Function Repository Resource:

IsogonalConjugate

Get the isogonal conjugate of a point with respect to a triangle or tetrahedron

Contributed by: Ed Pegg Jr

ResourceFunction["IsogonalConjugate"][simplex,p]

gives the isogonal conjugate of point p with respect to the triangle or tetrahedron simplex.

Details

The simplex may be a triangle defined by three 2D points, a tetrahedron defined by four 3D points or an n-simplex defined by n+1 points.

Given a triangle ΔABC, the isogonal conjugate of P is constructed by reflecting the lines PA, PB and PC about the angle bisectors and taking the intersection.

Given a tetrahedron ABCD, the isogonal conjugate of P is constructed by reflecting the lines PA, PB, PC and PD about the angle bisectors of the dihedral edges and taking the intersection.

In triangle ΔABC, the trilinear coordinates of a point are the ratio a:b:c of signed distances to the sides. The trilinear coordinates of the isogonal conjugate are the ratio a^-1:b^-1:c^-1.

In tetrahedron ABCD, the barycentric coordinates of a point are the ratio of signed volumes of the four tetrahedra points by replacing a vertex with the point a:b:c:d. Let ΔA be the squared face area of ΔBCD. The barycentric coordinates of the isogonal conjugate are then ΔA/a:ΔB/b:ΔC/c:ΔD/d.

Examples

Basic Examples (2)

Compute the isogonal conjugate of a point with respect to a given triangle:

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Find the isogonal conjugate of a point with respect to a triangle:

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tri = {{7, 2}, {4, 5}, {0, 1}};
p = {3, 3};
i = ResourceFunction["IsogonalConjugate"][tri, p]

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The isogonal conjugate of the isogonal conjugate is the original point:

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Show the triangle, point (green) and isogonal conjugate (blue):

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The centers of the incircle and excircles are the only four self-conjugate points:

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The product of trilinear coordinates for a point and isogonal conjugate gives three identical values:

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baryP = ResourceFunction["BarycentricCoordinates"][tri, p];
baryI = ResourceFunction["BarycentricCoordinates"][tri, i];
sides = Reverse[EuclideanDistance @@ # & /@ Subsets[tri, {2}]];
trilinearP = (baryP/sides);
trilinearI = (baryI/sides);
trilinearP trilinearI

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Scope (3)

Compute the isogonal conjugate of a Point with respect to a Triangle:

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Compute the isogonal conjugate of a Point with respect to a Tetrahedron:

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Compute the isogonal conjugate of a 5D point with respect to a simplex:

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Find the Monge point of a semi-random tetrahedron:

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randomtetra = Select[ResourceFunction["SignedPermutations"][12 {1, 1, 1}], EvenQ[Count[Sign[#], -1]] &] + RandomSample[Tuples[{1, 0, -1}, {3}], 4];
mongefull = ResourceFunction["Monge"][randomtetra];
monge = First[mongefull]

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Find the isogonal conjugate of the Monge point:

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The circumsphere of the face reflections of a point is centered at the isogonal conjugate (and vice-versa):

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reflectM = Flatten[ResourceFunction["ReflectPoints"][#, {monge}] & /@ Subsets[randomtetra, {3}], 1];
reflectI = Flatten[ResourceFunction["ReflectPoints"][#, {isog}] & /@ Subsets[randomtetra, {3}], 1];
Graphics3D[{Sphere[#, 1] & /@ randomtetra,
{Red, Sphere[monge, 1], Sphere[#, 1/2] & /@ reflectM}, {Green, Sphere[isog, 1], Sphere[#, 1/2] & /@ reflectI},
Opacity[.5], Tetrahedron[randomtetra], Opacity[.2],
Circumsphere[reflectM], Circumsphere[reflectI]}, Boxed -> False]

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Neat Examples (2)

For a tetrahedron, calculate the five insphere and exsphere points:

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These five points are all equal to their isogonal conjugates:

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Three other points are their own isogonal conjugate in a tetrahedron:

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Show the 16 collinear sets of three points formed by four tetrahedral points and eight self-conjugates:

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pp = Join[tetra, five, same];
lines = ResourceFunction["FindExtraordinaryLines"][pp];
ordinary = Complement[Subsets[Range[12], {2}], Flatten[Subsets[#, {2}] & /@ lines, 1]];
Graphics3D[{Sphere[#, 40] & /@ pp, Line[pp[[#]]] & /@ lines, Opacity[.5], Tetrahedron[tetra], Green, InfiniteLine[pp[[#]]] & /@ ordinary}, Boxed -> False]

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Find intersection points and repeat to obtain 50 lines:

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other = Complement[First /@ Select[RegionIntersection /@
Subsets[
Join[InfiniteLine[pp[[#]]] & /@ ordinary, InfiniteLine[pp[[#]]] & /@ lines], {2}], Head[#] === Point &], pp]; qq = Join[pp, other];
lines2 = ResourceFunction["FindExtraordinaryLines"][qq];
Graphics3D[{Sphere[#, 40] & /@ qq, Line[qq[[#]]] & /@ lines2, Opacity[.5], Tetrahedron[tetra]}, Boxed -> False, SphericalRegion -> True]

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The following triangle has nice Euler line points for the circumcenter, centroid, nine-point center and orthocenter:

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The Jerabek hyperbola is the locus of the isogonal conjugates of points on the Euler line. Use GroebnerBasis to derive the implicit equation:

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$jerabek = GroebnerBasis[{\[FormalX], \[FormalY]} == ResourceFunction["IsogonalConjugate"][ tri, (1 - \[FormalU]) euler[[ 1]] + \[FormalU] euler[[-1]]], {\[FormalX], \[FormalY]}, \[FormalU]][[1]]$

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The isogonal conjugates of the Euler points are the orthocenter, symmedian, Kosnita point and circumcenter:

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A graphic of the triangle with the Euler line (blue), circumcenter|circumcircle|perpendicular bisectors (red), centroid|medians (cyan), nine-point center|circle (brown) and orthocenter|altitudes (green) and the isogonal conjugate of the Euler line, the Jerabek hyperbola:

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$mids = Mean /@ Subsets[Reverse[tri], {2}]; otherthree = Mean[{#, euler[[4]]}] & /@ tri; Legended[ Graphics[{EdgeForm[Black], White, Thick, Triangle[tri], Blue, InfiniteLine[Take[euler, 2]], Thin, Red, InfiniteLine[{euler[[1]], #}] & /@ mids, Circumsphere[tri], Disk[euler[[1]], 0.2], Cyan, InfiniteLine /@ Transpose[{tri, mids}], Disk[euler[[2]], 0.2], Brown, Circle[euler[[3]], EuclideanDistance[euler[[3]], mids[[1]]]], Disk[#, 0.1] & /@ otherthree, Disk[euler[[3]], 0.2], Green, InfiniteLine[{euler[[4]], #}] & /@ tri, Disk[euler[[4]], 0.2], ContourPlot[ jerabek == 0 // Evaluate, {\[FormalX], -15, 13}, {\[FormalY], -14, 14}][[1]]}], PointLegend[{Red, Cyan, Brown, Green}, {"circumcenter", "centroid", "nine-point center", "orthocenter"}, LegendMarkers -> ConstantArray[Graphics[Disk[]], 3]]]$

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Version History

1.1.0 – 22 August 2022
1.0.0 – 19 August 2022

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License