Rigid Geometric Transformations

The special case for θ = π/2 radians is known as Pythagoras’ theorem. The definitions that follow focus on three-dimensional space. Two-dimensional geometry can be derived as a special case when the third coordinate of every point is set to zero. A Cartesian reference system for three-dimensional space is a point in space called the origin and three mutually perpendicular, directed lines though the origin called the axes. The order in which the axes are listed is fixed, and is part of the definition of the reference system. The plane that contains the second and third axis is the first reference plane. The plane that contains the third and first axis is the second reference plane. The plane that contains the first and second axis is the third reference plane. It is customary to mark the axis directions by specifying a point on each axis and at unit distance from the origin. These points are called the unit points of the system, and the positive direction of an axis is from the origin towards the axis’ unit point. A Cartesian reference system is right-handed if the smallest rotation that brings the first unit point to the second is counterclockwise when viewed from the third unit point. The system is left-handed otherwise. The Cartesian coordinates of a point in three-dimensional space are the signed distances of the point from the first, second, and third reference plane, in this order, and collected into a vector. The sign for coordinate i is positive if the point is in the half-space (delimited by the i-th reference plane) that contains the positive half of the i-th reference axis. It follows that the Cartesian coordinates of the origin are t = (0, 0, 0)T , those of the unit points are the vectors ex = (1, 0, 0)T , ey = (0, 1, 0)T , and ez = (0, 0, 1)T , and the vector p = (x, y, z)T of coordinates of an arbitrary point in space can also be written as follows: