The starting point of this manuscript is classical rigid body rotation. As it is well known, it contradicts basis of relativity since infinite speed is reached at infinite distance from the rotation center O. In order to fix this problem, a phenomenological circle-based construction using Euclidian trigonometry is first described: the relativistic rigid body rotation. The physical Eulerian acceleration implied by this geometrical construction then sketches future links with Maxwell’s equation and Lense-Thirring effect. More importantly, relativistic rigid body rotation is shown to be compatible with Lorentz transformation and brings new geometrical interpretations of time and space intervals.