In the Eye of the Beholder Projective Geometry

How it All Started  During the time of the Renaissance, scientists and philosophers started studying “the world around them.”  This inspired artists to try to create what the eye sees on canvas  They ran into a problem; how to portray depth on a flat surface  Artists came to the realization that their problem was geometric and started researching mathematical solutions  This is what led into what we know today as projective geometry

What is Projective Geometry?  Originated from the principles of perspective art  Central principle: two parallel lines meet at infinity  A branch of geometry dealing with properties and invariants of geometric figures under projection.  “Higher geometry”  “Geometry of position”  “Descriptive geometry”  Projective Geometry is non-Euclidean, however it can be thought of as an extension of Euclidean geometry…and this is why…

The Extension  The “direction” of each line is included in the line itself as an “extra” point  A horizon of directions corresponding to coplanar lines is thought of as a line  Because of this, two parallel lines will meet on a horizon as long as the possess the same direction  In essence directions=points at infinity and horizons=lines at infinity  All points and lines are treated equally

“The Axioms of Projective Geometry” With the extension now the axioms become easier to understand: 1. Every line contains at least 3 points 2. Every two points, A and B, lie on a unique line, AB 3. If lines AB and CD intersect, then so do lines AC and BD, assuming that A and D are distinct from B and C

Euclidean or non-Euclidean?  The reason that a line contains at least three points is easy to see when thinking of Euclidean space and then adding to that points and lines at infinity. The third point is considered the direction of the line  The second axiom has a similar form of Euclid’s fifth postulate  There are numerous other examples of how closely related projective and Euclidean geometry are  It seems ironic then, that Projective geometry is considered non-Euclidean

Influential People  Filippo Brunelleschi ( )  first person to study intensively  Leone Battista Alberti ( )  screen images are called “projections”  How are the images related?  painting a picture as the canvas was a window or screen  if an object is viewed from different locations, the “screen images” or “projections” are different  Study of projections termed projective geometry

Famous Artists Piero della Francesca ( ) Leonardo da Vinci ( ) Albrecht Durer ( )

Activity 1 Another way of thinking about this involves using a light source instead of your eyes Things that change:  Distance  Angle measure  Curvature Things that don’t change:  Straight lines

German and French Followers  Gerard Desargues ( )  projected conic sections and circles are always conic section  Jean Victor Poncelet ( )  influential book on projective geometry in a very unlikely place

Activity 2 Railroads are a classic way of demonstrating perspective. Let’s construct a set of receding railroad tracks!

Principle of Duality  Principles in projective geometry occur in dual pairs by interchanging “line” and “point” where appropriate.  “Two points determine exactly one line.”  Duality interchanges “line” and “point.”  “Two lines determine exactly one point.” These statements are duals of each other.

Mystic Hexagram  Blaise Pascal ( )  Hexagram inscribed in a conic section  Hexagram circumscribed about a conic section  Duality proves this automatically!

A hexagram can be inscribed in a conic section if and only if the points (of intersection) determined by its three pairs of opposite sides lie on the same (straight) line.

Applying Projective Geometry Used to describe natural phenomena such as:  tension between central forces and peripheral influences  organic developments The forms of buds of leaves and flowers, pine cones, eggs, and the human heart can all be described as path curves. When a single parameter interacts with these path curves, growth measures are formed, which are representations of organic forms. Without projective geometry, there would be no other way to mathematically describe these forms. If negative values were used for the parameter, the inversions then represent vortexes of water and air.

Timeline – Brunelleschi’s first study of projective geometry – Alberti’s projection studies – Desargues’ “discovery” of projected conic sections and circles always being conic – Poncelet’s influential book – Pascal’s Mystic Hexagram

References  Weisstein, Eric W. “Projective Geometry.” From Math World-A Wolfram Web Resource. y.html y.html y.html  Projective Geometry. Wikipedia. November 25, _geometry  Berlinghoff, William P. and Gouvea Fernando Q., Math Through the Ages. Oxton House Publishers, 2002.

Thank You! Lisa Carey and Sara Smith