Euler's Formula for Curvature

Differential Geometry · Axiom Academy

LESSON Euler's Formula for Curvature Understanding how normal curvature varies with direction on a surface 1. Principal Directions and Curvatures At each point on a surface, there exist two special orthogonal directions called principal directions where the normal curvature achieves its maximum and minimum values. These principal directions form a natural coordinate system on the surface. Any other direction can be described by an angle θ measured from the first principal direction. 2. Euler's Formula for Normal Curvature Euler discovered that the normal curvature in any direction θ can be expressed as a weighted combination of the principal curvatures. This elegant formula shows that: When θ = 0° (first principal direction): κ(0) = κ₁ When θ = 90° (second principal direction): κ(90°) = κ₂ For intermediate angles, the curvature smoothly interpolates between κ₁ and κ₂ 3. Derivation Using Principal Directions To understand why Euler's formula works, consider a unit tangent vector v making angle θ with the first principal direction e₁. The normal curvature is given by the second fundamental form: κ(θ) = II( v , v ). Since e₁ and e₂ are principal directions, the second fundamental form is diagonal: Expanding this quadratic form yields Euler's formula directly. 4. How Curvature Varies Between Extremes The function κ(θ) describes a smooth variation as θ ranges from 0° to 360°. The pattern depends on the relative magnitudes of κ₁ and κ₂.

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