parametrizations of curves & curvature

Read the pdf file.

Reparametrization doesn’t change geometric properties at a coordinate point of the image of a curve.

Taylor expansion up to second order:
f (x + ∆x) = f (x) + f ′ (x)∆x + f ′′ (x)∆2 x
It measures the deviation of f (x + ∆x) from f (x) along y axis, which consists of
two parts, tangent part and “parabola approximation” part, as shown in Fig. 2.

Curvature measure the degree of deviation of f (x + ∆x) from f (x) along the
normal direction, which is perpendicular to tangent vector n.


曲率用来衡量曲线与切线、曲面与切面的偏移。
椭圆的曲率
不是简单的 1/轴长
γ(t) = (p cos t, q sin t)

在p轴顶点，t=0, k =p/(q^2)

在q轴顶点，       k =q/(p^2)

椭球面

τ (θ, ϕ) = (a sin θ cos ϕ, b sin θ sin ϕ, c cos θ)

在顶点τ (0, 0)处，做过z轴的切面，得到曲率最大的和最小的两个椭圆。

令 a = 4;

b = 1;

c = 2;

曲率分别为 k1 = 2; k2 = 1/8

在顶点处做横切面，取两个主方面和反z轴构成右手准则坐标系，该椭球面的抛物面近似为

1/2 (k_1 x^2 + k_2 y^2)

曲率的作用：
1. 如上所述，在光滑曲面上一点P，取其两个主方向和准线构成右手准则坐标系，该处的局部结构近似为二次曲面
1/2 (k_1 x^2 + k_2 y^2)

2. 对于平面曲线上一点P，取其切线和准线构成坐标系，该处的局部结构也近似为抛物线，不过注意
只能为 1/2  k  s^2 而不是 1/2  k x^2
见这儿