### 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: $R_S = |\mathbf{x} - \mathbf{a}| = \sqrt{\rho^2 + (\dot{\rho} / \tau)^2}$.