Curvature#
TODO
Definition: Curvature
\[\kappa(t) = \frac{1}{s'(t)} ||\mathbf{T}'(t)||\]
\(s\) is arclength function
\(\mathbf{T}\) is unit tangent vector
Example: Circle Curvature
\[\frac{1}{r}\]
Theorem: Curvature in \(\mathbb{R}^2\)
\[\kappa(t) = \frac{|x'(t)y''(t)-y'(t)x''(t)|}{(x'(t)^2+y'(t)^2)^{\frac{3}{2}}}\]
Definition: Signed Curvature
\[\tilde{\kappa}(t) = \frac{x'(t)y''(t)-y'(t)x''(t)}{(x'(t)^2+y'(t)^2)^{\frac{3}{2}}}\]
If arclength parametrization, then
\[\tilde{\kappa}(t) = x'(t)y''(t)-y'(t)x''(t)\]
Proof
TODO
Theorem
\[\tilde{\kappa}(t) = \frac{f''(t)}{(1+f'(t)^2)^{\frac{3}{2}}}\]
Proof
TODO