Geodesic distance
High school
Consider two points $p$ and $q$ on a surface, such as two cities on the Earth. The geodesic distance between $p$ and $q$ is the length of the shortest path from $p$ to $q$ that stays on the surface.
Graduate (Riemannian manifolds)
Consider a connected Riemannian manifold $M$ and two points $p, q \in M$. The geodesic distance between $p$ and $q$ is the infimum of the lengths of the piecewise smooth curves in $M$ from $p$ to $q$:
\[d(p, q) = \inf \setdef{\int_0^1 \norm{\gamma^{\prime}(t)} \mathrm{d}t \suchthat \gamma(0) = p, \gamma(1) = q}.\]Examples
Worked examples
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The north pole $p = (0, 0, 1)$ and the point $q = (1, 0, 0)$ on the equator of the unit sphere in $\reals^3$
: $\frac{\pi}{2} \approx 1.57$
High school
The geodesic distance on the unit sphere from the north pole to a point on the equator is the length of a quarter of a great circle, $\frac{2 \pi}{4} = \frac{\pi}{2}$.
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The points $p = (0, 0)$ and $q = (100, 100)$ in $\reals^2$
: $100 \sqrt{2}$
High school
In the plane, the geodesic distance between $p$ and $q$ equals the Euclidean distance, because the straight segment from $p$ to $q$ stays in the plane.
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The points $p = (0, 0)$ and $q = (2, 0)$ in $\reals^2$
: $2$
High school
In the plane, the geodesic distance between $p$ and $q$ equals the Euclidean distance, because the straight segment from $p$ to $q$ stays in the plane.
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The points $p = (1, 1)$ and $q = (4, 5)$ in $\reals^2$
: $5$
High school
In the plane, the geodesic distance between $p$ and $q$ equals the Euclidean distance, because the straight segment from $p$ to $q$ stays in the plane.
Properties
- The geodesic distance is at least the Euclidean distance through the surrounding space, because a straight segment is the shortest of all paths, including those that leave the surface.
- In the plane (or in $\reals^n$), the geodesic distance is the Euclidean distance, because the straight segment stays in the space.
- A shortest path is called a geodesic. Geodesics need not be unique: every half great circle from the north pole to the south pole is a shortest path.
Distinctions involving this term
- Types of distances: A metric measures how far apart two points are, and the same two points can be at different distances under different metrics. The Euclidean distance is the straight-line distance, the taxicab distance travels along the axes, and the Chebyshev distance takes the largest difference in one coordinate. The geodesic distance is the length of the shortest path that stays on a surface, such as the great-circle distance on the Earth. The discrete metric only says whether two points are equal.