All notes
Geometry

### SO(3)

The 3D rotation group SO(3), where SO stands for special orthogonal, consists of all 3D rotations about the origin in $R^3$ under the operation of composition. Why is it about the origin? Because the origin in euclidean space establishes a one-to-one correspondence between points and vectors.

The set of all rotations is a group because:

• Composing of rotations is closed.
• Every rotation has a unique inverse.
• The identity map is a rotation.
• Associative property:
In addition, the rotation group has a natural manifold structure for which the group operations are smooth, so it is in fact a Lie group.

A distance-preserving transformation which reverses direction is called improper rotation.

### The Cauchy Crofton formula

The formula is a classic result of integral geometry relating the length of a curve to the measure of all intersecting lines.

What is the measure of all lines?
Writing any line $l$ in $R^2$ in a polar coordinate $\{(p, \theta)|0\le\theta\le 2\pi, p\ge 0\}$, where $p$ is non-negative distance from origin to the line, and $\theta$ is the angle between positive x-axis and the other line perpendicular to $l$ and passing through origin, ranging from $0$ to $2\pi$.
Then the measure of all lines is defined as:
$$m = \int \int dp d\theta$$

Now the Cauchy-Crofton theorem becomes: $L(c)=1/2 \int \int n(p,\theta) dp d\theta$, where $n(p, \theta)$ is the number of intersections between the line given by $(p, \theta)$ and the curve $c$.

How to prove it? Start from a straight line segment. Without losing its generality, we assume the line segment $c$, with length $l$, is located at x-axis and its middle point is at origin. We fix the $\theta$ and find all the $p$ in lines $(p, \theta)$ which intersect with $c$. It is easily to get that $0\le p \le 1/2l|\cos(\theta)|$. Therefore, the measure $$m = \int \int dp d\theta = \int_0^{2\pi} (\int_0^{1/2l|\cos{\theta}|} dp) d\theta = \int_0^{2\pi} 1/2 |\cos{\theta}|d\theta = 2l.$$ As for any curve other than straight line, it is always approximated by several line segments, and the lengths of these line segments are in turn calculated by Crofton formula.

### Rectifiable curve

In short, a rectifiable curve is a curve of finite length.