Chapter 2 • Plane Curves: Global Properties



2.1: Basic Properties


Definition 2.1.1: Closed Curve


Circle


Ellipse


Lissajous Figure


Flower


Cardioid


Definition 2.1.1: Simple Curve


Circle


Ellipse


Cardioid (Simple for k ≥ 1 or k ≤ –1. For –1 < k < 1, the cardioid is not simple.)





2.2: Rotation Index


Proposition 2.2.1: Rotation Index of a Curve


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Example 2.2.2: Figure-Eight Curve


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Definition 2.2.3: Degree of a Mapping


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2.3: Isoperimetric Inequality


Theorem 2.3.1: Isoperimetric Inequality


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2.4: Curvature, Convexity, and the Four-Vertex Theorem


Definition 2.4.2: Convex


Circle


Ellipse


Cardioid (Convex for k ≥ 2 or k ≤ –2. For –2 < k < 2, the cardioid is not convex.)


Page 54 and Figure 2.6: Height Function


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Theorem 2.4.5: Four-Vertex Theorem


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