Regular curves

A regular curve has a nonzero velocity, allowing local tangent geometry.

Different ways to understand it

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Arc length
Integrating speed measures a sufficiently regular curve's length.
Coordinate geometry
Coordinates translate geometric conditions into equations and algebraic calculations.
Regular surfaces
Regular surface charts have linearly independent coordinate tangent vectors.
Tangent spaces
Tangent vectors at a point form a linear space describing infinitesimal directions.
Covariant derivatives
A connection differentiates vector fields while accounting for changing tangent spaces.