Given a geodesic curve, an affine parameterization for that curve is a parameterization by a parameter t such that the parametric equations for the curve satisfy the geodesic equation.

Put another way, if one picks a parameterization of a geodesic curve by an arbitrary parameter s and sets u^μ = dx^μ∕ds, then we have

for some function f. In general, the right hand side of this equation does not equal zero — it is only zero in the special case where t is an affine parameter.

The reason for the name “affine parameter” is that, if t₁ and t₂ are affine parameters for the same geodesic curve, then they are related by an affine transform, i.e. there exist constants a and b such that

Conversely, if t is an affine parameter, then at + b is also an affine parameter.

From this it follows that an affine parameter t is uniquely determined if we specify its value at two points on the geodesic or if we specify both its value and the value of dx^μ∕dt at a single point of the geodesic.