### Differential geometry notes 3

#### by exactnature

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:

and

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:

where denotes the dot product and denotes the vector product.

* 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:

3)

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.