I think what you are looking for is given in the first chapter of the epic text "Spinors and Spacetime" by Wolfgang Rindler and Roger Penrose. Or at least it is **if what you're asking for is a simple and clear geometric construction that illustrates the effects of Lorentz transformations on the bulk (3+1) geometry.**

[Fingers crossed .... **IMHO** the use of this image and the ones below is covered under "fair use". If it is deleted blame the copyright regime.]

To cut a long story short, you identify points on the celestial sphere $\mathcal{S}^-$ with a light ray as seen by an observer at the center of the sphere. A stereographic projection allows to map points on $\mathcal{S}^-$ to the complex plane $\mathbb{C}$. The action of Lorentz boosts on the observer translates into the action of an $SL(2,\mathbb{C})$ element on the points of $\mathcal{S}^-$. The result is shown in the figure below:

I do not know of any other constructions which so vividly illustrates the geometrical effects of Lorentz transformations. I have left out many details for which I once again recommend the amazing text by Penrose and Rindler.

All hail the copyright gods.

**Edit:** In response to the comments, I answered the question the best I could understand it. I've emphasized the relevant sentence in the first para.

This post imported from StackExchange Physics at 2014-04-01 17:37 (UCT), posted by SE-user user346