Differential geometry notes 5

by exactnature

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}.

 

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