Computational neuroscience and machine learning

Month: April, 2014

Differential geometry notes 6

Natural equations of a curve: Is it possible to characterize a curve in a manner independent of the choice of coordinates, except for the position of the curve in space, that is, to within congruent transformations? For example, we have seen that $s$, $\kappa$ and $\tau$ are quantities of this sort (i.e. independent of the choice of coordinates). The following theorem, called the fundamental theorem of the theory of curves, answers this question.

Theorem 20.1: Let $\kappa(s)$ and $\tau(s)$ be continuous functions of a real variable $s$ defined in an interval $I: 0\leq s \leq a$. Then there exists one and only one arc $\mathbf{x}(s)$ of a curve, determined up to a direct congruent transformation, with arc length $s$ whose curvature and torsion are given by the functions $\kappa(s)$ and $\tau(s)$.

Involutes and evolutes: Tangents to a curve $\mathbf{x}(s)$ generate a surface called the tangent surface. The tangent surface can be represented as:

$\mathbf{y}(s,u) = \mathbf{x}(s) + u \mathbf{t}(s)$

Involutes of a curve are curves on the tangent surface that are orthogonal to the generating tangents. Thus the involutes can be represented as:

$\mathbf{z}(s) = \mathbf{x}(s) + u(s)\mathbf{t}(s)$

where $u(s)$ is chosen such that the orthogonality condition is satisfied, i.e. $\mathbf{\dot{z}} \cdot \mathbf{t} = 0$. Because:

$\dot{\mathbf{z}} = \mathbf{t} + \frac{d}{ds}(u \mathbf{t})$ $= \mathbf{t} + u \kappa \mathbf{p} + \dot{u} \mathbf{t}$

it follows that $1 + \dot{u}=0$ or $u(s) = c-s$ where $c$ is constant. Thus, the involutes can be expressed as:

$\mathbf{z}(s) = \mathbf{x}(s) + (c-s) \mathbf{t}(s)$

If $C$ is an involute of $C^*$, then $C^*$ is called an evolute of $C$. It can be shown that the evolutes of a curve $\mathbf{x}(s)$ can be represented in the form:

$\mathbf{y}(s) = \mathbf{x}(s) + \rho(s) \big[ \mathbf{p}(s) + \mathbf{b}(s) \cot \alpha(s) \big]$

where $\alpha(s) = \int_0^s \tau(\sigma) d\sigma + k^*$ and $k^*$ is a constant.

Differential geometry notes 5

Definition 19.1: Let a curve $C$ be represented by $\mathbf{x}(s)=\big(\alpha_1(s),\alpha_2(s),\alpha_3(s)\big)$ and $C^*$ be represented by $\mathbf{x}(s^*)=\big(\beta_1(s^*),\beta_2(s^*),\beta_3(s^*)\big)$ where $s$ and $s^*$ are the arc length parameters of the respective curves. $C$ is said to have contact of order $m$ with $C^*$ at a point $P_0$ if at point $P_0$:

$\alpha_i(s) = \beta_i(s^*)$    and    $\frac{d^{\mu} \alpha_i}{ds^{\mu}} = \frac{d^{\mu} \beta_i}{ds^{*\mu}}$                 ($\mu = 1,\ldots, m$ and $i = 1,2,3$)

and if also derivatives of order $m+1$ at $P_0$ exist and $\frac{d^{m+1} \alpha_i}{ds^{m+1}} \neq \frac{d^{m+1} \beta_i}{ds^{*m+1}}$.

Definition 19.2: A curve $C$ has contact of order $m$ with a surface $S$ at a point $P_0$ if there exists at least one curve $C^*$ on $S$ which has a contact of order $m$ with $C$ at $P_0$ and there does not exist a curve on $S$ which has a contact of order greater than $m$ with $C$ at $P_0$.

In this definition, a surface can be informally thought of as a point set satisfying $G(x_1,x_2,x_3)=0$.

Theorem 19.1: At any of its points, a curve has a contact of second order with its corresponding osculating plane.

