How to Make a Point Orbit a Line, 3D [closed]

StackOverflow https://stackoverflow.com/questions/22926361

  •  29-06-2023
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Question

Essentially, i want to find the equation of the ring that circles a point in space, this ring is perpendicular to a normal away from this point.

I have a line, in the form of 2 points, L1, L2;

I have the normal from L1->L2, N;

I have the plane that is normal to L1->L2, which L1 lies on. ax + by + cz = d;

I have the radius away from L1, R;

-> I want to make a point V, orbit this line around point L1;

I think I have to make a circular equation in this plane with L1 as the origin. I have no idea how to plot a 2d equation onto a 3d plane.

Or maybe someone knows how to do do this another way, cross products or something?

Was it helpful?

Solution

This problem actually requires a nontrivial solution. Suppose you have U = normalize(L2 - L1) and two unit vectors V and W such that U, V, W are pairwise orthogonal.

Then f(a) = L1 + R * (V * cos(a) + W * sin(a)) for angles a is the equation for the circle you want.

How can you find W given U and V? W can just be their cross product.

How can you find V given U? This is where it's not straightforward. There are a whole circle of such V that could be chosen, so we can't just solve for "the" solution.

Here's a procedure for finding such a V. Let U = (Ux, Uy, Yz).

If Ux != 0 or Uy != 0, then V = normalize(cross(U, (0,0,1)))
Else if Ux != 0 or Uz != 0, then V = normalize(cross(U, (0,1,0)))
Else U = 0, error

Note: You can negate W if you want your point to cycle in the opposite direction.

OTHER TIPS

You may use Rodrigues' Rotation Formula (try to find better description)

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