What does the "interior angle of the loop" mean for a curve that self-intersects?

Question

How does one measure that angle? I've looked but couldn't find a definition for it anywhere.

Answer 1

The double point $p=X(s_1)=X(s_2)$ divides the curve $X$ into two loops, one of which we call $\Gamma.$ This $\Gamma$ is then a closed curve that is smooth everywhere except at p, where it necessarily must have a sharp corner (since the self-intersection is transversal). The angle $\alpha$ is simply the interior angle at this corner, as illustrated here:

To be more rigorous, $\alpha$ can be defined as the angle between the tangent vectors $X'(s_1)$ and $-X'(s_2).$ Note that if the loop $\Gamma$ was smooth then we would have $\alpha = \pi;$ so the angle defect (or equivalently, the point mass of curvature) at $p$ is $k_p = \pi - \alpha.$ Thus the claim $$\int_\Gamma k = \pi + \alpha$$ can be rewritten as $$k_p + \int_\Gamma k = 2\pi,$$ which is the Theorem of Turning Tangents; i.e. the familiar fact that the total curvature of a simple closed curve is $2\pi.$