# Questions tagged [rigid-transformation]

Rigid transformations are mappings from one reference frame to another reference frame in the Euclidean space $\mathbb R^n$. They comprise translation, rotation (and sometimes reflection). More formally, rigid transformation are elements of the matrix Lie group SE($n$), the specially Euclidean group. If reflection is included, these proper rigid transformations are elements of the matrix Lie group E($n$), the Euclidean group.

126 questions
33 views

20 views

### Finite subgroups of motoins always fix some point on the plane.

Let $G$ be a finite subgroup of the group of motoins $M$ on the plane. Then there exists a point on the plane which is left fixed by every element of $G$. The proof of this is sketched by our ...
39 views

### motion of a rigid cube

A rigid cube is in motion. At the time depicted in the figure the face $ABCD$ is vertical, the velocity of vertex $A$ is vertical down with value $v$, the velocity of vertex $C$ is vertical up with ...
50 views

### why preserving norm is equivalent to preserving inner product in rigid body transformation

Define rigid body motion as following transformation $g$ $$g:\mathbb R^3 \to \mathbb R^3$$ such that $$|g(v)|=|v|,\forall v \in \mathbb R^3$$ $$g(u) \times g(v) = g(u\times v)$$ according to ...
19 views

### A slight confusion about a particular step in Euclidean Transformations (Computer Vision)

On page 392 of Computer vision by Simon Prince (http://web4.cs.ucl.ac.uk/staff/s.prince/book/book.pdf), equation 15.2 has the following expression for the Euclidean transformation in homogeneous ...
12 views

419 views

### Spline interpolation in SE(3)

Given: a sequence ${A_i}_{i=1}^n$ of elements of the special Euclidean group, $SE(3)$; a sequence of times ${t_i}_{i=1}^n$, where $t_1 \leq t_2 \leq ... \leq t_n$; instantaneous body frame velocity ...
94 views

### Find a rotary reflection in 3D space between two given tetrahedra

My problem is the following: Given two congruent tetrahedra (same side lengths but not equally oriented), I have to find the symmetry plane, the axis and the angle of a rotary reflection that goes ...
47 views

### Convention when screw has negative displacement and angle

Any rigid body transformation can be expressed as a screw rotation about and translation along an axis. The screw is usually defined by a line (a unit direction and point on the line), a displacement, ...
700 views

### Prove that every rotation is equivalent to two successive reflections (in 3D)

Prove that a rotation about any axis by a finite angle is equivalent to successive reflections in two different planes. Here's what I tried: I assumed two reflection planes (passing through the ...
99 views

### What transformation maps $y=\frac{1}{4}x-2$ to $y=-3x+6$?

What transformation maps $y=\frac{1}{4}x-2$ to $y=-3x+6$? I have tried many things, rotating around the origin, reflecting about common lines, and nothing seems to work. Any help is appreciated. ...
Suppose I have a functional $$E=\int F(y_{1,1},..y_{1,n},y_{2,1}\ldots,y_{n,n})d\boldsymbol{x}\,,$$ where \$\boldsymbol{y}:\mathbb{R}^{n}\to\mathbb{R}^{n},\,\boldsymbol{y}(\boldsymbol{x})=\left(y_{...