# Isomorphism of orientation preserving rigid motions

1. Feb 18, 2010

### kp266

Find an isomorphism from the subgroup of GL2(C) of the form
$$\begin{pmatrix} a & b\\ 0 & 1 \end{pmatrix} ,\left | a \right |=1$$

to the group of orientation preserving rigid motions.

*The problem is from Artin's Algebra Chapter5

2. Feb 18, 2010

### torquil

How about using the argument (angle) of a to represent rotation of the rigid object around the z-axis, and use the real and imaginary parts of b to represent an arbitrary translation along the (x,y) plane? These are orientation preserving motions of rigid objects in R^3.

Torquil

3. Feb 18, 2010

### Redbelly98

Staff Emeritus
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Last edited by a moderator: Apr 24, 2017
4. Feb 19, 2010

### wofsy

$$\begin{pmatrix} a & b\\ 0 & 1 \end{pmatrix} ,\left | a \right |=1$$

goes to az + b.