In the Eye of the In the Eye of the BeholderBeholder

Projective GeometryProjective Geometry

How it All StartedHow it All Started

During the time of the Renaissance, scientists and During the time of the Renaissance, scientists and philosophers started studying “the world around them.”philosophers started studying “the world around them.”

This inspired artists to try to create what the eye sees on This inspired artists to try to create what the eye sees on canvascanvas

They ran into a problem; how to portray depth on a flat They ran into a problem; how to portray depth on a flat surfacesurface

Artists came to the realization that their problem was Artists came to the realization that their problem was geometric and started researching mathematical geometric and started researching mathematical solutionssolutions

This is what led into what we know today as projective This is what led into what we know today as projective geometrygeometry

What is Projective Geometry?What is Projective Geometry?

Originated from the principles of perspective artOriginated from the principles of perspective art Central principle: two parallel lines meet at infinityCentral principle: two parallel lines meet at infinity A branch of geometry dealing with properties and A branch of geometry dealing with properties and

invariants of geometric figures under projection.invariants of geometric figures under projection. ““Higher geometry”Higher geometry” ““Geometry of position”Geometry of position” ““Descriptive geometry”Descriptive geometry” Projective Geometry is non-Euclidean, however it Projective Geometry is non-Euclidean, however it

can be thought of as an extension of Euclidean can be thought of as an extension of Euclidean geometry…and this is why…geometry…and this is why…

The ExtensionThe Extension

The “direction” of each line is included in the line The “direction” of each line is included in the line itself as an “extra” pointitself as an “extra” point

A horizon of directions corresponding to A horizon of directions corresponding to coplanar lines is thought of as a linecoplanar lines is thought of as a line

Because of this, two parallel lines will meet on a Because of this, two parallel lines will meet on a horizon as long as the possess the same horizon as long as the possess the same directiondirection

In essence directions=points at infinity and In essence directions=points at infinity and horizons=lines at infinityhorizons=lines at infinity

All points and lines are treated equallyAll points and lines are treated equally

““The Axioms of Projective The Axioms of Projective Geometry”Geometry”

With the extension now the axioms become With the extension now the axioms become easier to understand:easier to understand:

1.1. Every line contains at least 3 pointsEvery line contains at least 3 points

2.2. Every two points, A and B, lie on a Every two points, A and B, lie on a unique line, ABunique line, AB

3.3. If lines AB and CD intersect, then so do If lines AB and CD intersect, then so do lines AC and BD, assuming that A and D lines AC and BD, assuming that A and D are distinct from B and Care distinct from B and C

Euclidean or non-Euclidean?Euclidean or non-Euclidean?

The reason that a line contains at least three The reason that a line contains at least three points is easy to see when thinking of Euclidean points is easy to see when thinking of Euclidean space and then adding to that points and lines at space and then adding to that points and lines at infinity. The third point is considered the infinity. The third point is considered the direction of the linedirection of the line

The second axiom has a similar form of Euclid’s The second axiom has a similar form of Euclid’s fifth postulatefifth postulate

There are numerous other examples of how There are numerous other examples of how closely related projective and Euclidean closely related projective and Euclidean geometry are geometry are

It seems ironic then, that Projective geometry is It seems ironic then, that Projective geometry is considered non-Euclideanconsidered non-Euclidean

Influential PeopleInfluential People

Filippo Brunelleschi (1377-1446)Filippo Brunelleschi (1377-1446) first person to study intensivelyfirst person to study intensively

Leone Battista Alberti (1404-1472)Leone Battista Alberti (1404-1472) screen images are called “projections”screen images are called “projections” How are the images related?How are the images related? painting a picture as the canvas was a window or painting a picture as the canvas was a window or

screenscreen if an object is viewed from different locations, the if an object is viewed from different locations, the

“screen images” or “projections” are different“screen images” or “projections” are different

Study of projections termed Study of projections termed projective geometryprojective geometry

Famous ArtistsFamous Artists

Piero della Francesca(1410-1492)

Leonardo da Vinci(1452-1519)

Albrecht Durer(1471-1528)

Activity 1Activity 1

Another way of thinking about this involves using a Another way of thinking about this involves using a light source instead of your eyeslight source instead of your eyes