Lemma 19.2: Let $C$ be a curve with the allowable representation $\mathbf{x}(s)=\big( \alpha_1(s),\alpha_2(s), \alpha_3(s) \big)$ of class $r\geq m$ with arc length parameter $s$. Let $C$ have a point $P_0: s=s_0$ in common with a surface $S$ which has a representation $G(x_1,x_2,x_3)=0$ of class $r \geq m$. Then, $C$ has a contact of order $m$ with $S$ at point $P_0$ iff the function:

$p(s) = G(\alpha_1(s),\alpha_2(s),\alpha_3(s))$

and its derivatives with respect to $s$ up to and including the $m$-th derivative vanish at $P_0$:

$p(s_0) = 0$,            $\frac{d^\mu p}{ds^\mu}|_{s=s_0} = 0$     ($\mu = 1, 2, \ldots, m$),          $\frac{d^{m+1} p}{ds^{m+1}}|_{s=s_0} \neq 0$.

Theorem 19.3: Let $C$ be a curve of class $r \geq m+1$ which has contact of order $m$ with a surface $S$ of class $r$ at point $P_0$. If $m$ is even, $C$ pierces $S$ at $P_0$. If $m$ is odd, then in a sufficiently small neighborhood of $P_0$, $C$ lies on one side of $S$.

Contact between a sphere and a curve: A sphere $S$ of radius $R$ and center $\mathbf{a}$ can be represented as:

$G(x_1,x_2,x_3) = (\mathbf{x}-\mathbf{a})\cdot (\mathbf{x}-\mathbf{a}) - R^2 = 0$.

(1) For a curve $C$ to have a first-order contact with $S$ at a point $P$:

$\frac{dp}{ds} = \frac{d}{ds}\bigg[ G(x_1(s),x_2(s),x_3(s)) \bigg] = 2 (\mathbf{x}-\mathbf{a}) \cdot \mathbf{\dot{x}} = 2 (\mathbf{x}-\mathbf{a}) \cdot \mathbf{t} = 0$

Thus the center lies in the normal plane and can be written as

$\mathbf{a} = \mathbf{x} + \alpha \mathbf{p} + \beta \mathbf{b}$ (because the normal plane is spanned by $\mathbf{p}$ and $\mathbf{b}$).

Here $\alpha$ and $\beta$ are real numbers. Thus, there is a two-parameter family of spheres satisfying the first-order contact condition.

(2) For $C$ to have a second-order contact with $S$ at $P$, we must in addition have:

$\frac{d^2p}{ds^2} = 2 (\mathbf{t}\cdot \mathbf{t} + (\mathbf{x}-\mathbf{a})\cdot \mathbf{\dot{t}}) = 2 (1 + (\mathbf{x}-\mathbf{a}) \cdot \kappa \mathbf{p})= 0$.

In this case, $\mathbf{a} = \mathbf{x} + \rho \mathbf{p} + \beta \mathbf{b}$ where $\rho$ is the radius of curvature and $\beta$ is an arbitrary real number. Thus, there is a single-parameter family of spheres satisfying the second-order contact condition.

(3) For $C$ to have a third-order contact with $S$ at $P$, we must have (in addition to the conditions for first- and second-order contact):

$\frac{d^3p}{ds^3} = 2(\kappa \mathbf{t}\cdot \mathbf{p} + (\mathbf{x}-\mathbf{a})\cdot (\dot{\kappa} \mathbf{p} + \kappa \mathbf{\dot{p}}) ) = 2 (\mathbf{x} - \mathbf{a}) \cdot (\dot{\kappa} \mathbf{p} - \kappa^2 \mathbf{t} + \kappa \tau \mathbf{b}) = 0$. It can be shown that in this case $\mathbf{a}$ has to be:

$\mathbf{a} = \mathbf{x} + \rho \mathbf{p} + \frac{\dot{\rho}}{\tau} \mathbf{b}$

This unique sphere is called the osculating sphere of $C$ at point $P$. The radius of the osculating sphere is given by: $R_S = |\mathbf{x} - \mathbf{a}| = \sqrt{\rho^2 + (\dot{\rho} / \tau)^2}$.

