Differential geometry notes 3
by Emin Orhan
Let be a parametric representation of a curve . Two points of , say and , determine a straight line . If tends to , then tends to the tangent to at .
Let us ask similarly for the limit position of a plane passing through three points , , of when both and tend to .
Let , and be the parametric representations of , , respectively. The chords and are given by the vectors . These two vectors span . is thus also spanned by and thus by the vectors:
Using the Taylor expansion:
Thus as , and .
Osculating plane: The plane spanned by and is called the osculating plane of the curve at . For any point on the osculating plane: where:
Principal normal: The intersection of the osculating plane with the corresponding normal plane is called the principal normal.
Unit principal normal vector: The unit vector is called the unit principal normal vector. This vector lies both in the osculating and the normal planes.
Curvature: the length of the vector is called the curvature at point :
or in terms of any arbitrary parametric representation:
. This is equal to:
Primes denote derivatives with respect to , dots denote derivatives with respect to the natural parameter .
Radius of curvature: The reciprocal of curvature is called the radius of curvature:
Center of curvature: The point on the positive ray of the principal normal at distance from . The position vector of is given by .
Binormal: To every point of the curve, we associate three orthogonal unit vectors:
1) (unit tangent vector)
2) (unit principal normal vector)
The third vector is the vector product of these two unit vectors:
This is called the unit binormal vector (make sure you understand why this vector is a unit vector and it is orthogonal to the other two vectors).
Moving trihedron: The triple is called the moving trihedron of the curve.
- and span the osculating plane: .
- and span the normal plane: .
- and span the rectifying plane: .
Torsion: The vector is orthogonal to and (make sure you understand why), consequently it lies in the principal normal. Thus, we may set: or
is called the torsion of the curve at point . In terms of and its derivatives, torsion is given by:
The torsion, roughly speaking, measures the magnitude and direction of deviation of a curve from the osculating plane in the neighborhood of a point of the curve.