Hyperbolic triangle relations between the sides "s" and the angles "A" of an equilateral triangle (a triangle where all sides have the same length and all angles are equal). The relations are: Whether "C" is a right angle or not, the following relationships hold: The hyperbolic law of cosines is as follows: Its dual theorem is There is also a "law of sines": and a four-parts formula: For hyperbolic trigonometry: Hyperbolic triangle In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called "sides" or "edges" and three points called "angles" or "vertices".
Euler angles from −1 to 1, and let "β" = arccos("z"). Other properties of Euler angles and rotations in general can be found from the Geometric algebra(GA). GA is a higher level abstraction, in which the quaternions are an even subalgebra. The principal tool in GA is the rotor formula_42 where formula_43angle of rotation, formula_44rotation axis (unitary vector) and formula_45pseudoscalar (trivector in formula_46) It is possible to define parameters analogous to the Euler angles in dimensions higher than three. The number of degrees of freedom of a rotation matrix is always less than the dimension of the matrix squared. That is, the