I have a sphere with its center at $(0,0,0)$ and 2 vectors $\vec a$ and $\vec b$ and both of them have a length of spheres radius

$\Vert \vec a \Vert = \Vert \vec b \Vert =R$

Angle between these vectors ($\alpha$) is also known.

If $\vec a$ circles around the $\vec b$ keeping the angle $\alpha$ unchanged we are kind of getting a cone inside the sphere like in this picture.

How can I calculate the coordinates of $\vec a$ if it would rotate $\beta$ degrees around the $\vec b$ ?

  • $\begingroup$ there exists a separate rotation matrix for this purpose in linear algebra. $\endgroup$ – tp1 Jun 19 '17 at 19:49

To make thinks simpler lets assume that this is a unit sphere.

$p = (a\cdot b) b$ is the projection of $a$ onto $b$

$v = a - (a\cdot b) b$ is orthogonal to $b$

$p+v = a$

$u = b\times a$ is orthogonal to both $v$ and $b$

$\|u\| = \|v\|$

the circle: $p+v\cos t + u\sin t$

  • $\begingroup$ Thanks for the answer man, can you explain what (a * b) b means? First part looks like a dot product? $\endgroup$ – etrusks Jun 19 '17 at 20:39
  • 1
    $\begingroup$ $a\cdot b$ is the Euclidean inner (dot) product. $(a\cdot b)b$ is the vector $b$ scaled by $(a\cdot b).$ $a\times b$ is the vector (or cross) product. $\endgroup$ – Doug M Jun 19 '17 at 20:40
  • $\begingroup$ Last equation represents resulting vector to the circle and t is same as my $\beta$? $\endgroup$ – etrusks Jun 19 '17 at 20:47
  • $\begingroup$ Thanks a lot man, I tested it out and it gives me nice results :) Very much appreciated :) $\endgroup$ – etrusks Jun 19 '17 at 20:59

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