Differential geometry notes 4

Formulae of Frenet: These describe the derivatives of the moving trihedron of a curve in terms of the moving trihedron itself:

$\begin{bmatrix} \mathbf{\dot{t}} \\ \mathbf{\dot{p}} \\ \mathbf{\dot{b}} \end{bmatrix}$ $= \begin{bmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{bmatrix}$ $\begin{bmatrix} \mathbf{t} \\ \mathbf{p} \\ \mathbf{b} \end{bmatrix}$

Note that the coefficients inside the skew-symmetric matrix need not be constant, hence the system described by this equation is not necessarily linear.

Definition 16.1: A vector $\mathbf{d}$ is called a rotation vector of a rotation if it has the following properties:

1. $\mathbf{d}$ has the direction of the axis of rotation.
2. The sense of $\mathbf{d}$ is such that the rotation has the clockwise sense if one looks from the initial point of $\mathbf{d}$ to its terminal point.
3. The magnitude $|\mathbf{d}|$ of $\mathbf{d}$ equals the angular velocity $\omega$ of the rotation, that is, the velocity of points at distance 1 from the axis of rotation.

Theorem 16.2: The rotation vector of the trihedron of a curve $C: \mathbf{x}(s)$ of class $r \geq 3$ with non-vanishing curvature, when a point moves along $C$ with constant velocity 1 is given by the expression:

$\mathbf{d} = \tau \mathbf{t} + \kappa \mathbf{b}$

This vector is called the vector of Darboux.

The curves traced out by the terminal points of $\mathbf{t}(s)$, $\mathbf{b}(s)$ and $\mathbf{p}(s)$ on the unit sphere as the trihedron of a curve moves are called the tangent, binormal and principal normal indicatrix respectively. The linear elements of these curves $ds_T$, $ds_P$ and $ds_B$ are given by:

$ds_T^2 = \kappa^2 ds^2$

$ds_P^2 = (\tau^2 + \kappa^2) ds^2$

$ds_B^2 = \tau^2 ds^2$

Hence, we can write $ds_P^2 = ds_T^2 + ds_B^2$ (Equation of Lancret).

Shape of a curve in the neighborhood of any of its points: We can Taylor expand the representation of the curve around $s = 0$ (note that the zero point can be chosen arbitrarily):

$\mathbf{x}(s) = \mathbf{x}(0) + \sum_{v = 1}^3 \frac{s^v}{v!} \frac{d^v \mathbf{x}(0)}{ds^v} + o(s^3)$

We have $\mathbf{\dot{x}} = \mathbf{t}$, $\mathbf{\ddot{x}}=\mathbf{\dot{t}} = \kappa \mathbf{p}$, and $\mathbf{\dddot{x}} = \mathbf{\ddot{t}}= \dot{\kappa}\mathbf{p} + \kappa \mathbf{\dot{p}} = \dot{\kappa} \mathbf{p} - \kappa^2 \mathbf{t} + \kappa \tau \mathbf{b}$. We can choose the coordinate system such that $\mathbf{t}(0)=(1,0,0)$, $\mathbf{p}(0)=(0,1,0)$, $\mathbf{b}(0)=(0,0,1)$.

Plugging these expressions in the Taylor expansion, we get:

$x_1(s) = s - \frac{\kappa_0^2}{3!}s^3 + o(s^3)$

$x_2(s) = \frac{\kappa_0}{2}s^2 + \frac{\dot{\kappa_0}}{3!}s^3 + o(s^3)$

$x_3(s) = \frac{\kappa_0 \tau_0}{3!}s^3 + o(s^3)$

where $\kappa_0$ and $\tau_0$ are the values of the curvature and torsion at $s=0$. If we keep only the leading terms, we get:

$\mathbf{x}(s) \approx (s, \frac{\kappa_0}{2} s^2, \frac{\kappa_0 \tau_0}{6}s^3)$              ($\kappa_0 >0$, $\tau_0 \neq 0$)