Problem:
Given is is the centerpoint $C$ of a circle and two extra reference points $R$ and $E$, which are laying on the circle. While as the line $\overline{CE}$ is defining the zero point on the circle of a given degree $α$ (clockwise rotation). Point $F$ is also on the very same circle. Therefore all of the points are on one in space rotated plane, on the circle. Line $\overline{CF}$ is $α$ degrees away from line $\overline{CE}$. How can I get the XYZ coordinates of Point $F$? ($R ≠ E$) Fig. 1
Fig. 1: Illustration of described problem
- Note that $\overline{CR}$ is NOT necessarily parallel to $\overline{CF}$
- $\overline{CR}$ to $\overline{CE}$ not necessarily orthogonal
- all Points lay in one 3d circular plane
- in Fig. 1 a second circle is drawn orthogonal to the XY Plane just for better relation understanding (not a projection)
- The points $E$, $C$ and $R$ are defining the circles rotation in space, while $F$ is the point to move around on this circle (with a specific degree $\alpha$)
Hint:
In 3D there can be infinite points like F such that $\angle ECF=\alpha$ and also $\angle RCF=\beta$. They are on two conic surface vertex on C with axis CE and CR respectively. The intersection of these two surfaces makes two lines. Any point like F on these lines is such that $\angle ECF=\alpha$ and $\angle RCF=\beta$.So you have to find equations of these conic surfaces.For this take for example xy plane. Draw a line angle with EC as $\alpha$ . Now revolve it about CE and find the equation of produced surface. Now draw a line which make angle $\beta$ with RC and revolve it about RC anf find it' equation. The system of thses two equations give the equations of lines resulting by intersection.Now to find coordinates of F you solve this system of equations:
$\begin{cases} y=mx \\x^2+y^2=r^2 \end {cases}$
Where $m=tan (\alpha)$ and r is length of CF. So in adition to the angle you have you must also have r.Find x or y from first equation, put it in second equation and find x and y of point F.