Combining vectors with complex numbers: curves of constant curvature (1)

向量结合复数：常曲率曲线(1) 苏剑林 · 2011-06-19 · Mathematics vector curvature complex number

In an earlier article in the vector series], we cleverly combined physics with vectors to derive the radius of curvature of a curve (both in the plane and in space) as

$$R=\frac{v^2}{a_c}=\frac{|\dot{\vec{r}}|^3}{|\dot{\vec{r}}\times \ddot{\vec{r}}|}\tag{1}$$

The curvature is then the derivative of the radius of curvature: $\rho=\frac{1}{R}$. Let's turn the question around: is a circle the only plane curve with constant curvature?

The answer seems obvious, but we still need to prove it.

Since we're only considering the planar case, let's set $\dot{\vec{r}}=(v cos\theta,v sin\theta)=z=ve^{i\theta}$, and substituting into (1) we get

$\frac{\dot{\theta}}{v}=\rho$————(2) more

Note that here we've used the derivative notation $\dot{\theta}$, but haven't yet specified which variable we're differentiating with respect to. We notice that if $\dot{\theta}=\frac{d\theta}{dv}$, then (2) becomes easy to solve. So we simply stipulate that a dot above a function denotes differentiation with respect to the variable v. This is a kind of "decide after the fact" approach: since the variable of differentiation here is arbitrary, we're free to choose it in whatever way makes things easiest to solve.

With this, the general solution of (2) is: $$\theta=1/2 \rho v^2+C_1\tag{2}$$

Going back to the x, y variables, we have: $$\frac{dz}{dv}=ve^{i\theta}=ve^{i(1/2 \rho v^2+C_1)}\tag{3}$$

so that

$$\begin{aligned}z=\int ve^{i\theta}=ve^{i(1/2 \rho v^2+C_1)}dv \\ =\frac{1}{\rho} e^{i(1/2 \rho v^2+C_1)}d(1/2 \rho v^2+C_1)=\frac{1}{\rho} e^{i(1/2 \rho v^2+C_1)}+C_2\end{aligned}\tag{4}$$

Choosing an appropriate translation so that $C_2=0$, we get

$$z=\frac{1}{\rho} e^{i(1/2 \rho v^2+C_1)}\tag{5}$$

It's easy to check that $|z|=\frac{1}{\rho}$. Hence, this is indeed a circle.

Note that here the "complex number" plays a purely "formal" role — it's simply a tool for combining operations on orthogonal coordinates into a single unified computation, rather than being a number in the ordinary sense. In the next article we'll explore the analogous problem for space curves and surfaces.