In the Renaissance, many European artists became very interested in the problem of representing three dimensional scenes "illusionistically", as if seen through a window. A very simple element of a scene they might have wanted to depict would be a sidewalk made of square blocks, extending off from the viewer to the distance.[br][br]A child might approach this problem, reasonably enough, with something like the following:

They know that the blocks are square, and squares have four right angles and four equal sides, so they draw them with those properties.[br][br]Renaissance artists realized though that the image we might trace on a window of such a sidewalk does not match this picture. [br][list][*]The sides of the sidewalk don't appear parallel, but instead seem to converge to a point in the infinite distance, commonly called a vanishing point. [/*][*]Also, the dimensions of the blocks don't stay constant, but seem to get smaller and smaller, the more distant the block we look at becomes.[/*][/list][br]An attempt to draw an image that matches these observations unsystematically might look more like this:

The development of the vanishing point was a crucial step in the right direction, but it left a lot to be understood. How exactly should the distances between the horizontal segments shrink as we look further along the sidewalk?[br][br]Some artists took a rule of thumb that this diminution should be a geometric progression: each gap should be a fixed ratio (say 2/3) of the gap below it. This would look like the following:

Leon Battista Alberti developed a construction which give a precise and correct spacing. By adding an imaginary side view, he is able to make marks at the correct levels on his canvas.

A hexagon can be inscribed in a conic section if and only if the intersections of the pairs of opposite sides are collinear.

The dual version of this theorem:[br][br]A hexagon can be circumscribed about a conic section if and only iff the lines determined by the pairs of opposite vertices pass through a single point.