Curves

(Section 1) Curves

(subsection 1.2.) Parametrized Curves

(Def) R^{3} is denoted the set of triples (x,y,z) of real numbers. A real function of a real variable is differentiable (or smooth) if it has, at all points, derivatives of all orders.

(Def) A parametrized differentiable curve is a differentiable map \alpha : I \to R^{3} of an open interval I = (a,b) of the real line R into R^{3}. The differentiable means that \alpha is a correspondence which maps each t \in I into a point \alpha(t) = (x(t), y(t), z(t)) \in R^{3} which the function x(t), y(t), z(t) are differentiable.

(Def) the vector (x’(t), y’(t), z’(t)) = \alpha ‘(t) \in R^{3} is called the tangent vector (or velocity vector) of the curve \alpha at t. The image set \alpha (I) \subset R^{3} is called the trace of \alpha.

(Def) Let u = (u_{1}, u_{2}, u_{3}) \in R^{3} and define its norms (or length) by |u| = \sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2}}. Let v = (v_{1}, v_{2}, v_{3}) \in R^{3}, and let \theta , 0 \le \theta \le \pi be the angle formed by the segments 0u and 0v. the inner product u \cdot v is defined by u \cdot v = |u||v| cos \theta.

(prop)

Assume that u and v are nonzero vectors. Then u \cdot v = 0 iff u is orthogonal to v.

u \cdot v = v \cdot u.

\lambda(u \cdot v) = \lambda u \cdot v = u \cdot \lambda v.

u \cdot (v + w) = u \cdot v + u \cdot w.

(Def) Let e_{1} = (1,0,0) , e_{2} = (0,1,0) , and e_{3} = (0,0,1) . e_{i} \cdot e_{j} = 1 if i = j and that e_{i} \cdot e_{j} = 1 if i = j and that e_{i} \cdot e_{j} = 0 if i \neq j , where i , j = 1,2,3. Thus, by writing u = u_{1}e_{1} + u_{2}e_{2} + u_{3}e_{3} , v = v_{1}e_{1} + v_{2}e_{2} + v_{3}e_{3} , we obtain u \cdot v = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}.

(subsection 1.3.) Regular Curves ; Arc length

(Def) Let \alpha : I \to R^{3} be a parametrized differentiable curve. For each t \in I where \alpha ‘(t) \neq 0, there is a well-defined straight line which contains the point \alpha (t) and the vector \alpha ‘(t) . This line is called the tangent line to \alpha at t. We call any point where \alpha’(t) = 0 a singular point. We restrict our attention to curves without singular points.

(Def) A parametrized differentiable curve \alpha : I \to R^{3} is said to be regular if \alpha ‘(t) \neq 0 for all t \in I.

Given t \in I, the arc length of a regular parametrized curve \alpha : I \to R^{3} , from the point t_{0}, is by definition s(t) = \int_{t_{0}}^{t} |\alpha ‘(t)| dt, where |\alpha ‘(t)| = \sqrt{(x’(t))^{2} + (y’(t))^{2} + (z’(t))^{2}} is the length of the vector \alpha ‘(t).

(Def) Given the curve \alpha parametrized by arc length s \in (a,b) , we may consider the curve \beta defined in (-b, -a) by \beta(-s) = \alpha (s) , which has the same trace as the first one but is described in the opposite direction. We say that these two curves differ by a change of orientation.

(subsection 1.4.) The Vector product in R^{3}

(Def) Two ordered basis e = {e_{i}} and f = {f_{i}} , i = 1,…,n, of an n-dimensional vector space V has the same orientation if the matrix of change of basis has positive determinant. We denote this relation by e ~ f.

e~f is an equivalence relation. Each of equivalence classes determined by the above relation is called an orientation of V. In the case V = R^{3}, there exists a natural ordered basis e_{1} = (1,0,0) , e_{2} = (0,1,0) , and e_{3} = (0,0,1). We call the orientation corresponding the positive orientation of R^{3}, the other one being the negative orientation.

(Def) Let u, v \in R^{3} . The vector product of u and b is the unique vector u \wedge v \in R^{3} characterized by (u \wedge v) \cdot w = det (u,v,w) for all w \in R^{3}. Here det (u,v,w) means that if we express u,v, and w in the natural basis {e_{i}} , u = \sum u_{i}e_{i} , v = \sum v_{i}e_{i} , w = \sum w_{i}e_{i} , i = 1,2,3, then det(u,v) = \begin{vmatrix} u_{1}& u_{2} &u_{3} \ v_{1}& v_{2} &v_{3} \ w_{1} &w_{2} &w_{3} \end{vmatrix}. Usually u \wedge v is written u \times v and denoted as cross product.

(prop) u \wedge v = - v \wedge u.

u \wedge v depends linearly on u and v; for any real numbers a,b, we have (au + bw) \wedge v = au \wedge v + bw \wedge v.

u \wedge v = 0 iff u and v are linearly dependent.

(u \wedge v) \cdot u = 0, (u \wedge v) \cdot v = 0.

