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Given a triangle formed by vertices $v_{0},v_{1},v_{2}$, a point inside of it, $P$, with barycentric coordinates $u,v,w$ and the sub-triangle $S$ formed by the midpoint of triangle edge ($m_{01},m_{12},m_{02}$):

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What are the barycentric coordinates of $P$ relative to the subtriangle $S$? I.e., is there a simple formula to compute them given the midpoint formulation of the sub-triangle?

Xaldew
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1 Answers1

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It is straightforward to show that the coordinates of the triangle vertices and the midpoints of its edges are related by $v_0 = m_{01} + m_{02} - m_{12}$ (similarly for other vertices). Therefore, if $P = uv_0 + vv_1 + wv_2$, then $P = (u+v-w)m_{01} + (u+w-v)m_{02} + (v+w-u)m_{12}$.

If $u+v+w = 1$, then $(u+v-w) + (u+w-v) + (v+w-u) = 1$ as well.

Answer Provider
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