Severely Theoretical

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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)