Things that change:Things that change: DistanceDistance Angle measureAngle measure CurvatureCurvature

Things that don’t change:Things that don’t change: Straight linesStraight lines

German and French FollowersGerman and French Followers

Gerard Desargues (1593-1662)Gerard Desargues (1593-1662) projected conic sections and circles are projected conic sections and circles are

always conic sectionalways conic section

Jean Victor Poncelet (1788-1867)Jean Victor Poncelet (1788-1867) influential book on projective geometry in a influential book on projective geometry in a

very unlikely placevery unlikely place

Activity 2Activity 2

Railroads are a Railroads are a classic way of classic way of demonstrating demonstrating

perspective. Let’s perspective. Let’s construct a set of construct a set of receding railroad receding railroad

tracks!tracks!

Principle of DualityPrinciple of Duality

Principles in projective Principles in projective geometry occur in dual geometry occur in dual pairs by interchanging pairs by interchanging “line” and “point” where “line” and “point” where appropriate.appropriate. ““Two points determine Two points determine

exactly one line.” exactly one line.” Duality interchanges Duality interchanges

“line” and “point.”“line” and “point.” ““Two lines determine Two lines determine

exactly one point.” These exactly one point.” These statements are statements are dualsduals of of each other.each other.

Mystic HexagramMystic Hexagram

Blaise Pascal (1623-1662)Blaise Pascal (1623-1662) Hexagram inscribed in a conic sectionHexagram inscribed in a conic section Hexagram circumscribed about a conic Hexagram circumscribed about a conic

sectionsection Duality proves this Duality proves this automaticallyautomatically!!

A hexagram can be inscribed in a conic section if and only A hexagram can be inscribed in a conic section if and only if the points (of intersection) determined by its three pairs of if the points (of intersection) determined by its three pairs of

opposite sides lie on the same (straight) line.opposite sides lie on the same (straight) line.

Applying Projective GeometryApplying Projective Geometry

Used to describe natural phenomena such as:Used to describe natural phenomena such as: tension between central forces and peripheral influencestension between central forces and peripheral influences organic developmentsorganic developments

The forms of buds of leaves and flowers, pine cones, eggs, The forms of buds of leaves and flowers, pine cones, eggs, and the human heart can all be described as path and the human heart can all be described as path curves. When a single parameter interacts with these curves. When a single parameter interacts with these path curves, growth measures are formed, which are path curves, growth measures are formed, which are representations of organic forms. Without projective representations of organic forms. Without projective geometry, there would be no other way to geometry, there would be no other way to mathematically describe these forms. If negative values mathematically describe these forms. If negative values were used for the parameter, the inversions then were used for the parameter, the inversions then represent vortexes of water and air.represent vortexes of water and air.

TimelineTimeline

1377-1446 – Brunelleschi’s first study of 1377-1446 – Brunelleschi’s first study of projective geometryprojective geometry

1404-1472 – Alberti’s projection studies1404-1472 – Alberti’s projection studies

1593-1662 – Desargues’ “discovery” of 1593-1662 – Desargues’ “discovery” of projected conic sections and circles projected conic sections and circles always being conicalways being conic

1788-1867 – Poncelet’s influential book1788-1867 – Poncelet’s influential book

1623-1662 – Pascal’s Mystic Hexagram1623-1662 – Pascal’s Mystic Hexagram

ReferencesReferences

Weisstein, Eric W. “Projective Geometry.” From Weisstein, Eric W. “Projective Geometry.” From Math World-A Wolfram Web Resource. Math World-A Wolfram Web Resource. http://mathworld.wolfram.com/ProjectiveGeometrhttp://mathworld.wolfram.com/ProjectiveGeometry.htmly.html

Projective GeometryProjective Geometry. Wikipedia. November 25, . Wikipedia. November 25, 2006. http://en.wikipedia.org/wiki/Projective 2006. http://en.wikipedia.org/wiki/Projective _geometry_geometry

Berlinghoff, William P. and Gouvea Fernando Berlinghoff, William P. and Gouvea Fernando Q., Q., Math Through the AgesMath Through the Ages. Oxton House . Oxton House Publishers, 2002. Publishers, 2002.

Thank You!Thank You!

Lisa Carey and Sara SmithLisa Carey and Sara Smith