a cube map to?

If you roll a sheet of paper so left and right touch, then bend the cylinder so its ends also touch, you can see the surface of a 2D rectangle maps onto the surface of a 3D torus, a doughnut. I was ...
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### How to find the center of an (scaled) ellipse?

This question is an extension of How to find the center of an ellipse?. The solution there works well, but in Javascript the floating point calculations are not that accurate. The workaround is to "...
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### Geometry Reflection Notation

The following are exercises from The Four Pillars of Geometry; I'm not sure what they are stating, for example I don't know what the addition of prime (an apostrophe) to a line means. There are no ...
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### Why is “glide symmetry” its own type?

Artin's Algebra pages 155 & 156 list the types of symmetry of a plane figure as: Reflective Rotational Translational Glide He then goes on to say "Figures such as wallpaper patterns may have ...
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### Please explain this definition of symmetry

I am reading an article "Differential Equations: Not just a bag of tricks" in the mathematics magazine. The author has given elementary examples of symmetry ($y=x^2$ symmetric about $y$ axis, $y=x^3$ ...
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### Do translations form a normal subgroup in a general Euclidean plane?

Do translations form a normal subgroup of the group of rigid motions in a general Euclidean plane with no underlying field? This is a question that has puzzled me for the past few days. In the ...
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### Understanding a Lemma from J. Aarts Plane and Solid Geometry

I read this recent question on proving that the identity is never the product of an odd number of reflections. After googling a bit, I found a Lemma in J. Aarts Plane and Solid Geometry, but I didn't ...