Vectors and angles and math, oh my

Factual Questions
1

This problem is a bit esoteric, and hard to explain without pictures, but I’ll give it my best shot.

We are given a plane in 3-space, specified by its normal vector and a point that it contains. An object-- say, a car-- is sitting upon the plane, such that its nose is pointing at a given compass heading.

What we want to find is the orientation of this object, using Euler angles-- roll, pitch, yaw. As noted above, the yaw is a given. How do we calculate an appropriate roll and pitch that allow the car to sit flush against the plane?

2

Looks to me as if the equations at ROTATION SEQUENCES AND EULER ANGLES might be useful; he even solves some of the systems.

3

Thanks for the link, but it’s a lot more general than what I’m looking for. I’m trying to find a specific algorithm, but it’s not

I’ve already coded the stuff that handles most everything to do with rotation. It all works as long as I have something that completely specifies the 3d rotation-- a matrix, a quaternion, a set of Euler angles, whatever.

The problem is that, in the situation I specified above, only the heading (yaw) is known. I have the normal vector of the plane I’m parallel to, but I have not been able to correctly use that to extract pitch and roll.

Come on, people… just because I’ve been a non-posting lurker for a few months is no reason to abandon me in my time of need! =b^)

4

Why don’t you just use the heading of the car on the plane and the spherical coordinate angles of the normal vector? I’m not sure if those work out to a standard set of Euler angles, but they’d be equivalent. Or are you looking for the formulae for converting a Cartesian vector to spherical?

5

Never mind, I’ve got a correct solution now; it’s decidedly sub-optimal, but it does work. I do cross products to find a pair of orthogonal vectors in the horizon plane, and the rotated vectors in the terrain plane; from there, obtaining each angle just involves a dot product and an arc-cosine.

If anyone’s interested, I’ll post the algorithm on the web someplace.

6

Never mind, I’ve got a correct solution now; it’s decidedly sub-optimal, but it does work. I do cross products to find a pair of orthogonal vectors in the horizon plane, and the rotated vectors in the terrain plane; from there, obtaining each angle just involves a dot product and an arc-cosine.

If anyone’s interested, I’ll post the algorithm on the web someplace.