Page 240 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 20: Curves

The planes spanned by pairs of vectors in the Frenet frame are given special names: Notes
(1) T and N — the osculating plane;

(2) N and B — the normal plane;
(3) T and B — the rectifying plane.
We see that to second order the curve stays within its osculating plane, where it traces a parabola

2
y = (k/2) s . The projection onto the normal plane is a cusp to third order: x  (3 /2)y  2 /3 . The
projection onto the rectifying plane is to second order a line, whence its name.
Here are a few simple applications of the Frenet frame.

Theorem 2. Let  be a regular curve with k  0. Then  is a straight line.

Proof. Since T'  k  0, it follows that T is constant and  is linear.
Theorem 3. Let  be a regular curve with k > 0, and  = 0. Then  is planar.


Proof. Since B’ = 0, B is constant. Thus the function      (0) . B vanishes identically:
 (0) 0,   T B  0.


It follows that remains in the plane through (0) perpendicular to B.
Theorem 4. Let be a regular curve with k constant and  = 0. Then  is a circle.

Proof. Let b =  + k N. Then
-1
1
'   T  ( kT   B)  0.

k
1

Thus,  is constant, and     k . It follows that  lies in the intersection between a plane and
a sphere, thus  is a circle.
20.3 Plane Curves

20.3.1 Local Theory

2
Let :[a,b]   be a regular plane curve parametrized by arclength, and let k be its curvature.
Note that k is signed, and in fact changes sign (but not magnitude) when the orientation of  is
reversed. The Frenet frame equations are:


e  ke , e   ke 1
2
2
1
2
Proposition 2. Let :[a,b]   be a regular curve with '  1. Then there exists a differentiable

function :[a,b]   such that
e = (cos , sin ). ...(3)
1
Moreover,  is unique up to a constant integer multiple of 2, and in particular (b) – (a) is
independent of the choice of . The derivative of  is the curvature:  = k.

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