Lecture notes - Institut für Mathematik - TU Berlin

TOPICS IN DISCRETE DIFFERENTIAL GEOMETRY AND VISUALIZATION 5Then det J = 1 and, since N × f u = −N u and N × f v = N v , we getN × df = dN ◦ ( −1 0 1 0−1 0) = df ◦ A ◦ (0 1 )and hence K = det A = − det J = −1.The map f is an asymptotic line parametrization, because f uu = N ×N uu and f vv = −N × N vv is tangent to f.□Theorem 3.8 above justifies the following definition.3.9. Definition. A smooth map f : M ⊂ R 2 → R 3 is called a K–surface(shorthand for surface of negative constant Gaussian curvature) if thereexists N : M → S 2 such thatf u = N × N u and f v = −N × N v .3.10. Remark. Note that this definition allows that f and N have singularities,i.e., points at which f or N fails to be an immersion.4. Discrete Lorentz Harmonic MapsIn 3.7 we saw that Lorentz harmonic maps in S 2 are, away from singularpoints, the same as weak Chebyshev nets in S 2 . That impliesthat small coordinate quadrilaterals are spherical parallelograms. Thisproperty translates easily into the discrete situation, and as we knowfrom Theorem 3.8 it also captures a characterizing property of K–surfaces.4.1. Spherical parallelograms. Let N d , N r , N u , N l ∈ S 2 be a nondegenerated spherical quadrilateral whose edges are shorter than π.2Such four points form a spherical parallelogram (a quadrilateral suchthat opposite edges have the same length) if and only if (N u + N d ) ×(N l + N r ) = 0, i.e., rotation by π about (N u + N d ) or (N l + N r ) mapsthe quadrilateral onto itself.The restriction to edges shorter than π ensures that the entire quadrilaterallies in one2hemisphere.4.2. Definition. A discrete map N : Z 2 → S 2 is called Lorentz harmonicif and only ifand all edges are shorter than π 2 .(N u + N d ) × (N l + N r ) = 04.3. Notation. For discrete maps subscripts denote points in Z 2 . Specialsubscripts are d = (m, n), u = (m + 1, n − 1), l = (m, n − 1),r = (m + 1, n) for some m, n ∈ Z. The letters express the fact that ifone reflects Z 2 at the line that intersects the first coordinate axes atthe angle π (i.e., displays discrete uv–coordinates in xt–coordinates)4then the vertices of the coordinate quadrilaterals are naturally identifiedby their positions (d)own, (u)p, (l)eft, and (r)ight. Often, e.g,, inDefinition 4.2, equations are ment to hold for all admissible m, n ∈ Z.