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   NCB Deposit # 76

Jeffrey M. Groah
Montgomery College
Conroe, Texas


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Animated conics

A Power Point Presentation and Animation

The Conics and Stereographic Projection

To Infinity and Beyond

Power Point presentation
Click below to see the Power Point slide show.

Power Point Slide Show


Johannes Kepler said, “A parabola is an ellipse with a focus at infinity.” On its face, this statement is nonsense. What does it mean for a focus to be “at” a point that is not defined? In this animation, points on the plane are associated with points on a sphere by stereographic projection. The north pole of the sphere corresponds to the point at infinity. This is the one point compactification of the plane. Curves in the plane correspond to curves on the sphere. In particular, a family of ellipses in the plane with one focus that tends to infinity is displayed. On the sphere, when the focus is at the point at infinity, the curve corresponds exactly to a parabola in the plane. The focus on the sphere is then sent beyond infinity, over the top. The corresponding curves in the plane are hyperbolas. In this sense, hyperbolas are ellipses where one focus has been sent to infinity and then back from the other side, just as Kepler described. Using the one point compactification, we can make sense of Kepler’s descriptions of the conic sections.

For more animations please see < >.

For more on Kepler please see < >.

Note:  There are many spellings of the name of Pavel Sergeiivich Aleksandrov (1896-1982) as it must be translated from the Cyrillic alphabet.  In researching this material we found Alexandroff and Alexandrov as well as Aleksandrov. Aleksandrov is used by Gardner, Katz and Weisstein.

Howard Eves, An Introduction to the History of Mathematics, 6th ed., Saunders College Publishing, p. 324.
R. J. Gardner, "Geometric Tomography."  Not. Amer. Math. Society, 42, 422-429, 1995.
Victor J. Katz, A History of Mathematics, 2nd ed., Addison Wesley Longman, 1998, p. 832.
Katz writes of the chain of famous names associated with the birth of Algebraic Topology.  Noether, Aleksandrov, Eilenberg and Mac Lane are all joined in this whole new  20th century field of study.
"When in the course of our lextures (in Göttingen in 1926 and 1927) she first became acquainted with a systematic construction of combinatorial topology, she immediagely observed that it would be worthwhile to study directly the groups of algebraic complexes and cycles of a given polyhedron and the subgroup of the cycle group consisting of cycles homologous to zero ...."
Aleksandrov's memorial to Emmy Noether (1882-1935)

Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 1999.

Kepler introduced the word "focus" into the geometry of conics.  Indeed, it has been written that if the Greeks had not perused the conics, Kepler could not have superseded Ptolemy.  But he is primarily remembered for his three laws, a cornerstone of astronomy.

Kepler stamp
Kepler is honored on many stamps.

"I am writing a book for my contemporaries - or does not matter - for posterity.  It may be that my book will wait for a hundred years for a reader.  Has not God waited 6,000 years for an observer?"

Johannes Kepler, Harmony of the Worlds,  1619

Brahe and Kepler statue
Both Brahe and Kepler are honored in Prague . . . .
Street in Prague
and for a time, both
lived on this same street.

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