Suppose I have a triangle defined by 3 unit vectors {$v_1, v_2, v_3$} in a 3 dimensional complex inner product space.

What would be the centre of such a triangle? I guess it should be something like $(v_1+v_2+v_3)/3$.

How about if the 3 vectors define a spherical triangle on the unit sphere? Is it just a case of taking the vector in the paragraph above and projecting it onto the unit sphere? Are there any other possible candidates which I imagine may make use of the inner products of the 3 vectors?

Thanks for your help,


EDIT ----

So to perhaps make my question clearer, how would I go about finding the incenter of a spherical triangle?

  • $\begingroup$ Note that three such vertices define two spherical triangles. Alternatives are possible but depend on your intention, just as you have in the planar case e.g. the center of circumscribed circle, the center of the inscribed circle and so on. $\endgroup$ – Hagen von Eitzen Jan 21 '13 at 17:05
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    $\begingroup$ Related to Calculating a spherical polygon centroid at the GIS SE. $\endgroup$ – MvG Jan 22 '13 at 6:26
  • $\begingroup$ @HagenvonEitzen. So say I wanted to calculate the center of the inscribed circle. Would this then be $[|v_2-v_3|v_1+|v_1-v_3|v_2+|v_2-v_1|v_3]$ (normalised to ensure it lies on the unit sphere)? Would I be right in thinking that projecting back through the origin would give the center for the other spherical triangle that's been defined? $\endgroup$ – Stan Jan 22 '13 at 13:57

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