Geodesic Triangles

Differential Geometry · Axiom Academy

Exploring how triangles behave on curved surfaces and what their angles reveal about curvature 1. Triangles with Geodesic Edges A geodesic triangle is formed by connecting three points on a surface using geodesics—the shortest paths between points. On a flat plane, geodesics are straight lines. On curved surfaces, geodesics follow the surface's natural geometry. The key property: each edge of a geodesic triangle is a locally distance-minimizing curve connecting two vertices. 2. Measuring Angles at Vertices At each vertex of a geodesic triangle, we measure the angle between the two incident geodesic edges. This angle is measured in the tangent plane at the vertex—the flat space that best approximates the surface at that point. Let the three interior angles be denoted α, β, and γ. The sum of these angles reveals important information about the surface's curvature. On a flat plane (zero curvature), geodesic triangles are ordinary Euclidean triangles with straight edges. The fundamental theorem of Euclidean geometry states that the interior angles always sum to exactly π radians. This serves as our baseline: when the angle sum equals π, we have flat (Euclidean) geometry. 4. Positive Curvature: Sum > π On surfaces with positive curvature (like a sphere), geodesic triangles have angle sums greater than π . The excess angle above π is called the angular excess and is directly proportional to the triangle's area and the surface's curvature.

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