# Tagged Questions

36 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 ...
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 ...
26 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 ...
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 ...
107 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 ...
18 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 ...
54 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 ...
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 ...
35 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). ...
24 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 ...
35 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 ...
19 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 $P,Q,R\in\mathbb{R}^2$ ...
179 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 ...
126 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 ...
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 ...
30 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 ...
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 ...
27 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 ...
124 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? ...
65 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 ...
52 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: ...
78 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 ...
40 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 ...
39 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 ...
19 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 ...
198 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 ...
144 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.
34 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 ...
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 ...
63 views

### Möbius Transformation of Triangles

I understand that MÃ¶bius 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 ...
304 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 ...
78 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 ...
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 ...
124 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 ...
52 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 ...
42 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 ...
154 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 ...
149 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 ...
108 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 ...
93 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 ...