Given a point on a unit radius sphere (let's say at coords x=0, y=0.4, z=-0.9165 as shown in the attached diagram), and given a starting angle from 0 to 360 for which the point to travel, what math can I use to exactly 'cut' the sphere in half? For example, the attached diagram below shows the sphere cut exactly in half in two different ways according to the starting direction (blue line about 30 degrees (anti-clockwise from 3 o'clock), and orange line about 75 degrees). So input is: x, y, z, starting angle direction, and distance travelled around the sphere (as an angle). And output is newx, newy, and newz. What equation can I use?