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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:

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.

Radon transform