Chapter 3 • Curves in Space: Local Properties

3.1: Definitions, Examples, and Differentiation

Example 3.1.2: Lines

Example 3.1.3: Planar Curves

Example 3.1.4: Twisted Cubic

Example 3.1.5: Cylindrical Helix

Example 3.1.6: Conical Helix

Example 3.1.7: Curves on Hyperboloid of One Sheet

Example 3.1.8: Space Cardioid

3.2: Curvature, Torsion, and the Frenet Frame

Page 68: Definition of Unit Tangent Vector

Page 69: Definition of Principal Normal Vector

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Definition 3.2.2: Frenet Frame and Definition of Binormal Vector

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Definition 3.2.3: Space Curvature and Plane Curvature

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Definition 3.2.4: Torsion Function

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Example 3.2.5: Helices

Curvature of the circular helix

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Torsion of the circular helix

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Example 3.2.6: Space Cardioid

Frenet frame of the space cardioid

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Torsion of the space cardioid

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3.3: Osculating Plane and Osculating Sphere

Definition 3.3.1: Osculating Plane

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Proposition 3.3.3: Osculating Sphere

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3.4: Natural Equations

Theorem 3.4.1: Fundamental Theorem of Space Curves

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