ConicsThis book engages the reader in a journey of discovery through a spirited discussion among three characters: Philosopher, Teacher and Student. Throughout the book, Philosopher pursues his dream of a unified theory of conics, where exceptions are banished. With a helpful teacher and example-hungry student, the trio soon finds that conics reveal much of their beauty when viewed over the complex numbers. It is profusely illustrated with pictures, worked-out examples, and a CD containing 36 applets. Conics is written in an easy, conversational style, and many historical tidbits and other points of interest are scattered throughout the text. Many students can self-study the book without outside help. This book is ideal for anyone having a little exposure to linear algebra and complex numbers. |
Contents
with brief chapter descriptions | 1 |
Life at Infinity | 27 |
How to GiftWrap a Hyperbola | 45 |
The Cube | 57 |
A WellKept Secret | 65 |
Are Hyperbolas Really Ellipses? | 89 |
Stakes and Strings | 103 |
Tie each end of a piece of string to a thumb tack and push the tacks into a drawing board | 109 |
A Most Excellent Theorem | 183 |
In moving one of the above crossed ellipses precisely when the number of points | 211 |
Curvature | 235 |
Curvature of Conics | 269 |
Photons and Conics | 283 |
How Conics Solved a 2000YearOld Problem | 305 |
Waves and Conics | 341 |
Some Conics Formulas | 359 |
Directrices New and Old | 121 |
Conics in General Position | 143 |
A Beautiful Mathematical Universe | 169 |
A Quick Handshake | 379 |
Suggestions for Further Reading | 393 |
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Common terms and phrases
3-D slice actually algebraic algebraic curves analogue angle Ax² axis Bézout's theorem Chapter Applet complex circle cone conic coordinates corresponding cosh curve Cy² defines dehomogenization differential dimension directrix distance dot product eccentricity ellipse equation Euclidean example Figure foci focus formula Gaussian curvature geometric gives graph hemisphere homogenization horizontal hyperbola branch idea imaginary Interactive Applet intersection points line at infinity line segment look loop Maple mathematics means Minkowski plane Möbius strip moving multiplicity negative normal open sets orbit origin orthogonal parabola parametrization Pascal's theorem Philosopher photon picture point at infinity polynomial projective plane quadrilateral radius real circle rectangle rotation screen shows side sin² sinh sint smooth smooth manifold space sphere stretch Student surface symmetric t₁ tangent line Teacher there's tilt topological translate triangle unit circle vector vertex vertical x-axis y-axis y)-plane Y₁ zero