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
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
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.
As of this snapshot date, this entry was owned by rspuzio.