Ambient acceleration criterion for a surface geodesic

For a smoothly parametrized curve on an embedded surface in Euclidean space, the geodesic equations hold exactly when the ambient acceleration is normal to the surface.

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- Constant speed of an affinely parametrized geodesic Ambient acceleration criterion for a surface geodesic

Constant speed of an affinely parametrized geodesic

The velocity of a surface curve is tangent. If its acceleration is normal, then

\frac{d}{d t} ∣ \overset{γ}{˙} ∣^{2} = 2 \overset{γ}{˙} \cdot \overset{γ}{¨} = 0,

so every affinely parametrized geodesic has constant speed.

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