Differential Geometry - Computer Graphics and Geometric Modeling - page 632

Proof. For a proof of (1) using the exponential map which is defined later in this
section see [Thor79]. For (2) see [MilP77].

Theorem 9.10.11 implies that the first definition of a geodesic is equivalent to the
following one:

Third definition of a geodesic: A geodesic on a surface S in R 3 is just a regular
curve in S that has the property that locally it defines a curve of shortest length.

Moving on to property (3) of straight lines, it is easy to give examples that show
geodesics are neither unique nor exist in general. For example, there are an infinite
number of (in fact, minimal-length) geodesics between antipodal points on a sphere
and the surface R 2 - 0 (a punctured plane ) has no geodesic from (-1,0) to (1,0) (there
exist curves between the two points with length arbitrarily close to 2 but none of length
exactly 2). Although there may not be a geodesic between an arbitrary pair of points,
geodesics do exist if the points are not too far apart. The proof of this fact will also
show that finding geodesics is simply a matter of solving second order differential
equations.

9.10.12. Theorem. Let S be a surface in R 3 . Let p Œ S and v Œ T p ( S ), v π 0. Then
there is an e>0 and a unique constant speed geodesic g :(-e,e) Æ S with g(0) = p and
g¢(0) = v . (For v = 0, the unique “geodesic” would be the constant curve g(t) = p .)

Sketch of two proofs. For the first proof, we use a regular parameterization F of
a neighborhood of p in S and equation (9.71). Since we are looking for a geodesic,
the left-hand side of the equation vanishes. But the partials F 1 and F 2 are linearly
independent, being a basis of the tangent space. It follows that any solution g must
satisfy the equations

2

Â

i k

x ¢¢ + G x x ¢ ¢ = 0 , k = 1 2 , . (9.72)

k i j

ij , = 1

Conversely, one can show that any such solution satisfying our initial conditions will
solve our problem, namely, it must be a constant speed curve. One only has to appeal
to theorems about the existence and uniqueness of solutions to differential equations
to finish the proof. See [MilP77].
The proof we just sketched shows that the existence of geodesics is an intrinsic
property of surfaces that does not depend on any imbedding in R 3 . The second proof
does use the imbedding and normals to the surface. Its advantage is that it is more
direct and useful computationally. In practice, surfaces are presented via parameter-
izations anyway, so computing normals is not a problem.
Near p one can represent the surface S as the zero set of some function, that is,
we may assume that

- 1

() S = f 0

for some function f : U Æ R , where U is a subset of R 3 and the gradient of f does not
vanish on U . In this case, the Gauss map for S is given by