Chapter 1 • Plane Curves: Local Properties



1.1: Parametrizations


Example 1.1.1: Lines


figure1-1

Example 1.1.2: Circles


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Definition 1.1.3: Length


figure1-3


Example 1.1.9: Graphs of Functions


figure1-4


Example 1.1.10: Circles Revisited


figure1-5


Example 1.1.11: Ellipses


figure1-6


Example 1.1.12: Lissajous Figures


figure1-7


Example 1.1.13: Cycloids


figure1-8


Example 1.1.14: Heart Curve


figure1-9


Example 1.1.15: Polar Functions


figure1-11

Example 1.1.16: Cardioid


figure1-13


Problem 1.1.1: Epicycloid



Problem 1.1.3: Distance


figure13b     figure13c

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Problem 1.1.6: Distance

figure13d     figure13e

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1.2: Position, Velocity, and Acceleration


Example 1.2.4


Example 1.2.5


figure1-16     figure1-17     figure1-18


Definition 1.2.6: Unit Tangent Vector


figure1-19     figure1-20





1.3: Curvature


Definition 1.3.1: Unit Normal Vector


figure1-24     figure1-25

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Definition 1.3.2: Curvature


figure1-26     figure1-27     figure1-28

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Example 1.3.3: Circles


figure1-29     figure1-30     figure1-31

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Problem 1.3.12: Parallel Curve


figure31a

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Problem 1.3.15: Pedal Curves


figure31b 

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1.4 Osculating Circles, Evolutes, and Involutes


Definition 1.4.4: Osculating Circle and Definition 1.4.7: Evolute Curve



Definition 1.4.10: Involute Curve






1.5: Natural Equations


Theorem 1.5.2: Fundamental Theorem of Plane Curves


figure1-34     figure1-35     figure1-36

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