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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 $\alpha$)
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  • $\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$
    – José Carlos Santos
    Commented Dec 5, 2020 at 8:09
  • $\begingroup$ @JoséCarlosSantos Thanks! $\endgroup$
    – Paul
    Commented Dec 5, 2020 at 8:15
  • $\begingroup$ I think you must also have angle between RC and CF. $\endgroup$
    – sirous
    Commented Dec 5, 2020 at 9:30
  • $\begingroup$ @sirous Thank you for your suggestion. But I think you can calculate with the cosine Law!? Not necessarily correct. $\endgroup$
    – Paul
    Commented Dec 5, 2020 at 10:25
  • $\begingroup$ @sirous But do you know how to do it then? I am really out of ideas right now :( $\endgroup$
    – Paul
    Commented Dec 5, 2020 at 12:13

2 Answers 2

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enter image description here

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.

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  • $\begingroup$ Thank you very much for your input. I am really trying to understand what the illustration should display. Why are there cones around $CR$ and $CE$? And not ONE circle that is connecting $R$, $E$, $C$ and $F$? As all Point should lay in one plane right? $\endgroup$
    – Paul
    Commented Dec 5, 2020 at 16:47
  • $\begingroup$ If E, R and c are on one plane then F can not be on that plane. Otherwise there will a 2D orientation. $\endgroup$
    – sirous
    Commented Dec 5, 2020 at 18:11
  • $\begingroup$ I think it can. There might be a misunderstanding of what the problem is (most likely my mistake though). The Circle is defined by three points in space with c the middle point. Therefore those 3 points must laying in a rotated plane in space(defined by the circle). Now additionally one point on this -in space rotated- circle is taken. All points are laying on one rotated plane. This point isn't just randomly put on the circle. It can be found by an angle from a reference line created by the centerpoint and a reference point (both on the circle, as they were used to create the circle). $\endgroup$
    – Paul
    Commented Dec 5, 2020 at 18:50
  • $\begingroup$ Sorry about misleading instructions. I would be really happy if you could tell what to do better in the future. But is it now more clearly or still not understandable? Should I upload a overworked graphical visualisation? $\endgroup$
    – Paul
    Commented Dec 5, 2020 at 18:54
  • $\begingroup$ @Paul, I edited my answer and showed how to find coordinates of F.Good luck. $\endgroup$
    – sirous
    Commented Dec 6, 2020 at 6:49
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I am going to answer how such a problem can be solved in a different way:

All lines to the points must be converted to vectors. A vector axis perpendicular to the circles face must be created. Then the vector of $CE$ is taken as the vector to rotate around the created axis vector, by theta degrees(counter clockwise rotation). This rotation and the vector of the new position can be calculated with the Rodrigues' rotation formula! See also: Quaternion rotation

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