Thank you for those formulas. Even though my University says I've passed two
terms of Calculus, I've had some difficulty differentiating the bezier spline
formula myself.

Josh

John VanSickle wrote:

> Peter Popov wrote:
> >
> > On Fri, 31 Mar 2000 14:34:48 -0800, Josh English
> > <eng### [at] spiritonecom> wrote:
> >
> > >I am continuing to develop my bezier spline macros, but I am currently
> > >having a problem with banking the object to have a realistic motion
> > >along the curve. Does anyonw have any suggestions?
> >
> > Take the second derivative of the equation of motion you are using and
> > use (scaled to a proper range of course) to rotate the object around
> > the axis formed by the first derivative of aforementioned function. I
> > hope I know what I am talking about at this time of night :)
>
> Close enough for government work.  Take the 2nd derivative of the motion,
> which will be a vector inward of the curve.  From this subtract a vector
> representing the force of gravity; since gravity is down, the result will
> be mostly up, tilted by the amount of acceleration.  Normalize the
> result.  That's the new "up" vector for the object.
>
> Since a bezier spline uses the equation
>
> P(u)=(1-u)^3 * P_0 + 3*u*(1-u)^2 * P_1 + 3*u^2*(1-u) * P_2 + u^3 * P_3
>
> The first derivative ought to be
>
> dP/du= (-3+6*u-3*u^2) * P_0 + (3-12*u+9*u^2) * P_1
>   +(6*u-3*u^2)*P_2 + 3*u^2 * P_3
>
> and the second ought to be
>
> d^2P/du^2 = (6-6*u)* P_0 + (-12+18*u) * P_1 + (6-6*u) * P_2 + 6*u * P_3
>
> Now to get the object placed and banked, calculate the position along
> the spline, and the first two derivatives, and place them in the variables
> Location, Velocity, and Acceleration.
>
> The velocity becomes your object's forward vector:
>
> #local VecF=vnormalize(Velocity);
>
> From the acceleration, you require only the portion that is perpendicular
> to the direction of motion:
>
> #local VecA=Accleration-VecF*vdot(VecF,Accleration);
>
> From this, subtract the vector for gravity and normalize the result:
>
> #local Gravity=<0,-9.8,0>; // gravity is down at 9.8 m/sec^2
> #local VecU=vnormalize(VecA-Gravity);
>
> Then you get your right-hand vector for a full transformation:
>
> #local VecR=vnormalize(vcross(VecU,VecF));
>
> And then recalculate the up vector (in case the forward motion is
> going up or down a slope):
>
> #local VecU=vnormalize(vcross(VecF,VecR));
>
> and then invoke it wall with a matrix transform.  If your object
> is modeled so that z is forward and y is up,
>
> matrix < VecR.x, VecR.y, VecR.z,
>          VecU.x, VecU.y, VecU.z,
>          VecF.x, VecF.y, VecF.z,
>          Location.X,Location.Y,Location.Z >
>
> Hope this helps (and isn't full of bugs),
> John
> --
> ICQ: 46085459

--
Josh English
eng### [at] spiritonecom
"May your hopes, dreams, and plans not be destroyed by a few zeros."

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