How to get circle points in 3d given a radius and a vector orthogonal to the circle area?

Question

I already know how to get a point on a circle (here), but I need a circle in 3d which should be the orthogonal to a given vector.

I got:

• Angle in degree/radians
• Circle radius
• Orthogonal vector

I think, I need to rotate the 2d circle positions to be orthogonal to the given vector, but I do not how how to do that.

Answer 1

Let us denote the angle by $\theta$, the radius by $r$ and the orthogonal vector by $\vec a$ ("axis"); furthermore let us assume the length of $\vec a$ is 1.

First you must define which point is defined by $\theta=0$. If $\vec a$ is not parallel to the $xy$-plane then there is a unique vector $\vec v_0 = (x_0\, 0\, z_0)$ orthogonal to $\vec a$ with length $r$ and $x_0 > 0$. (Otherwise you need a different choice for $\vec v_0$. In 3D there is no obvious way to define such a $\vec v_0$ for any $\vec a$.)

Then all you need to do is rotate the vector $\vec v_0$ around $\vec a$ by the angle $\theta$. The formula is: $$ \cos \theta\ \vec v_0 + (1-\cos \theta)(\vec a \cdot \vec v_0)\vec a + \sin \theta\ \vec a \times \vec v_0$$