The Lorentz transformations is a set of equations that describe a linear transformation between a stationary reference frame and a reference frame in constant velocity. The equations are given by:
where represents the new x co-ordinate, represents the velocity of the other reference frame, representing time, and the speed of light.
On a Cartesian coordinate system, with the vertical axis being time (t), the horizontal axis being position in space along one axis (x), the gradients represent velocity (shallower gradient resulting in a greater velocity). If the speed of light is set as a 45° or 1:1 gradient, Lorentz transformations can rotate and squeeze other gradients while keeping certain gradients, like a 1:1 gradient constant. Points undergoing a Lorentz transformations on such a plane will be transformed along lines corresponding to where n is some number
Points undergoing a Lorentz transformation follow the green, conjugate hyperbola, where the vertical axis represents time,