Task Description
The SDL currently provides no way to compute the exact direction of a spline at a given location, even though mathematically this is a piece of cake: The firstorder derivative of any spline section gives you the “speed” as a vector function, and is trivial to compute for polynomial splines (which are behind all spline types that POVRay supports); the normalized “speed” vector, in turn, gives the “pure” direction.
For exact direction/speed computations, I propose to extend the SDL invocation syntax as follows to allow for evaluating a spline’s derivative:
SPLINE_INVOCATION:
SPLINE_IDENTIFIER ( FLOAT [, SPLINE_TYPE] [, FLOAT] )
or
SPLINE_INVOCATION:
SPLINE_IDENTIFIER ( FLOAT [, FLOAT] [, SPLINE_TYPE] )
where the second FLOAT will specify the order of derivative to evaluate (defaulting to 0). In order to compute the position, direction, and acceleration of an object traveling along a certain spline, one could then for instance use:
#declare S = spline { ... }
#declare Pos = S(Time);
#declare VSpeed = S(Time,1);
#declare VAccel = S(Time,2);
#declare Dir = vnormalize(VSpeed);
#declare Speed = vlength(VSpeed);
#declare AccelDir = vnormalize(VAccel);
#declare GForce = vlength(VAccel) / 9.81;
Alternatively, a mechanism may be devised to create a spline representing another spline’s derivative; however, it would be debatable whether the syntax should be parameterlike (being an added information that could be overridden again when creating other splines from such a derived spline), or operationlike (converting the spline), and in the latter case how it should affect spline type (and consequently control points); so the spline invocation parameter approach might be more straightforward, with less potential surprises for the user.
