Given a geodesic curve, an affine parameterization for that curve is a parameterization by a parameter 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 and sets
, then we have
for some function. In general, the right hand side of this equation does not equal zero -- it is only zero in the special case where is an affine parameter.
The reason for the name “affine parameter” is that, if and are affine parameters for the same geodesic curve, then they are related by an affine transform, i.e. there exist constants and such that
Conversely, if is an affine parameter, then is also an affine parameter.
From this it follows that an affine parameter 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
at a single point of the geodesic.
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