The vector product of u and v is a vector u \wedge v perpendicular to a plane generated by u and v, with a norm equal to the area of a parallelogram generated by u and v and a direction s.t. (u,v, \u \wedge v) is a positive basis.

(subsection 1.5.) The Local Theory of Curves Parametrized by Arc Length

(Def) Let \alpha : I \to R^{3} be a curve parametrized by arc length s \in I. The number |\alpha ‘’(s) | = k(s) is called the curvature of \alpha at s.

(Def) At points where k(s) \neq 0, a unit vector n(s) in the direction \alpha’’(s) is well defined by the equation \alpha’’(s) = k(s) n(s). n(s) is normal to \alpha’(s) and is called the normal vector at s. The plane determined by the unit tangent and normal vectors, \alpha’(s) and n(s), is called the osculating plane at s.

s \in I is a singular point of order 1 if \alpha’’(s) = 0. We denote t(s) = \alpha’(s) the unit tangent vector of \alpha at s. Thus, t’(s) = k(s)n(s).

The unit vector b(s) = t(s) \wedge n(s) is normal to the osculating plane and will be called the binormal vector at s. It measures how rapidly the curve pulls away from the osculating plane at s , in a neighborhood of s. b’(s) is parallel to n(s), and we may write b’(s) = \tau(s) n(s) for some function \tau (s).

(Def) Let \alpha : I \to R^{3} be a curve parametrized by arc length s s.t. \alpha’’(s) \neq 0 , s \in I. The number \tau(s) defined by b’(s) = \tau(s) n(s) is called the torsion of \alpha at s.

(Def) To each value of the parameter s, we have associated three orthogonal unit vectors t(s), n(s), b(s) . The trihedrom formed is referred to as the Frenet trihedron at s. We call the equations t’ = kn , n’ = -kt - \tau b, b’ = \tau n the Frenet formulas. The tb plane is called the rectifying plane, and the nb plane the normal plane. The lines which contain n(s) and b(s) and pass through \alpha(s) are called the principal normal and the binormal, respectively. The inverse R = 1/k of the curvature is called the radius of curvature at s.

(Fundamental Theorem of the Local Theory of Curves) Given differentiable functions k(s) >0 and \tau(s), s \in I, there exists a regular parametrized curve \alpha : I \to R^{3} s.t. s is the arc length, k(s) is the curvature, and \tau(s) is the torsion of \alpha. Moreover, any other curve \bar{\alpha} , satisfying the same conditions, differs from \alpha by a rigid motion ; that is , there exists an orthogonal linear map \rho of R^{3}, with positive determinant, and a vector c such that \bar{\alpha} = \rho \bullet \alpha + c.

(subsection 1.6.) The Local canonical form

(Def) Let \alpha : I \to R^{3} be a curve parametrized by arc length without singular points of order 1. We write the equations of the curve, in a neighborhood of s_{0}, using the trihedron t(s_{0}) , n(s_{0}), b(s_{0}) as a basis for R^{3}. WLOG , s_{0} = 0. Consider the taylor expansion \alpha(s) = \alpha(0) + s(\alpha’(0)) + \frac{s^{2}}{2} \alpha’’(0) + \frac{s^{3}}{6} \alpha’’’(0) + R where \lim_{s \to 0} R/s^{3} = 0. Let us now take the system Oxyz in such a way that the origin O agrees with \alpha(0) and that t = (1,0,0) , n = (0,1,0) , b = (0,0,1). Under these conditions, \alpha(s) = (x(s), y(s), z(s)) is given by x(s) = s - \frac{k^{2}s^{2}}{6} + R_{x} , y(s) = \frac{k}{2} s^{2} + \frac{k’s^{3}}{6} + R_{y} , z(s) = -\frac{k \tau}{6} s^{3} + R_{z} ,where R = (R_{x} , R_{y}, R_{z}) . This representation is called the local canonical form of \alpha, in a neighborhood of s = 0.

(prop) Existence of a neighborhood J \subset I of s = 0 s.t. \alpha(J) is entirely contained in the one side of the rectifying plane toward which the vector n is pointing.

The osculating plane at s is the limit position of the plane determined by the tangent line at s and the point \alpha(s + h) when h \to 0.

(subsection 1.7.) Global Properties of Plane Curves

(Def) A differentiable function on a closed interval [a,b] is the restriction of a differentiable function defined on an open interval containing [a,b].

A closed plane curve is a regular parametrized curve \alpha : [a,b] \to R^{2} s.t. \alpha and all its derivatives agree at a and b; that is, \alpha (a) = \alpha (b), \alpha’(a) = \alpha’(b) , …

The curve \alpha is simple if it has no further self-intersections.

We usually consider the curve \alpha : [0, l] \to R^{2} parametrized by arc length s; hence, l is the length of \alpha.

We assume that a simple closed curve C in the plane bounds a region of this plane that is called the interior of C. We assume that the parameter of a simple closed curve can be so chosen that if one is going along the curve in the direction of increasing parameters, the interior of the curve remains left. Such a curve will be called positively oriented.

(subsubsection A) The Isoperimetric Inequality.

