December 03, 2003

This is a reconstruction of the animation that led to five months of working with Dr. Kim Veltman in Italy and some of the richest learning experiences of my life.

In linear perspective, when parallel lines are projected onto a *picture plane* [1], the resulting *projections* [2] will intersect at a *vanishing point*. For lines parallel to the ground, the vanishing points will appear on the *horizon line* [3]. When drawing objects in perspective, the artist usually begins by drawing the horizon line and a couple vanishing points. Construction lines are drawn from those vanishing points to help the artist correctly locate the edges of buildings or other objects. Perspective drawings and paintings are sometimes classified by the number of vanishing points used in the construction.

I don't remember the exact definition of a vanishing point from Dr. Veltman's collection. It would have been similar to what I've written above: accurate and descriptive with a certain amount of precision in the use of language and inherently relative to other terms of perspective. But words don't suffice. A vanishing point demands illustration. I found my original diagram of a vanishing point painfully lacking. It looked basically like a stick figure and a pile of triangles. At the time I was very interested in animation and thought I could visualize a vanishing point more clearly with a little motion. I ended up with a stick figure and a *moving* pile of triangles which was much better. ;-)

Without further adieu (with the caveat that an SVG plugin is required) . . . *drumroll* . . . (click **Go**):

It is probably not self evident as to why Kim was so impressed by such a simple animation. Computers as an Historical Tool for Mathematics, Science and Art will help clarify why this is an interesting contribution to the history of perspective. There are some other hints in another article which I quote below.

Back to the pictures of infinity. Notice how as the projected lines grow ever more slowly towards the vanishing point before leaping at the very end. This is a direct consequence of geometry and infinity. The ground lines are growing at a constant rate whereas their projections are growing in a ratio relative to the point of view and position of the picture plane. If I let it keep going it does eventually get fairly close to the end, but it's really not worth the wait. Someday I'll rearrange the animation so the projected lines grow at a constant rate. I'll still have to do some fudging towards the end as the ground lines approach the limits of the SVG coordinate system.

Speaking of coordinate systems there's another detail worth mentioning if you're inclined to look at the code behind the animation. SVG uses a two-dimensional coordinate system whereas the scene I'm depicting is three-dimensional. For this animation I'm just using the 2D coordinates to calculate intersections. In the AutoCAD version I had the advantage of a three dimensional coordinate system for my code. That arrangement more accurately reflects the actual geometry at work when light reflects off of objects and into our eyes. For simplicity my stick figure is a cyclops. In both of these ways I'm actually continuing a very long tradition in the history of perspective and geometry in general. :-)

Here's how Kim described it in the introduction to Computers and Renaissance Perspective

A long tradition of Euclidean geometry developed two-dimensional conventions of representation to the extent that they were part of the legitimation process in mathematics. As a result, Renaissance treatises on perspective evidence a basic paradox: they use abstract two-dimensional conventions to display the principles of a new three-dimensional method of representing space. This is achieved by folding different planes (usually a lateral view and or a ground view) into a single plane (usually a frontal view). This procedure of folding back (technically termed ribaltimento in Italian and rabattement in French), makes most of the diagrams in the early treatises virtually incomprehensible to the untrained eye, all the more so because the reader is confronted with a completed construction which usually gives no visual clues concerning the steps taken to get there. One can identify the steps taken in arriving at an end product in any of these constructions; one can reconstruct these steps and theoretically it would be possible to print these, except that the cost of including so many diagrams makes this alternative prohibitively expensive. All of which helps explain why these treatises have never been studied systematically.

Many of the animations I created in Siena were folding the two-dimensional images into their three-dimensional origins. In one, I animated the steps Piero della Francesca described to construct a perspective image of a pentagon. No small task to understand a centuries old Italian text on a geometric construction. When I was done animating his instructions I re-did the animation in a way that I thought made the technique he was describing more clear. If I remember correctly, I then folded the resulting drawing into its three-dimensional arrangement and animated something of a proof of the technique by projecting lines from the point of view to the pentagon on the ground and showing that constructed image aligns correctly. When I get more tuits I'll see if I can repeat that stunt. :-)

[1,2] See my illustration of perspective basics

[3] I still haven't illustrated a horizon line anywhere.

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