0
votes
0answers
26 views

Reflection about y=-5

What would be the reflection matrix when reflected about line y=-5. E.g. line segement with endpoints (-1,3) & (6,-2) will become a line segment with endpoints (-1,-13) & (6,-8). I have tried ...
0
votes
1answer
21 views

Transformation of a surface normal

I'm taking a university level course in discrete geometrics and graphical programming, and I'm having trouble understanding this exercise. Let p be a point in R^3, n a surface normal, and M a ...
0
votes
0answers
10 views

Affine Transformation and Continuous Deformation

How do these two concepts relate? Thus far I have a (what I think is a) good intuitive idea of a continuous deformation- the visual basically looks like the boundary being stretched so that it never ...
0
votes
1answer
30 views

how to find triangular point from a side

i have two triangles. Say a , b , c and p, q, r and the projection of the abc to pqr a - > p b - > q c - > r here known point values are a b c p q and r unknown. $\overline{PR}=\overline{AC}$ ...
2
votes
2answers
37 views

Do rotations of one point around all arbitrary axes form a sphere?

Correct me if I am wrong but assume I have a point in 3D which I would like to rotate around all arbitrary axes fixed at common origin. Then this is true that all orbits circled by rotated point will ...
0
votes
1answer
22 views

Translation of basis for a vector space on the specified distance

In the Euclidean space $XYZ$ is a basis $X_1Y_1Z_1$ defined that is specified by the vectors $\overrightarrow {O_1X_1}$, $\overrightarrow {O_1Y_1}$ and $\overrightarrow {O_1Z_1}$. How to calculate ...
1
vote
0answers
41 views

Decompose distortion affected homography matrix

I am working on a system that finds homography between images taken by moving (shaking) camera with rolling shutter and map. The map is orthogonal image of flat 2D plane and the camera images are ...
1
vote
1answer
24 views

Transformation shifts parallelogram to trapezoid - fairly simple

We are given the region $D= {\{(x,y) | 1 \leq x-y \leq 2, x \leq 0, y\leq 0\} \subseteq \mathbb R^2}$ I drew this region on a piece of paper, it resembles an infinite parallelogram on the third ...
1
vote
1answer
115 views

Analytic geometry - rotation + translation

In $K=O\vec{e_1}\vec{e_2}\vec{e_3}$ I have to find the analytical representation of the screw motion( rotation + translation) $\psi$ with a rotation axis $g$ given by the points $A(5,-4,3)$ and ...
0
votes
0answers
19 views

Find the reference point required to transform scale two elements uniformly

This is actually a programming issue I am having but the answer is rooted in matrix mathematics so this seems like the best place to ask it. I am no mathematician so I apologise if some of my concepts ...
1
vote
1answer
66 views

Calculating the adjustment translation to be applied after rotating and scaling so that operations pivot about a given point.

I have a matrix for transforming an image into a target frame. The matrix is a function of a scale, $s$ rotation angle, $\theta$, and a translation that is applied after rotating, $tx, ty$. The ...
0
votes
0answers
15 views

Generate rotations about X & Y axes between certain 3D vectors

Given a semi-arbitrary 3D vector (the z will always be positive for my purposes), how could I find rotation about the X and Y axis? Alternatively, how might I simplify an XYZ rotation to the X and Y ...
0
votes
1answer
37 views

Transforming a curve on an arc to a line

I have a function, actually a point cloud, (similar to a sine wave) on an arc with a known radius of curvature. I need to remove the curvature to regenerate the original function (or point cloud). ...
0
votes
0answers
28 views

Variations of transformation of inversion?

Is there a transformation analogous to inversion, that is based on something other than circle (or sphere in higher dimensions), and has some interesting properties or applications? The motivation ...
0
votes
2answers
37 views

How does the transformation on a point affect the normal at that point?

