Differentiable Curves

Curves in

Differentiable Curve

A (parametrized) differentiable curve is an infinitely differentiable map of an interval of the real line into . Say, . The variable is called the parameter of the curve. is called the tangent vector of the curve at . And the image is called the trace of the curve.

e.g. The map is not a parametrized differentiable curve since it is not differentiable at .

Singular Point and Regular Curve

Let be a differentiable curve. A point is called a singular point of order of if . Specifically, a point is called a singular point (of order ) if . A curve is called regular if it has no order singular points.

Arc Length

The arc length of a regular curve from is defined as where is the Euclidean norm on . And write the whole arc length as .

Proposition

The parameter is itself the arc length of the curve if and only if for all .

Proof If for all , then . Conversely, if arc length is , then for all .

Remark

In this case, we write the tangent vector as .

Curvature

Curvature

Let be a regular curve parametrized by arc length. The curvature of at a point is defined as And the inverse of curvature is called the radius of curvature.

e.g. If is a straight line, then and so everywhere. Conversely, if everywhere, then is a straight line. A circle of radius has curvature .

Normal Vector

Let be a regular curve parametrized by arc length. The normal vector of at a point with nonzero curvature is defined as a unit vector:

Proposition

The normal vector is orthogonal to the tangent vector .

Proof Since is parametrized by arc length, we have . So . Then .

Lemma

The derivative of normal vector is orthogonal to .

Proof We have Moreover, compute the derivative: Therefore, substituting back, we have $$< n(s), n’(s)> = \frac{1}{|\gamma”(s)|^{3}} \left(< \gamma”(s),\gamma'''(s) >- < \gamma'''(s),\gamma”(s) > \right) = 0$$$\square$

Osculating Plane, Binormal Vector

Let be a regular curve parametrized by arc length. At with nonzero curvature, the plane spanned by the unit tangent and normal vectors, and , is called the osculating plane at . The binormal vector of at is defined as the unit vector orthogonal to the osculating plane by cross product:

Proposition

The derivative of binormal vector is parallel to .

Proof We may write as is parallel to . So is orthogonal to . Moreover, we have Hence is orthogonal to . That is is parallel to .

Torsion

Torsion

Let be a regular curve parametrized by arc length. The torsion of at a point with nonzero curvature is defined as

Geometric Interpretation of Torsion

Intuitively, the torsion measures how the curve twists around the binormal vector , and measures how rapidly the curve pulls away from the osculating plane. Specifically, if is a plane curve, that is the trace of is contained in a plane, then the plane of the curve agrees with the osculating plane everywhere and the torsion is zero. Converse is not true in general.

Rectifying Plane, Normal Plane

Given a regular curve parametrized by arc length, the rectifying plane at a point is spanned by the tangent vector and the binormal vector . The normal plane at is spanned by the normal vector and the binormal vector . The lines parallel to and pass through are called principal normal. The lines parallel to and pass through are called binormal.

Frenet Formulas

A regular curve satisfies the following Frenet formulas:

Rigid Transformation

A rigid transformation of a subset is a transformation that preserves the Euclidean distance between any two points in . Indeed, it is a composition of a translation, rotation, and reflection.

Proposition

Rigid transformations of a curve is a composite of an orthogonal transformation in with positive determinant, and a translation. i.e. is a rigid transformation of if and only if there exists an orthogonal linear map with and a vector such that for all .

Fundamental Theorem of Curves

Given differentiable functions and , with . Then there exists a unique regular curve , up to a rigid transformation, such that is the arc length of and the curvature and torsion of are and respectively.

Global Properties of Plane Curves

Isoperimetric Inequality

Let be a regular simple closed curve with length , and area enclosed by . Then the isoperimetric inequality holds: with equality if and only if is a circle.