Geodesics
Outline
1. Geodesics
A curve γ on a surface S is called a geodesic if
1. γ has unit speed, and
2. γ̈ is normal to S at each point.
For example:
• The geodesics on a sphere are great circles.
• The geodesics on a cylinder are straight lines, circular cross sections, and helices that
wind around the cylinder.
2. Finding Geodesics
There are two major tricks for finding geodesics on a surface:
1. Any straight line on a surface is a geodesic.
2. If a surface intersects a plane at right angles, then the curve of intersection is a geodesic.
In the second case, the geodesic is called a normal section of the surface. Many surfaces of
interest have normal sections:
• Let S be a generalized cylinder C × R, where C is a plane curve. Then each horizontal
cross section C × {z0 } is a normal section of S.
• Let S be a surface of revolution r = f (z). Then each meridian curve (i.e. curve of
constant θ) is a normal section of S.
• Let S be a surface of revolution r = f (z), and let z0 be a z-value such that f 0 (z0 ) = 0.
Then the circular cross-section of S at this z-value is a normal section of S.
One can prove that a planar curve on a surface is a geodesic if and only if it is either a straight
line or a normal section.
3. Properties of Geodesics
Given a point on a surface, there is essentially one geodesic through that point in any given
direction:
Theorem. Let S be a surface, let p ∈ S, and let t be a unit tangent vector to S at p. Then
there exists a unique geodesic γ passing through p whose tangent vector at p is t.
It follows from the uniqueness in the above theorem that two different geodesics can never be
tangent at a point. Another important property of geodesics is that they are shortest-length
paths:
Theorem. Let γ : [a, b] → S be a geodesic. Then there exists an > 0 so that γ is the
shortest path on S from γ(a) to γ(a + ).
Moreover, any shortest-length path must be a geodesic:
Theorem. Let S be a surface, let p, q ∈ S. Then any path from p to q on S with minimum
length must be a geodesic.
Note that there may not exist a minimum-length path from p to q. Hence, this theorem does
not guarantee that there exists a geodesic connecting any two points on a surface.
Because geodesics are locally minimum-length, they must be preserved by local isometries:
Theorem. If f : S1 → S2 is a local isometry and γ is a geodesic on S1 , then f ◦γ is a geodesic
on S2 .
4. Geodesic Triangles
A region R on a surface is said to be simply-connected if it is homeomorphic to an open
disk. Such a region cannot have any “holes” or other topological features.
A geodesic triangle on a surface is a simply-connected region bounded by three geodesic
segments. Such a region has three interior angles, whose sum is determined by the following
theorem:
Theorem. Let T be a geodesic triangle on a surface S, and let α, β, and γ be the interior
angles of T . Then
ZZ
K dA
α+β+γ−π =
T
where K is the Gaussian curvature of S.
This is a simple case of the Gauss-Bonnet Theorem. The quantity α + β + γ − π is called
the angle excess of the triangle. As a special case, we have:
Theorem. Let T be a geodesic triangle on the unit sphere, with interior angles α, β, and γ.
Then the area of T is equal to α + β + γ − π.
That is, the angle excess of any spherical triangle is equal to the area. A similar theorem
holds for the hyperbolic plane: the angle excess of any hyperbolic triangle is the negative of
the area.
Practice Problems
1. Let C be the cylinder x2 + y 2 = 1.
(a) Find the length of the shortest path on C from the point (1, 0, 0) to the point (0, 1, 1).
(Hint: Use a local isometry to unroll the cylinder.)
(b) Find parametric equations for the shortest geodesic segment between these two points.
2. Let γ be the geodesic on the unit sphere satisfying
γ(0) =
1
(1, 2, 2)
3
and
γ̇(0) =
1
(−2, 2, −1).
3
Find a formula for γ(t).
3. In each of the following parts, let S1 and S2 be the given surfaces, and let C = S1 ∩ S2 .
Determine whether C is a geodesic curve on (i) the surface S1 , and (ii) the surface S2 .
(a) S1 is the unit sphere, and S2 is the plane z = x.
(b) S1 is the paraboloid z = r2 , and S2 is the plane x = y.
(c) S1 is the paraboloid z = r2 , and S2 is the cone z = r with the origin removed.
(d) S1 is the catenoid r = cosh z, and S2 is the cylinder x2 + y 2 = 1.
(e) S1 is the helicoid z = θ, and S2 is the cylinder x2 + y 2 = 1.
(f) S1 is the parabolic cylinder z = x2 , and S2 is the plane x = 1.