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.

### 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 $\omega > 0$.