Spherical trigonometry

intersection of the sphere and a plane which passes through the center point of the sphere

branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons

shortest distance between two points along the surface of a sphere

overview about the solution of triangles

formula for the great-circle distance between two points on a sphere

spherical triangle that can be used to tile a sphere

relation between the sides and angles of spherical triangles

In mathematics, a versor is a quaternion whose norm is one, also known as a unit quaternion. Each versor has the form \ u = \exp(a\mathbf{r}) = \cos a + \mathbf{r} \sin a, \qquad \mathbf{r}^2 = -1, \qquad a \in [0,\pi]\ , where the condition \ \mathbf{r}^2 = -1\ means that \ \mathbf{r}\ is an algebraic imaginary unit. There is a sphere of imaginary units in the quaternions. Note that the expression for a versor is just Euler's formula for the imaginary unit \ \mathbf{r} ~. If \ a = \tfrac{\pi}{2}\ (when \ a\ is a right angle), then \ u = \mathbf{r}\ , and it is called a right versor.

Group realized geometrically by reflections across the sides of a triangle

theorem in geometry

a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles.