In the Eye of the Beholder

1
In the Eye of the Beholder

• Projective Geometry

2
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

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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
  geometryand this is why

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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 directionspoints at infinity and
  horizonslines at infinity
• All points and lines are treated equally

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The Axioms of Projective Geometry

• With the extension now the axioms become easier
  to understand
• Every line contains at least 3 points
• Every two points, A and B, lie on a unique line,
  AB
• If lines AB and CD intersect, then so do lines AC
  and BD, assuming that A and D are distinct from B
  and C

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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 Euclids
  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

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Influential People

• Filippo Brunelleschi (1377-1446)
• first person to study intensively
• Leone Battista Alberti (1404-1472)
• 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

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Famous Artists
Piero della Francesca (1410-1492)
Albrecht Durer (1471-1528)
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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 dont change
• Straight lines

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German and French Followers

• Gerard Desargues (1593-1662)
• projected conic sections and circles are always
  conic section
• Jean Victor Poncelet (1788-1867)
• influential book on projective geometry in a very
  unlikely place

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Activity 2

• Railroads are a classic way of demonstrating
  perspective. Lets construct a set of receding
  railroad tracks!

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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.

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Mystic Hexagram

• Blaise Pascal (1623-1662)
• Hexagram inscribed in a conic section
• Hexagram circumscribed about a conic section
• Duality proves this automatically!

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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.
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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.

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Timeline

• 1377-1446 Brunelleschis first study of
  projective geometry
• 1404-1472 Albertis projection studies
• 1593-1662 Desargues discovery of projected
  conic sections and circles always being conic
• 1788-1867 Poncelets influential book
• 1623-1662 Pascals Mystic Hexagram

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References

• Weisstein, Eric W. Projective Geometry. From
  Math World-A Wolfram Web Resource.
  http//mathworld.wolfram.com/ProjectiveGeometry.ht
  ml
• Projective Geometry. Wikipedia. November 25,
  2006. http//en.wikipedia.org/wiki/Projective
  _geometry
• Berlinghoff, William P. and Gouvea Fernando Q.,
  Math Through the Ages. Oxton House Publishers,
  2002.

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Thank You!

• Lisa Carey and Sara Smith