Say I have a point in 3D with coordinates $\begin{bmatrix} p_1 \\ p_2 \\p_3 \end{bmatrix}$ and the normal on the point with coordinates $\begin{bmatrix} n_1 \\ n_2 \\n_3 \end{bmatrix}$. Now I apply ...
0
votes
1answer
21 views

Non constant function of two points invariant under Affine transformation proof

Here is the question; Prove that there does not exist any nonconstant function of pairs of distinct points $P,Q\in\mathbb{R}^2$ or of triples of distinct non collinear points $P,Q,R\in\mathbb{R}^2$ ...
0
votes
1answer
207 views

Transformation of ellipsoid to sphere

So I need to find an volume-preservating mapping from an ellipsoid to a ball (solid sphere). (Specifically: x^2/9 + y^2 + z^2 <= 3, but I'd rather understand the general case than just get how to ...
1
vote
2answers
138 views

Transformation matrix from quadrilateral to rectangle

There exists a rectangle somewhere in space with some orientation. A camera from the coordinate center point is looking along the z axis and is seeing the rectangle as a quadrilateral (due to ...
5
votes
3answers
69 views

I'm looking for the name of a transform that does the following (example images included)

I'm in the usual situation that if I would know what the name of the thing was, then I could find the answer. Since I dont know the name, here is what I'm looking for: Suppose I have the following ...
0
votes
1answer
32 views

How can I calculate the origin of a scale transformation, given the starting and ending coords and dimensions?

Background: I have two sets of coordinates/dimensions. One for the red rectangle and one for the blue rectangle, as shown below. The blue rectangle is quite simply the red rectangle transformed by ...
0
votes
1answer
20 views

general rotations

Let $R$ be the rotation about the point $(1,0)$ by an angle of $45$ degrees. By using matrix methods: Find the image of the line $2x-3y+1=0$ under $R$ I would really appreciate it if someone ...
0
votes
0answers
29 views

Find angle-preserving transformation matrix given 2 points

I asked a similar question yesterday about finding an affine transform matrix given the same 2 points in both coordinate systems. I was told that there was only a unique solution, if the scaling was ...
0
votes
1answer
134 views

Find 2D affine transform matrix given a pair of points

I have the coordinates of two points in an initial 2d coordinate system and the corresponding coordinates in a target system. Is is possible to determine the affine transform matrix from these values? ...
1
vote
2answers
66 views

Does congruence guarantee length conversion?

Suppose that a linear transformation $M:R^2 \rightarrow R^2$ maps a triangle $ABC$ to a congruent triangle $A'B'C'$ ($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$ are not colinear, and $A,B,C\neq O$) Is it ...
0
votes
1answer
53 views

Describing Cartesian transformations in Cylindrical Polar Coordinates

I have a question about converting functions defined in Cartesian coordinates to a cylindrical polar system. The particular coordinate transformation that I'm reading about is: \begin{equation} ...
1
vote
1answer
81 views

Rotation formalisms in three dimensions

I'm little bit confused. The Rotations are described by various means Direction Cosines Matrix (DCM); Euler Angles; Euler Axis/Angle; Quaternion. What is the difference between them. How I can ...
0
votes
1answer
43 views

What is/are the algebraic equation(s) for transforming a unit square into a specific parallelogram?

Goal: To transform a unit square into a parallelogram in which (a) the diagonals are parallel to the unit square's diagonals, (b) the longest diagonal is equal in length to either of the unit square's ...
2
votes
1answer
46 views

Translating and scaling a line based on two grabbed points

Say there is a line segment going from 0 to 10, now imagine that point 7 and 8 are 'grabbed' and translated to respectively 6 and 11. The effect would this would be that the line segment get's scaled ...
0
votes
0answers
20 views

How to find the values of transformstion and its center/axis? [duplicate]

I have system of two planes. Each plane is defined by three points, so I have their equations. One of these plane is stable, I can't perform any transformation. The second one is modifiable - rotation ...
1
vote
2answers
203 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
0
votes
3answers
164 views

If two sides of a triangle are equal, and the angle between them is $60^\circ$, prove the third side is equal to the first two sides.

In other words, given points $A$ and $X$. Rotate $X$ $\,-60^\circ$ around $A$ to get point $X'$. How would you prove $XX' = AX = AX'$? I know this is true.
1
vote
0answers
36 views

Hyperbolic analogue of the Euclidian Reflection across a straight line

I have already solved that $\Phi(z)=z$ in the geodesic $g$, but I am stuck on this part of the problem: Let $g$ be a complete geodesic of $H^2$, which is a semicircle of radius $R$ centered at ...
1
vote
1answer
22 views

Fractional linear maps and geodesics.

