Page 194 - Geometric Modeling and Algebraic Geometry

196 I. Ivrissimtzis and H.-P. Seidel
10.7 Appendix

A. The eigenvectors of Z n
2
By Proposition 2 the two characters of Z 2 are
χ 0 : χ 0 (0) = 1,χ 0 (1) = 1 (10.30)

and
χ 1 : χ 1 (0) = 1,χ 1 (1) = −1 (10.31)
n
By Proposition 3 the character χ h corresponding to the element h of Z is given by
2
n

χ h (g): e ihg (10.32)
i=1
where e ihg = −1 iff δ i (h)= δ i (g)=1 and e ihg =1 otherwise.
n
Finally, we can use Proposition 1 to compute the eigenvectors of the Z -circulant
2
matrices.
B. Exact computation of the limit shape of the prism
To compute the exact limit shape of A, B we write the polygons a 1 C 1 +
2
C n−1 and b 1 C 1 + b n−1 C n−1 in parametric form. We notice that they are both
a n−1
2 2 2
planar polygons inscribed on ellipses, thus their vertices lie on the curves
(10.33)
c a + r au cos θu a + r av sin θv a
and
(10.34)
c b + r bu cos θu b + r bv sin θv b
respectively, where c a , c b are the centers of the ellipses, u a , v a and u b , v b are
orthonormal vectors on E A ,E B and θ = 2πj ,j =0, 1,...,k − 1.
k
Then Eq.(10.29) gives the limit shape as
(c a + c b )+cos θ(r au u a + r bu u b ) + sin θ(r av v a + r bv v b ) (10.35)

which is again the equation of an ellipse.

References

1. Darboux, G.: Sur un probl´ eme de g´ eom´ etrie ´ el´ ementaire. Darboux Bull. (1878)
2. Berlekamp, E., Gilbert, E., Sinden, F.: A polygon problem. Am. Math. Mon. 72 (1965)
233–241
3. Bachmann, F., Schmidt, E.: n-Gons. University of Toronto Press (1975)
4. Davis, P.J.: Circulant matrices. Wiley-Interscience. (1979)
5. Fisher, J., Ruoff, D., Shilleto, J.: Perpendicular polygons. Am. Math. Mon. 92 (1985)
23–37