Differential geometry notes 3

by exactnature

Let \mathbf{x}(t) be a parametric representation of a curve C. Two points of C, say P and P_1, determine a straight line S. If P_1 tends to P, then S tends to the tangent to C at P.

Let us ask similarly for the limit position of a plane E passing through three points P, P_1, P_2 of C when both P_1 and P_2 tend to P.

Let \mathbf{x}(t), \mathbf{x}(t+h_1) and \mathbf{x}(t+h_2) be the parametric representations of P, P_1, P_2 respectively. The chords PP_1 and PP_2 are given by the vectors \mathbf{a}_i = \mathbf{x}(t+h_i) - \mathbf{x}(t). These two vectors span E. E is thus also spanned by \mathbf{v}^{(i)} = \mathbf{a}_i / h_i and thus by the vectors:

\mathbf{v}^{(i)} and \mathbf{w}= \frac{2(\mathbf{v}^{(2)}-\mathbf{v}^{(1)})}{h_2-h_1}

Using the Taylor expansion:

\mathbf{v}^{(i)} = \mathbf{x^\prime}(t) + \frac{1}{2}h_1 \mathbf{x^{\prime \prime}}(t) + o(h_1)

\mathbf{w} = \mathbf{x^{\prime \prime}}(t) + o(1)

Thus as h_i \rightarrow 0, \mathbf{v}^{(i)} \rightarrow \mathbf{x^\prime}(t) and \mathbf{w} \rightarrow \mathbf{x^{\prime \prime}}(t).

Osculating plane: The plane spanned by \mathbf{x^\prime}(t) and \mathbf{x^{\prime \prime}}(t) is called the osculating plane of the curve at P. For any point \mathbf{z} on the osculating plane: |(\mathbf{z}-\mathbf{x})\mathbf{x^\prime} \mathbf{x^{\prime \prime}}|=0 where:

|\mathbf{a}\mathbf{b}\mathbf{c}| = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix}    a_1 & b_1 & c_1 \\    a_2 & b_2 & c_2 \\    a_3 & b_3 & c_3    \end{vmatrix}

where \cdot denotes the dot product and \times 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 \mathbf{p}(s) = \frac{\mathbf{\dot{t}}(s)}{|\mathbf{\dot{t}}(s)|} is called the unit principal normal vector. This vector lies both in the osculating and the normal planes.

Curvature: the length of the vector \mathbf{\dot{t}}(s) is called the curvature at point \mathbf{x}(s):

\kappa(s) = |\mathbf{\dot{t}}(s)| = \sqrt{\mathbf{\ddot{x}}(s)\cdot \mathbf{\ddot{x}}(s)}                   (\kappa >0)

or in terms of any arbitrary parametric representation:

\kappa(s) = |\mathbf{\dot{x}}\times \mathbf{\ddot{x}}| = |\mathbf{x^\prime} \times \mathbf{x^{\prime \prime}}|\big( \frac{dt}{ds}\big)^3. This is equal to:

\kappa(s) = \frac{\sqrt{ (\mathbf{x^\prime}\cdot \mathbf{x^\prime}) (\mathbf{x^{\prime \prime}} \mathbf{x^{\prime \prime}}) - (\mathbf{x^\prime} \cdot \mathbf{x^{\prime \prime}})^2 }}{(\mathbf{x^\prime}\cdot \mathbf{x^\prime})^{3/2}}

Primes denote derivatives with respect to t, dots denote derivatives with respect to the natural parameter s.

Radius of curvature: The reciprocal of curvature is called the radius of curvature:

\rho(s) = \frac{1}{\kappa(s)}

Center of curvature: The point M on the positive ray of the principal normal at distance \rho(s) from \mathbf{x}(s). The position vector of M is given by \mathbf{z} = \mathbf{x} + \rho \mathbf{p} = \mathbf{x} + \rho^2 \mathbf{\dot{t}}.

Binormal: To every point of the curve, we associate three orthogonal unit vectors:

1) \mathbf{t}(s) = \mathbf{\dot{x}}(s)  (unit tangent vector)

2) \mathbf{p}(s) = \frac{\mathbf{\dot{t}}(s)}{|\mathbf{\dot{t}}(s)|} = \rho(s) \mathbf{\ddot{x}}(s) (unit principal normal vector)

The third vector is the vector product of these two unit vectors:

3) \mathbf{b}(s) = \mathbf{t}(s) \times \mathbf{p}(s)

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 \mathbf{t}, \mathbf{p}, \mathbf{b} is called the moving trihedron of the curve.

  • \mathbf{t} and \mathbf{p} span the osculating plane: (\mathbf{z} - \mathbf{x})\cdot \mathbf{b} = 0.
  • \mathbf{p} and \mathbf{b} span the normal plane: (\mathbf{z} - \mathbf{x})\cdot \mathbf{t} = 0.
  • \mathbf{t} and \mathbf{b} span the rectifying plane: (\mathbf{z} - \mathbf{x})\cdot \mathbf{p} = 0.

Torsion: The vector \mathbf{\dot{b}} is orthogonal to \mathbf{t} and \mathbf{b} (make sure you understand why), consequently it lies in the principal normal. Thus, we may set: \mathbf{\dot{b}}(s)=-\tau(s) \mathbf{p}(s) or \tau(s) = -\mathbf{p}(s) \cdot \mathbf{\dot{b}}(s)

\tau(s) is called the torsion of the curve at point \mathbf{x}(s). In terms of \mathbf{x}(s) and its derivatives, torsion is given by:

\tau = \rho^2 |\mathbf{\dot{x}}\mathbf{\ddot{x}}\mathbf{\dddot{x}}| = \frac{|\mathbf{\dot{x}}\mathbf{\ddot{x}}\mathbf{\dddot{x}}|} {\mathbf{\ddot{x}}\cdot \mathbf{\ddot{x}}}.

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.