Consider the fractional linear map $\Phi(z)=(az+b/cz+d)$ where $a$, $b$, $c$ $d$ are real numbers with $ad-bc=1$. Suppose in addition that $|a+d| > 2$ a) show that if $c \ne 0$, there exists ...
3
votes
1answer
64 views

Möbius Transformation of Triangles

I understand that Möbius transformations are angle preserving transformations. Knowing this, my professor asked us to think about how the image of equilateral triangle is not an equilateral triangle ...
2
votes
2answers
327 views

Are there non-affine matrices?

Matrices are useful for proving statements like The ratio between the areas of a parallelogram and the quadrilateral formed by joining their midpoints is $2$. The ratio between the volumes ...
1
vote
1answer
84 views

How to apply perspective transform to Bezier curve?

I found that both Bezier curves and B-splines are described with a formula $p(t)=\sum\limits_{i=0}^d B^i_m p_i$ but in the case of B-splines $B^i_m$ are B-spline blending functions, while for Bezier ...
0
votes
1answer
82 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 ...
-1
votes
1answer
128 views

Transformations of the complex plane

I was reading my text and just had some questions about transformations: (1) Are all line preserving transformations linear transformations? Why? I want to say yes... but I feel like the answer is ...
0
votes
1answer
53 views

3D coordinate Transformation

I am currently trying to align two bodies which do not have similar sizes and shapes. But both of these two bodies share some keynodes(similar nodal position with 0.1% error difference). How can I ...
2
votes
0answers
43 views

Bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line

Prove, that a bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line. There exists an elementary proof? I know this question can be found here ...
0
votes
1answer
157 views

Inversion of Circles

I'm studying for my exam and one of the questions I am stuck on is: Show that under inversion in the unit circle a circle with centre C and radius r inverts into a circle centre ...
1
vote
1answer
156 views

Geometry of Complex Numbers

Write down in the form ${Z}\rightarrow{AZ+B}$ the following transformations of the complex plane: (a) translation in the direction $(2,-3)$ (b) rotation about (0,1) through $\pi/4$ I know from my ...
2
votes
2answers
112 views

Intuition for Geometric Transformations

I've been making a lot of effort over the past few hours to gain some intuition into the art of geometric transformation but to little avail. I would really like to be able to look at a transformation ...
0
votes
2answers
95 views

Transfomation of one coordinate system to a another

I have a molecule with one coordinate system ( denote as x,y,z ) where the origin is center of mass of the molecule. I have to define another coordinate system (p,q,r) for a local motion. (shown in ...
1
vote
1answer
141 views

Using absolute coordinates in 2D affine transformation matrix

In my 2D animation program I have a sprite which transformation is described by a 2D affine transformation matrix (SVGMatrix): $$ \begin{bmatrix} a & c & e \\ b & ...
0
votes
0answers
126 views

formula for transforming the interior point of a 2d bounding box when the box is stretched by moving a single corner of the box

I would like to find out a formula (mathematical equation(s)) for adjusting the position of any point enclosed in a rectangle when its corner point is repositioned such that (1) all other corner ...
1
vote
0answers
70 views

Determining pose of an object in 3d space

Given a 3D model of an object centred at the origin, if I place a camera at position (x,y,z) and make it face the origin, from the image rendered the object appears ...
0
votes
2answers
293 views

How to enlarge a circle?

if you are given a circle with equation $(x-a)^2 + (y-b^2) = r^2$ and it is enlarged by a factor of $3$ what would the new equation be? Would you put $2x$ an $2y$ in the place of $y$?
8
votes
1answer
196 views

Mirror anamorphosis for Escher's Circle Limit engravings?

You are probably familiar with "mirror anamorphosis," the rendering in a painting of a distorted figure that can be undistorted by viewing in an appropriately tilted or curved mirror. The skull in the ...
0
votes
1answer
222 views

Transform between cartesian coordinate system and abstract coordinate system

I am trying to find a transformation that takes me between Cartesian coordinates and a pseudo-coordinate-system I have developed which is described as follows: Please first see the diagram below. ...