A model in ordinary Euclidean geometry is presented to illustrate the Lorentz transformation formulas of special relativity. In one version of the model, the space-coordinate x in the direction of motion, the time-coordinate τ = ct, and the velocity coordinate β = v/c are represented in a cylindrical coordinate system rθz with r = τ, θ = arcsin β, and z = x. An alternative version is given by r = x, θ = arcsin β, and z = τ. In both versions an event is represented as a straight generatrix of a one-sheet hyperboloid with axis in the z direction. If the angle θ instead is taken to be a spatial angle in a Euclidean space-time world, where light lines take the role of the event-lines in the above xtv-space, it is shown to follow from the above model and Huygens' principle that it would be possible for moving observers to make Lorentz-invariant observations despite the Euclidean metric.