I'm new to ng posting, so let me know if i screw something up.  :)

Long background description of the project.  If interested read it :)
otherwise I have a math question at the bottom.

I've been working on a system for scripting car-motion.  It's
geometry-based.  ... Based on the idea that when turning, the car will
revolve around some point which is located at the intersection of where the
normals (if extended) of all four wheels interrsect.   Each wheel is then
tangent to the overall axis of rotation and... tracks along the ground
rather than skids.  That point of course always be inline with the back
axle.

The steering angle of the car is defined by an imaginary middle front wheel.
If it is say... 45 degrees left, the angle of the left and right front
wheels are calculated separately.  The wheel on the inside of the turn will
be something steeper than 45 and vice versa for the wheel on the outside of
the turn.

This approach is very geometry based... as opposed to an approach involving
a simple physics model.   (I know of at least one commercial package for
animators which models forces and is specifically for people who want to
animate vehicles.)  But that's what I intended it to be.   I was shooting
for something backed by nice clean math....  (...in theory....however pretty
or ugly the the actual code looks is another story. :)    So far so good.

I wanted accelleration and smooth steering-angle change.   So I worked on a
simple scripting system where you can tell it initial speed, speed delta,
initial steering angle and steering delta.   ...And how long to do it.   Any
time one of the variables changes, it defines a new motion segment.
Realistic car motion can (well hopefully) be created by stringing together a
sequence of segments.

The problem is in the iterative approach calculating each segment.   I use
macros to move the car through many small movements for different steering
angles if the steering delta is non-zero.   (The final position of the car
is easy to calculate in one step as long as the steering angle for the
segment is constant.)  When an animation involves a long list of motion
segments, recalculating the movement of the car from the beginning for each
frame puts a real parsing burdon on  my almost antique computers. :)

So now I have a system that pre-processes all of the motion segments and
saves the results in a file.  For each frame only one motion segment (or
partial motion segment) will need to be calculated.  This works great for
final rendering but I still need to work on a generalized system for when
I'm doing test renders.  (I.e. one that can separately remember individual
segments rather than pre-calculating The Whole Thing.)

************************************************************************

In the spirit of clean mathematical models, I really want to do each motion
segment in one step using calculus.

I figured there are seven integrals I need to find:

*The first one gives the cumulative of the rate of change of the direction
the car is facing... and thus gives the final direction the car is facing
for the given segment.

*The second and third integrals find the x and y locations of the car.  For
the x position, take the cosine (or sine for y position) of the first
integral, multiply it by the current speed as speed changes.   Take the
integral of this (in terms of time of course).

*The other four integrals are of the speeds of each of the wheels.  Knowing
how far each wheel has travelled we know how many times each wheel has
turned.



The first integral is do-able.   The second and third might not be.  (4,5,6
&7 I haven't bothered with yet... at least they're not all completely
different....)   I'm having trouble integrating anything with a trig
function within a trig function...  Say  the integral of:
    cos( sin(x) )  dx
Seems like everything I try leads me in circles.
If it were something like:
    cos( sin(x) )  * cos(x)  dx
it would be easy with substitution.... but unfortunately...   :)

I've been looking for a way I can simplify the actual problem so that the
trig-within a trig function goes away... or something...

If mathematica can't do it, is that a bad sign?  I threw a few things into
the online mathematica integrator. http://integrals.wolfram.com/
Unfortunately it wasn't able to do them.

Help?  Suggestions?   Is there a way to approximate these integrals w/o
iteration?

-Charles

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