Chapter 1 • Plane Curves: Local Properties



1.1: Parametrizations


Example 1.1.1: Lines


figure1-1

Example 1.1.2: Circles


figure2-2


Definition 1.1.3: Length


figure1-3


Example 1.1.9: Graphs of Functions


figure1-4


Example 1.1.10: Circles Revisited


figure1-5


Examples 1.1.11, 1.1.12, 1.1.14, 1.1.15, and 1.1.16


figure1-6


Example 1.1.13: Cycloids


figure1-8


Problem 1.1.1: Distance from a Point to a Line


figure13b     figure13c

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Problem 1.1.4


Problem 1.1.5: Distance from a Point to a Curve

figure13d     figure13e

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Problem 1.1.6: Epicycloid



Problem 1.1.7: Hypocycloid


Problem1_1_7






1.2: Position, Velocity, and Acceleration


Definition of Velocity Vector, p.14


Example 1.2.4


Example 1.2.5


figure1-16     figure1-17     figure1-18


Definition 1.2.7: Unit Tangent Vector


figure1-19


Example 1.2.10


figure1-39





1.3: Curvature


Definition 1.3.1: Unit Normal Vector



Definition 1.3.2: Curvature


figure1-26     figure1-28

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Example 1.3.3



Problem 1.3.16: Parallel Curve



Problem 1.3.19: Pedal Curves


figure31b 

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1.4 Osculating Circles, Evolutes, and Involutes


Definition 1.4.3: Osculating Circle and Definition 1.4.6: Evolute Curve



Definition 1.4.9: Involute Curve






1.5: Natural Equations


Theorem 1.5.2: Fundamental Theorem of Plane Curves



p.44, Taylor Series of Curves





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