Morse and Morse-Smale Complexes - Mathematical Tools for Shape Analysis and Description - page 75

C H A P T E R 10

Morse and Morse-Smale
Complexes

e intuition behind Morse and Morse-Smale complexes was given by Maxwell [ 136 ]:

“Hence each point of the earth's surface has a line of slope, which begins at a certain
summit and ends in a certain bottom. Districts whose lines of slope run to the same
bottom are called basins or dales. ose whose lines of slope come from the same
summit may be called, for want a better name, hills. Hence, the whole earth may be
naturally divided into basins or dales, and also, by an independent division, into hills,
each point of the surface belonging to a certain dale and also to a certain hill.”

e decomposition of the surface into its hills corresponds to the ascending, Morse com-
plex. e decomposition of the surface into its dales corresponds to the descending, Morse com-
plex. If we overlap the decompositions on hills and on dales, we obtain a Morse-Smale decom-
position.
Morse and Morse-Smale complexes describe a shape by decomposing it into cells of uni-
form behavior of the gradient flow of a real function f applied on the shape, and by encoding the
adjacencies among these cells in a complex which describes both the topology and the geometry
of the gradient of f .
Morse and Morse-Smale complexes were introduced in computer graphics for the analysis
of two-dimensional scalar fields and their use has been extended to handle three-dimensional
scalar fields and more generic shapes. e theory behind is more general, however, with connec-
tions to the theory of dynamical systems [ 156 ]. e difference between Morse and Morse-Smale
complexes concerns the properties of the function f which is used to study the shape, as it will be
explained later on. Alternatively, this decomposition can be interpreted as having been obtained
by joining the critical points of the function f by lines, in the case of a two-dimensional scalar
field (or surfaces in the case of a three-dimensional scalar field), of steepest ascent or descent of
the gradient.

10.1 BASIC CONCEPTS
Let M be a smooth compact n -manifold without boundary, and let f WM!R be a smooth
Morse function. Let us also assume that M is embedded in R n or that a Riemannian metric is
defined on M . Morse complexes are induced by the partition induced by the integral lines of the