(Def) The area A bounded by a positively oriented simple closed curve \alpha (t) = (x(t), y(t)), where t \in [a,b] is an arbitrary parameter : A = -\int_{a}^{b} y(t) x’(t) dt = \int_{a}^{b} x(t) y’(t) dt = \frac{1}{2} \int_{a}^{b} (xy’ – yx’) dt.

(The Isoperimetric Inequality) (Thm 1) Let C be a simple closed plane curve with length l, and let A be the area of the region bounded by C. Then l^{2} – 4 \pi A \ge 0 , and equality holds iff C is a circle.

This applies to C^{1} curves, that is, curves \alpha(t) = (x(t), y(t)) , t \in [a,b] for which we require only that the functions x(t), y(t) have continuous first derivatives. It holds for piecewise C^{1} curves, which is continuous curves that are made up by a finite number of C^{1} arcs.

(subsubsection B) The Four-Vertex Theorem

(Def) Let \alpha : [0,l] \to R^{2} be a plane closed curve given by \alpha(s) = (x(s), y(s)) . Since s is the arc length, the tangent vector t(s) = (x’(s), y’(s)) has unit length. The tangent indicatrix t : [0,l] \to R^{2} that is given by t(s) = (x’(s), y’(s)) is a differentiable curve, the trace of which is contained in a circle of radius 1.

Define a global differentiable function \theta : [0,l] \to R by \theta(s) = \int_{0}^{s} k(s) ds . Since \alpha is closed, this angle is an integer multiple I of 2 \pi ; that is , \int_{0}^{l} k(s) ds = \theta(l) - \theta (0) = 2 \pi I. This integer I is called the rotation index of the curve \alpha.

(prop)(Theorem of Turning Tangents) The rotation index of a simple closed curve is \mp 1, where the sign depends on the orientation of the curve.

(Def) A regular, plane curve \alpha : [a,b] \to R^{2} is convex if, for all t \in [a,b] , the trace \alpha([a,b]) of \alpha lies entirely on one side of the closed half-plane determined by the tangent line at t.

A vertex of a regular plane curve \alpha : [a,b] \to R^{2} is a point t \in [a,b] where k’(t) = 0.

(The Four-vertex Theorem) (Thm 2) A simple closed convex curve has at least four vertices.

(Lemma) Let \alpha : [0,1] \to R^{2} be a plane closed curve parametrized by arc length and let A,B, and C be arbitrary real numbers. Then \int_{0}^{1} (Ax + By + C) \frac{dk}{ds} ds = 0 where the functions x = x(s), y = y(s) are given by \alpha(s) = (x(s), y(s)) , and k is the curvature of \alpha.

(subsubsection C) The Cauchy-Crofton Formula

(Def) Let C be a regular curve in the plane. We look at all straight lines in the plane that meet C and assign to each such line a multiplicity which is the number of its intersection points with C.

A straight line in the plane can be thought as a point in a plane given by two parameters \rho and \theta which determines a line. Define a measure (area) of a subset of straight lines in the plane, by the area of certain plane.

(The Cauchy-Crofton Formula) (Thm 3) Let C be a regular plane curve with length l. The measure of the set of straight lines (counted with multiplicities) which meet C is equal to 2l.

(Def) A rigid motion in R^{2} is a map F : R^{2} \to R^{2} given by (\bar{x}, \bar{y}) \to (x,y) where x = a + \bar{x} cos \phi - \bar{y} sin \phi , y = b + \bar{x} sin \phi + \bar{y} cos \phi.

(Prop 1) Let f (x,y) be a continuous function defined in R^{2}. For any set S \subset R^{2}, define the area A of S by A(S) = \int \int_{S} f(x,y) dx dy (on only those sets for which the abouve integral exists). Assume that A is invariant under rigid motions; that is , if S is any set and \bar{S} = F^{-1}(S), where F is the rigid motion, we have A(\bar{S}) = \int \int_{S} f(\bar{x} , \bar{y}) d \bar{x} d \bar{y} = \int \int_{S} f(x,y) dx dy = A(S). Then f(x,y) = const.

Jacobian of a rigid motion is 1, and the rigid motions are transitive on points of the plane; that is, given two points in the plane there exists a rigid motion taking one point into the other.

(Def) In the set of all straight lines in the plane \mathcal{L} = {(p, \theta) \in R^{2} ; (p,\theta) ~ (p , \theta + 2k\pi) and (p, \theta) ~ (-p, \theta \mp \pi)}, from the prop 1, define the measure of a set \mathcal{S} \subset \mathcal{L} as \int \int_{\mathcal{S}} d \rho d \theta in the same way as prop 1.

(App) If a curve is not rectifiable but the \int \int n d \rho d \theta (Let n = n(\rho, \theta) be the number of intersection points of the straight line (\rho , \theta)) has a meaning, this can be used to determine the length of such a curve.

Consider a family of parallel straight lines s.t. two consecutive lines are at a distance r. Rotate this family by angles of \pi/4, 2 \pi/4. 3 \pi/4 in order to obtain four families of straight lines. Let n be the number of intersection points of a curve with all these lines. Then \frac{1}{2} nr \frac{\pi}{4} is an approximation to length of C.