Differential Geometry (2026 Spring)
《微分几何》彭家贵、陈卿
Curves.
Problem 13(1)
弧长参数：三角换元（$x’=\cos\theta(s),y’=\sin\theta(s).$）
Problem 15: curves on sphere
$\gamma(s)\subset S^2$ satisfies $<t,\gamma(s)-P_0>,$ thus $\gamma(s)-P_0 \in span{n,b}.$
Problem 16: approximation of curves at some point
$$r(s) = r(0) + (s-\frac{\kappa^2s^3}{6})t + (\frac{\kappa s^2}{2}+\frac{\kappa’s^3}{6})n +{\frac{\kappa\tau s^3}{6}}b + \mathcal{o}(s^3).$$
Setup: Frenet Framework ${r;t,n,b}$, $P_0 = r(0), P = r(s). d(P,L) = <P-P_0,n>. d(P_0,P)^2 = x^2+y^2+z^2.$
LHS = $\lim_{s\to 0} \frac{2*(\frac{\kappa}{2})s^2 + \mathcal{O}(s^3)}{s^2 + \mathcal{O}(s^3)} = \kappa(P_0). $