Section 1.7B The Four-Vertex Theorem Matthew Hendrix Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition The integer I is called the rotation index of the curve α where I is an integer multiple of 2π; that is, Z 1 k(s) dt = θ(`) − θ(0) = 2πI 0 Definition 2 Let α : [0; `] ! R be a plane closed curve given by α(s) = (x(s); y(s)). Since s is the arc length, the tangent vector t(s) = (x0(s); y 0(s)) has unit length. The tangent indicatrix is 2 0 0 t : [0; `] ! R , which is given by t(s) = (x (s); y (s)); this is a differentiable curve, the trace of which is contained in the circle of radius 1. Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition 2 Let α : [0; `] ! R be a plane closed curve given by α(s) = (x(s); y(s)). Since s is the arc length, the tangent vector t(s) = (x0(s); y 0(s)) has unit length. The tangent indicatrix is 2 0 0 t : [0; `] ! R , which is given by t(s) = (x (s); y (s)); this is a differentiable curve, the trace of which is contained in the circle of radius 1. Definition The integer I is called the rotation index of the curve α where I is an integer multiple of 2π; that is, Z 1 k(s) dt = θ(`) − θ(0) = 2πI 0 Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition 2 A vertex of a regular plane curve α :[a; b] ! R is a point t 2 [a; b] where k0(t) = 0. Definition 2 A regular, plane (not necessarily closed) curve α :[a; b] ! R is convex if, for all t 2 [a; b], the trace α([a; b]) of α lies entirely on one side of the closed half-plane determined by the tangent line at t. Matthew Hendrix Section 1.7B The Four-Vertex Theorem Definition 2 A regular, plane (not necessarily closed) curve α :[a; b] ! R is convex if, for all t 2 [a; b], the trace α([a; b]) of α lies entirely on one side of the closed half-plane determined by the tangent line at t. Definition 2 A vertex of a regular plane curve α :[a; b] ! R is a point t 2 [a; b] where k0(t) = 0. Matthew Hendrix Section 1.7B The Four-Vertex Theorem Theorem (Four-Vertex) A simple closed convex curve has at least four vertices. Lemma (Four-Vertex) 2 Let α : [0; 1] ! R be a plane closed curve parametrized by arc length and let A, B, and C be arbitrary real numbers. Then Z 1 dk (Ax + By + C) ds = 0 0 ds where functions x = x(s), y = y(s) are given by α(s) = (x(s); y(s)), and k is the curvature of α. Theorem (Turning Tangents) The rotation index of a simple closed curve is ±1, where the sign depends on the orientation of the curve. Matthew Hendrix Section 1.7B The Four-Vertex Theorem Lemma (Four-Vertex) 2 Let α : [0; 1] ! R be a plane closed curve parametrized by arc length and let A, B, and C be arbitrary real numbers. Then Z 1 dk (Ax + By + C) ds = 0 0 ds where functions x = x(s), y = y(s) are given by α(s) = (x(s); y(s)), and k is the curvature of α. Theorem (Turning Tangents) The rotation index of a simple closed curve is ±1, where the sign depends on the orientation of the curve. Theorem (Four-Vertex) A simple closed convex curve has at least four vertices. Matthew Hendrix Section 1.7B The Four-Vertex Theorem Theorem (Turning Tangents) The rotation index of a simple closed curve is ±1, where the sign depends on the orientation of the curve. Theorem (Four-Vertex) A simple closed convex curve has at least four vertices. Lemma (Four-Vertex) 2 Let α : [0; 1] ! R be a plane closed curve parametrized by arc length and let A, B, and C be arbitrary real numbers. Then Z 1 dk (Ax + By + C) ds = 0 0 ds where functions x = x(s), y = y(s) are given by α(s) = (x(s); y(s)), and k is the curvature of α. Matthew Hendrix Section 1.7B The Four-Vertex Theorem.