Perspective Drawing and Projective Geometry

### Perspective Drawing and Projective Geometry

The use of perspective began during the Renaissance. It changed the way we represented and visualized the world. Experiments with perspective drawing were completed when people with interdisciplinary interests (like math and art) were perhaps more common. In this 1525 woodcut, from "Unterweisung der Messung", by Albrecht Durer, the screw eye on the wall is the desired viewer's eye, the lute on the left is the object, the taught string is a light ray, and the picture plane is mounted on a swivel.

Leonardo Da Vinci and Brook Taylor researched the question of how to find the viewing distance of a painting, and Taylor's 1715 work was published in Linear Perspective: Or, a New Method of Representing Justly All Manner of Objects as They Appear to the Eye in All Situations London: R. Knaplock.

These mathematicians and artists found the precise mathematical rules for perspective drawing. College art history professor Sam Edgerton believes that the discovery of the rules of perspective ushered in the industrial revolution. The newfound ability to reveal depth in drawings led to unprecedented sharing of ideas.

What are the properties shared by two perspective views of the same scene?

Desargues explored conics as perspective deformations of a circle and the intersection of parallel lines at infinity. Desargues' Theorem is a famous result for projective geometry, which we'll explore in the lab.

Poncelet explored projective geometry including properties invariant under projection.

In Felix Klein's Erlangen programm, the group of transformations is how to study geometric objects - ie not the set of points making up the space, but the transformations of them. In this sense there is a hierarchy of geometries via an algebraic structure called a group, and projective geometry contains all the other geometries in this sense.

Euclidean transformations is contained in Similarity transformations (includes scalings) which includes projective transformations.

As the transformation groups become smaller and less general, the corresponding spatial structures become more rigid and have more invariants.

Spherical and hyperbolic transformations can separately be considered a part of projective transformations