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Rotations and Trajectories

  • Victor Gutenmacher
  • N. B. Vasilyev

Abstract

In this chapter we present some remarkable curves that are naturally generated as trajectories of points on a circle rolling along a straight line or along another circle. The most interesting properties of these curves are connected with tangents. We will start by investigating cycloids, which are the paths traced by a single point on a circle as the circle rotates along another curve. The reader may recall that, at the end of the Introduction, we revisited Problem 0.1 and encountered a curve real¬ized as the envelope of a family of lines. This envelope was a curve with four cusps, called an astroid. We will examine this fact in greater detail here, and we will also see why a spot of light in a cup formed by reflected rays has a characteristic singularity, a cusp. The devotee of classical geometry will find out about the connections between the nine-point circle of a triangle, its Wallace-Simson lines and their envelope, the Steiner deltoid, which is a cycloid with three cusps.

Keywords

Angular Velocity Linear Velocity Ball Bearing Stationary Circle Rolling Circle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Victor Gutenmacher
    • 1
  • N. B. Vasilyev
  1. 1.BrooklineUSA

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