Spherical triangle centers with vectors

02 May 2021

Tags: geometry , vectors , cartography

There are many triangle centers defined for Euclidean triangles. These triangle centers can also be defined in spherical geometry, although some Euclidean centers correspond to two or more spherical centers. See this paper for details. This post lists some spherical triangle centers with nice constructions in terms of vectors. Here’s an image of them together.

Some centers of a spherical triangle

Let the triangle have vertices \(\mathbf{\hat{a}}\), \(\mathbf{\hat{b}}\), and \(\mathbf{\hat{c}}\), oriented counterclockwise. See this earlier post for more details on spherical geometry using vectors.

Centroid (Vertex-Median point)

The simplest spherical triangle center is to take the Euclidean centroid of the points and then normalize onto the sphere. This is the point of intersection of medians.

\[\mathbf{\hat{z}} = \frac{\mathbf{\hat{a}} + \mathbf{\hat{b}} + \mathbf{\hat{c}}} {\sqrt{3 + 2~\mathbf{\hat{a}} \cdot \mathbf{\hat{b}} + 2~\mathbf{\hat{b}} \cdot \mathbf{\hat{c}} + 2~\mathbf{\hat{c}} \cdot \mathbf{\hat{a}}}}\]

The linked paper above calls this the vertex-median point, because not all the qualities of the Euclidean centroid carry over. For example, in spherical geometry, unlike Euclidean, the centroid is not (in general) the point that divides the triangle into three equal-area triangles.

The vertex-median point of a spherical triangle

Centroid (Area)

The point that divides the triangle into three equal-area triangles, as pictured, is more complicated to construct. See this separate post for details.

The area centroid of a spherical triangle

Circumcenter

The circumcenter of a Euclidean triangle is the point with equal distance to each vertex.

\[\mathbf{\hat{z}} = \frac{\mathbf{\hat{a}} \times \mathbf{\hat{b}}+ \mathbf{\hat{b}} \times \mathbf{\hat{c}} + \mathbf{\hat{c}} \times \mathbf{\hat{a}}} {\|\ldots\|}\]

The numerator is similar to the formula for the area of a Euclidean triangle as mentioned above, but more importantly it is the formula for the normal to a plane that passes through the three points.

The circumradius \(r\) is just the distance from \(\mathbf{\hat{z}}\) to any vertex, and is given by

\[\cos r = \frac{\left|\mathbf{\hat{a}}, \mathbf{\hat{b}}, \mathbf{\hat{c}} \right|} {\|\mathbf{\hat{a}} \times \mathbf{\hat{b}} + \mathbf{\hat{b}} \times \mathbf{\hat{c}} + \mathbf{\hat{c}} \times \mathbf{\hat{a}}\|}.\]
The circumcenter of a spherical triangle

Orthocenter

The orthocenter is the intersection of altitudes, extending the edges outside the triangle if needed. This seems to be the most likely of these centers to lie outside the triangle.

\[\mathbf{\hat{z}} = \frac{(\mathbf{\hat{c}} \cdot \mathbf{\hat{a}})(\mathbf{\hat{b}} \cdot \mathbf{\hat{c}}) \mathbf{\hat{a}} \times \mathbf{\hat{b}} + (\mathbf{\hat{a}} \cdot \mathbf{\hat{b}}) (\mathbf{\hat{c}} \cdot \mathbf{\hat{a}}) \mathbf{\hat{b}} \times \mathbf{\hat{c}} + (\mathbf{\hat{a}} \cdot \mathbf{\hat{b}}) (\mathbf{\hat{b}} \cdot \mathbf{\hat{c}}) \mathbf{\hat{c}} \times \mathbf{\hat{a}}}{\|\ldots\|}\]

Note that if any two edges have length \(\pi/2\), both these expressions are undefined. Geometrically, the altitude from the point where those two edges meet to the opposite edge is underspecified: every point on the opposite edge defines an altitude. Here are some sensible special definitions for that case:

The orthocenter of a spherical triangle

Incenter

The incenter is the intersection of the angle bisectors, and is given by:

\[\mathbf{\hat{z}} = \frac{\| \mathbf{\hat{b}} \times \mathbf{\hat{c}}\| \mathbf{\hat{a}} + \| \mathbf{\hat{c}} \times \mathbf{\hat{a}}\| \mathbf{\hat{b}} + \| \mathbf{\hat{a}} \times \mathbf{\hat{b}}\| \mathbf{\hat{c}}} {\|\ldots\|}\]

The incenter is also the center of a circle that is tangent to each edge, or the incircle. The radius of that circle is the inradius, and the points where it touches each edge are the intouch points. The inradius \(r\) is given by:

\[\sin r = \frac{|\mathbf{\hat{a}}, \mathbf{\hat{b}}, \mathbf{\hat{c}}|} {\begin{Vmatrix} \|\mathbf{\hat{b}} \times \mathbf{\hat{c}}\| \mathbf{\hat{a}} + \| \mathbf{\hat{c}} \times \mathbf{\hat{a}}\| \mathbf{\hat{b}} + \| \mathbf{\hat{a}} \times \mathbf{\hat{b}}\| \mathbf{\hat{c}}\end{Vmatrix}}\]
The incenter of a spherical triangle

In plane geometry, one can also define excircles, which are tangent to one edge of the triangle and the extension of the other two sides. In spherical geometry, the extension of the other two sides eventually meets in a point, forming a triangle adjacent to the original triangle. The excircles of the original triangle are just the incircles of the adjacent triangles formed by the extension of its sides. The points of contact between the excircles and the original triangle are the extouch points. In the figure below, the incircle and intouch points are blue and the excircles and extouch points are red.

The excenters and extouch points of a spherical triangle

Unlike in plane geometry, the area of the triangle formed by the intouch points (the intouch triangle, or Gergonne triangle) is not the same as the area of the triangle formed by the extouch points (the extouch triangle, or Nagel triangle).

It’s worth mentioning that the study of spherical triangle centers is much less well developed than that of planar triangle centers. There may be other ways to translate these definitions between geometries. And like I said earlier, some planar triangle centers correspond to multiple spherical centers. These definitions correspond to the paper I linked earlier: other papers may use other definitions.

A Python script to create the figures in this post is located here.

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