Severely Theoretical

Computational neuroscience and machine learning

Month: March, 2014

Differential geometry notes 3

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.

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Differential geometry notes 2

Definition 9.1: Let \mathbf{x}(t) (a \leq t \leq b) be an allowable parametric representation of an arc C with initial point A and terminal point B. Denote by l(Z) the length of a broken line of chords Z whose end points lie on C and correspond to the values a = t_0 < t_1 < \ldots < t_n = b. Let n \rightarrow \infty and \sigma(Z)= \underset{1 \leq v \leq n}{\max}(t_v - t_{v-1}) \rightarrow 0. If l(Z)\rightarrow s, then C is called rectifiable and s is called the length of C.

Theorem 9.1: Let \mathbf{x}(t) (a \leq t \leq b) be an allowable parametric representation of an arc C of a curve of class r (r times differentiable). Then C has length:

s = \int_{a}^{b} \sqrt{(\frac{dx_1}{dt})^2 + (\frac{dx_2}{dt})^2 + (\frac{dx_3}{dt})^2} dt = \int_{a}^{b} \sqrt{\mathbf{x^{\prime}} \cdot \mathbf{x^{\prime}} } dt

and s is independent of the choice of the allowable parametric representation.

Arc length: The function s(t) = \int_{t_0}^{t} \sqrt{\mathbf{x^{\prime}} \cdot \mathbf{x^{\prime}} } dt is called the arc length of C.

The arc length may be used as the parameter in a parametric representation of a curve: \mathbf{x}(s) is an allowable parametric representation. s is called the natural parameter. The choice of \mathbf{x}(s) as the parametric representation simplifies many investigations.

Unit tangent vector: The vector \mathbf{t}(s) = \underset{h\rightarrow 0}{\lim} \frac{\mathbf{x}(s+h)-\mathbf{x}(s)}{h} = \frac{d\mathbf{x}}{ds} = \mathbf{\dot{x}}(s) is called the unit tangent vector to the curve C at point \mathbf{x}(s).

The straight line passing through a point P of C in the direction of the corresponding unit tangent vector is called the tangent to the curve at P. The tangent can be represented in the form \mathbf{y}(u) = \mathbf{x} + u\mathbf{t}. Thus, \mathbf{y}(0) is the point of contact between C and the tangent.

The totality of all vectors bound at a point P of C which are orthogonal to the corresponding unit tangent vector lie in a plane. This plane is called the normal plane to C at P.

Differential geometry notes 1

These are some notes on differential geometry I took while studying Erwin Kreyszig’s Differential Geometry. I will keep posting these notes as I read the book.

Consider the vector function \mathbf{x}= [x_1(t), x_2(t),x_3(t)] for I: a \leq t \leq b. This defines a point set M in \mathbb{R}^3. t is called the parameter.  We make the following assumptions about this function:

  • x_i(t) for i=1,2,3 are r times continuously differentiable in I.
  • For every t \in I, there is an i such that \frac{dx_i}{dt} is different from zero.

A representation of \mathbf{x} satisfying these assumptions is called an allowable parametric representation. We may obtain different parametrizations using transformations such as:

t = t(t^{\ast})

The point set M is invariant under such transformations. We make the following assumptions about these transformations:

  • t is defined in the interval I^{\ast}: a^{\ast} \leq t^{\ast} \leq b^{\ast} and the range of values corresponding to the interval I^{\ast} lies in I.
  • t is r times continuously differentiable in I^{\ast}.
  • The derivative \frac{dt}{dt^{\ast}} is different from zero everywhere in I^{\ast}.

A transformation satisfying these assumptions is called an allowable parametric transformation. Allowable parametric representations can be grouped into equivalence classes by means of allowable parametric transformations: two allowable representations are equivalent if there is an allowable transformation transforming one into the other.

Definition 6.1: A point set in \mathbb{R}^3 which can be represented by the allowable representations of an equivalence class is called an arc of a curve.

Multiple point: If a certain point of M corresponds to multiple values of t, it is called a multiple point of the arc. An arc having no multiple points is called simple.

Definition 6.2: A point set is called a curve if it can be represented by an equivalence class of allowable representations whose interval I is not assumed to be closed or bounded, but which are such that one always obtains an arc of a curve if the values of the parameter t are restricted to any closed and bounded subinterval of I.

A curve is said to be closed if it possesses at least one allowable representation whose vector function is periodic: \mathbf{x}(t + \omega) = \mathbf{x}(t) for some \omega > 0.