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Triangles, Circles and Spheres

The ‘real world` has provided precious few opportunities for musings. The following is motivated by a question on Mathematica Stackexchange.

The following code aims to find the circumcentre of a triangle defined by 3 points in R^3.

cc[{p_, q_, r_}] := Module[
{v1, v2, b1, b2, norm, n1, n2, cntr, rad, un},
v1 = Normalize@(q - p);
v2 = Normalize@(r - p);
b1 = (p + q)/2;
b2 = (r + p)/2;
norm = Cross[v1, v2];
n1 = Normalize@Cross[norm, v1];
n2 = Normalize@Cross[norm, v2];
cntr = b1 + s n1 /. First@Solve[b1 + s n1 == b2 + t n2, {s, t}];
rad = Norm[p - cntr];
un = Normalize@Cross[norm, v1];
{cntr, rad, v1, un}
]

plcc[{p_, q_, r_}] := Module[{ctr, radius, un1, un2, p1, p2},
{ctr, radius, un1, un2} = cc[{p, q, r}];
p1 = Graphics3D[{{Black, PointSize[Large],
Point@{p, q, r}}, {Polygon[{p, q, r}]}, {Red, PointSize[Large],
Point@ctr}, {Opacity[0.3], Sphere[ctr, radius]}}];
p2 = ParametricPlot3D[
ctr + radius Cos[t] un1 + radius Sin[t] un2, {t, 0, 2 Pi}];
Show[p1, p2, Axes -> True]]

Testing on 100 random triples:

circumcenter

Peace to all…

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Categories: Mathematica, Mathematics
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