Book contents
- Frontmatter
- Contents
- List of Illustrations
- List of Tables
- Preface
- 1 The Euclidean Plane
- 2 Parametrized Curves
- 3 Classes of Special Curves
- 4 Arc Length
- 5 Curvature
- 6 Existence and Uniqueness
- 7 Contact with Lines
- 8 Contact with Circles
- 9 Vertices
- 10 Envelopes
- 11 Orthotomics
- 12 Caustics by Reflexion
- 13 Planar Kinematics
- 14 Centrodes
- 15 Geometry of Trajectories
- Index
10 - Envelopes
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- List of Illustrations
- List of Tables
- Preface
- 1 The Euclidean Plane
- 2 Parametrized Curves
- 3 Classes of Special Curves
- 4 Arc Length
- 5 Curvature
- 6 Existence and Uniqueness
- 7 Contact with Lines
- 8 Contact with Circles
- 9 Vertices
- 10 Envelopes
- 11 Orthotomics
- 12 Caustics by Reflexion
- 13 Planar Kinematics
- 14 Centrodes
- 15 Geometry of Trajectories
- Index
Summary
So far in this book we have been concerned solely with the geometry of a single parametrized curve z. But numerous situations give rise naturally to a family of curves (zλ), where the parameter λ ranges over some set ∧. In this chapter we restrict ourselves to one parameter families of curves (meaning that ∧ is an open interval of the real line) and study one feature of such a family, namely that it may have an ‘envelope’, roughly speaking a curve which at every point touches some curve in the family. A paradigm for the concept is the family of tangent lines to a given parametrized curve: in more detail, we start with a regular curve z, and for each parameter λ let zλ be the tangent line to z at λ: then z has the property that at every parameter λ there is a curve in the family (zλ) which touches it at that point. Our concern is with the reverse process, where one starts with a family of parametrized curves (zλ), and seeks a parametrized curve z with the property that at every point it touches some curve in the family. The key result of this chapter is the Envelope Theorem, which in principle enables one to find all possible envelopes of a given family of curves.
Envelopes
To make sense of the idea of an ‘envelope’ we require some formal definitions.
- Type
- Chapter
- Information
- Elementary Geometry of Differentiable CurvesAn Undergraduate Introduction, pp. 137 - 150Publisher: Cambridge University PressPrint publication year: 2001