Calculating rotation along a path

Question

I am trying to animate an object, let\'s say its a car. I want it go from point

x1,y1,z1

to point x2,y2,z2 . It moves to those points, but

Answer 1

Always pointing the car toward the destination point is simple and cheap, but it won't work if the car is following a curved path. In which case you need to point the car along the tangent line at its current location (see other answers, above).

Answer 2

You need to work out the initial orientation of the car, and the final orientation of the car at its destination, then interpolate between them to determine the orientation in between for the current timestep.

This article describes the mathematics behind doing the interpolation, as well as some other things to do with rotating objects that may be of use to you. gamasutra.com in general is an excellent resource for this sort of thing.

Answer 3

I think interpolating is giving the drift you are seeing. You need to model the way steering works .. your update function should 1) move the car always in the direction of where it is pointing and 2) turn the car toward the current target .. one should not affect the other so that the turning will happen and complete more rapidly than the arriving.

Answer 4

going from one position to another gives an object a velocity, a velocity is a vector, and normalising that vector will give you the direction vector of the motion that you can plug into a "look at" matrix, do the cross of the up with this vector to get the side and hey presto you have a full matrix for the direction control of the object in motion.

Answer 5

First off how do you represent the road?

I recently done exactly this thing and I used Catmull-Rom splines for the road. To orient an object and make it follow the spline path you need to interpolate the current x,y,z position from a t that walks along the spline, then orient it along the Frenet Coordinates System or Frenet Frame for that particular position.

Basically for each point you need 3 vectors: the Tangent, the Normal, and the Binormal. The Tangent will be the actual direction you will like your object (car) to point at.

I choose Catmull-Rom because they are easy to deduct the tangents at any point - just make the (vector) difference between 2 other near points to the current one. (Say you are at t, pick t-epsilon and t+epsilon - with epsilon being a small enough constant).

For the other 2 vectors, you can use this iterative method - that is you start with a known set of vectors on one end, and you work a new set based on the previous one each updateframe() ).

Answer 6

In general terms, the direction the car is pointing is along its velocity vector, which is the first derivative of its position vector.

For example, if the car is going in a circle (of radius r) around the origin every n seconds then the x component of the car's position is given by:

x = r.sin(2πt/n)

and the x component of its velocity vector will be:

vx = dx/dt = r.(2π/n)cos(2πt/n) 

Do this for all of the x, y and z components, normalize the resulting vector and you have your direction.