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2answers
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Find the matrix that represents a rotation clockwise around the origin by$ 30∘$ followed by a magnification by a factor of 4.

Find the matrix that represents a rotation clockwise around the origin by 30∘ followed by a magnification by a factor of 4. My attempt: I multiplied the magnification matrix $\left[ ...
0
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0answers
65 views

Using basic transformations to derive matrix for the reflection in a line?

Using basic transformations (translation, scaling and rotations), show all the steps to derive the transformation matrix for the reflection of points n the line : y = 3 - x I know that a directional ...
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0answers
24 views

Similarity Transformations

For the transformation x'= 2x + 3y + 1 , y'= 3x-2y +4 ( dealing in matrix form) a) Show that this is a Similarity , determine whether it is direct or opposite and find its ratio? b) Find an ...
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2answers
66 views

Definition of collineation

In a book I am reading on transformational Euclidean geometry, the author defines a collineation as a bijection of the plane which takes lines to lines -- that is, for a collineation $F$, if $L$ is a ...
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1answer
343 views

Finding equation of the image under a linear transformation

The equation of C is $x^2 + y^2 =1 $ How do I find the equation of the curve $C'=f(C)$ This is the image of $C$ under the linear transformation $f$ represented by the matrix $A=\begin{bmatrix}2 & ...
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0answers
43 views

Vector transforms

I have used ...
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1answer
86 views

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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0answers
21 views

Homogenous coordinates: are Wc and Z different?

In these two calculations: http://upload.wikimedia.org/math/b/2/3/b23832f1c400204a4011459712cd5603.png http://upload.wikimedia.org/math/f/5/f/f5f8c42659c83689fa35185403312899.png Is Wc essentially ...
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5answers
314 views

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 ...
3
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1answer
123 views

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 ...
0
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1answer
28 views

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 ...
2
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3answers
74 views

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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1answer
116 views

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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1answer
225 views

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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1answer
2k views

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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1answer
334 views

What are spatial Transformations?

What are spatial Transformations? Are Affine transformations also part of spatial transformations?
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1answer
2k views

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 ...
0
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1answer
492 views

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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1answer
121 views

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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0answers
61 views

Finding a basis for a set of points in plane and a lattice.

Suppose my domain is $\mathbb{R}^3$ where I have: a set of points $A = \left\lbrace x_i \right\rbrace_{i=1}^{n_1} $ and $n_1 > 3$. $A$ is contained in a plane (2-dimensional) $P_1$. i.e. $A ...
3
votes
1answer
486 views

Convert LLA (long, lat, alt) to flat earth model

I would like to divide the globe into 1000 $\times$ 1000 meter geodesic squares, and then map any long / lat to the applicable square. The altitude of each block would be the altitude of the earth at ...
4
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0answers
298 views

Average transformation matrix?

I have several estimates of the transformation matrix between two planes and some values that give some indication of the error involved in the estimate. How can I use this information to gain the ...
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2answers
3k views

How can I determine the scale factor of a pantograph from the ratio of the arms?

I know this is probably simple but I just can't see it. How can I determine the scale factor of a pantograph from the ratio of the arms?
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2answers
237 views

When do three points determine the same orientation?

I'm trying to understand the following claim: If $z_1,z_2,z_3,z_4$ are points (as complex numbers) on a circle, then $z_1,z_3,z_4$ and $z_2,z_3,z_4$ determine the same orientation iff ...
1
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0answers
66 views

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 ...
2
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0answers
132 views

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 ...
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3answers
220 views

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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2answers
9k views

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) , ...
2
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2answers
206 views

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 ...
2
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0answers
368 views

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 ...
2
votes
1answer
1k views

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 ...
4
votes
3answers
374 views

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 ...
3
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2answers
229 views

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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2answers
375 views

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 ...
2
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1answer
108 views

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 ...
4
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1answer
273 views

Why is the identity map never equal to the product of an odd number of reflections?

Suppose I have an some plane and an identity mapping on the points of the plane. I see that the identity can be expressed as a product of an even number of reflections, since any reflection has itself ...
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2answers
1k views

Rigid Motions - The product of two rotations around different points is equal to a rotation around a third point or a translation

I'm having some difficulty wrapping my head around rigid motions in a plane. In particular, I'm trying to solve this following problem: In a Euclidean plane, show that the product of two rotations ...
8
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5answers
1k views

On isometric affine transformations

Somewhat prompted by the discussions of Qiaochu Yuan and Aryabhata in this question, I realized that my understanding of linear/affine transformations thus far had been built on a convoluted series of ...
4
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4answers
557 views

What transformations of the plane are geometrically constructable (compass & straight edge)?

Congruence transformations (isometries) and similarity transformations (isometries + dilations) should be constructable. What about other affine transformations? Other conformal mappings? edit: by ...