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Geography and Mathematics.
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Sandra Lach Arlinghaus

     There are a number of elegant theorems about conics in the projective plane.  The animations below illustrate a few associated constructions. 

PASCAL'S THEOREM [Coxeter, 1961]:  If A, B, C, A', B', C' are six vertices of a hexagon inscribed in a conic, then the points of intersection of opposite sides of the hexagon, L, M, and N, are collinear (lying along line z).

Figure 1.  Pascal's Theorem.

BRIANCHON'S THEOREM [Coxeter, 1961]:  If a, b, c, a', b', c' are six sides of a hexagon circumscribed about a conic, then the lines joining opposite vertices (the diagonals) of the hexagon, l, m, and n, are concurrent (at the point Z).

Figure 2.  Brianchon's Theorem:  dual of Pascal's Theorem.

BRAIKENRIDGE-MACLAURIN CONSTRUCTION [Coxeter, 1961]:  Construction of a conic through 5 given points based on:

Fiugre 3a.  This construction is the converse of Pascal's Theorem.  Choose line z through L, leading to point C' on the conic through the 5 given points, A, B, C, A', B'.

Figure 3b.  This construction is independent of the choice of z.  Choose a different line line z through N.  It, too, will produce a point C' in a location different from that of Figure 3a but the new C' will also lie on the conic through the 5 given points!

Part of the motivation for creating these animations lies in offering helpful graphics to illuminate notation.  Another part of it is to build a foundation on which to continue ongoing research linking non-Euclidean geometries and the  compression of (geo)graphics [Arlinghaus and Batty, 2005] -- within the digital world but perhaps not beyond the "limits" of the Escher series! [Escher, Circle Limit series; kmz file link).
Click to download Google Earth KMZ file


Congratulations to all Solstice contributors.


Solstice:  An Electronic Journal of Geography and Mathematics
Volume XVIII, Number 2
Institute of Mathematical Geography (IMaGe).
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