Computing polar triangles with linear algebra

The previous post defined the polar triangle of a spherical triangle in terms of its sides and angles, but it doesn’t construct the triangle. It only specifies one of a family of triangles located anywhere on the sphere. How do we know that polar triangles exist? There is a geometric construction that starts with a spherical triangle and constructs a unique new spherical triangle, and one can show that this new triangle has the angles and sides described in the previous post. There is also a linear algebraic construction of polar triangles, and that’s the topic of this post. In a nutshell, take your sphere to be the unit sphere in 3D. Write the coordinates of each vertex as the columns of a matrix. Invert that matrix and normalize each row. Then the rows are the coordinates of the polar vertices. This process is an involution: the polar of the polar of a triangle is the original triangle. The following Python code illustrates the process described above. It begins with a triangle on…