How to orbit around the Z-axis in 3D

Question

Hello folks, I'm primarily a Flash AS3 dev, but I'm jumping into openframeworks and having trouble using 3D (these examples are in AS)

In 2D you can simulate an object orbiting a point by using Math.Sin() and Math.cos(), like so

function update(event:Event):void
{
    dot.x = xCenter + Math.cos(angle*Math.PI/180) * range;
    dot.y = yCenter + Math.sin(angle*Math.PI/180) * range;
    angle+=speed;
}

I am wondering how I would translate this into a 3D orbit, if I wanted to also orbit in the third dimension.

function update(event:Event):void
{
    ...
    dot.z = zCenter + Math.sin(angle*Math.PI/180) * range;
    // is this valid?
}

An help is greatly appreciated.

Answer 1

I would pick two perpendicular vectors v, w that define the plane in which to orbit, then loop over the angle and pick the proper ratio of these vectors v and w to build your vector p = av + bw.

More details are coming.

EDIT:

This might be of help

http://en.wikipedia.org/wiki/Orbit_equation

EDIT: I think it is actually

center + sin(angle) * v * radius1 + cos(angle) * w * radius2

Here v and w are your unit vectors for the circle.

In 2D they were (1,0) and (0,1).

In 3D you will need to compute them - depends on orientation of the plane.

If you set radius1 = radius 2, you will get a circle. Otherwise, you should get an ellipse.

Answer 2

If you are orbiting around Z axis, then you just do your first code, and leave Z coordinate as is.

Answer 3

If you just want the orbit to happen at an angled plane and don't mind it being elliptic you can just do something like z = 0.2*x + 0.2*y, or any combination you fancy, after you have determined the x and y coordinates.

Answer 4

If you are orbiting around the z-axis, you are leaving your z-coordinate fixed and changing your x- and y-coordinates. So your first code sample is what you are looking for.

To rotate around the x-axis (or y-axes), just replace x (or y) with z. Use Cos on whichever axis you want to be 0-degrees; the choice is arbitrary.

If what you actually want is to orbit an object around a point in 3d-space, you'll need two angles to describe the orbit: its elevation angle and its inclination angle. See here and here.
For reference, those equations are (where θ and φ are your angles)

x = x₀ + r sin(θ) cos(φ)
y = y₀ + r sin(θ) sin(φ)
z = z₀ + r cos(θ)