Projective Geometry

This is a sub-page of our page on Geometry.

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The sub-pages of this page are:

Projective Metrics

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Related KMR-pages:

The Euclidean Degeneration
Metric Geometry
Affine Geometry
Linear Algebra

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Books:

• Jürgen Richter-Gebert (2011), Perspectives on Projective Geometry – A Guided Tour Through Real and Complex Geometry, Springer, ISBN 978-3-642-17285-4.
• W. T. Fishback (1969), Projective and Euclidean Geometry, John Wiley & Sons, Inc.,
ISBN 13: 978-047126-053-0.
• H. Winroth (1999), Dynamic Projective Geometry, PhD thesis at NADA/KTH.

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Other relevant sources of information:

Projective geometry
Homogeneous coordinates
Cross ratio
Homography
Cayley transform
Collineation
Fundamental theorem of projective geometry
Pole and polar
Semilinear transformation
Cayley-Klein metric

Dynamic Projective Geometry (PhD-thesis of Harald Winroth, KTH, 1999)

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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History of projective geometry:

Felix Klein
Arthur Cayley
Julius Plücker
August Ferdinand Möbius
Jacob Steiner
Karl Georg Christian von Staudt
Joseph Diez Gergonne
Jean-Victor Poncelét
Blaise Pascal
Girard Desargues
Kepler’s idea for projective geometry
Piero della Francesca
Leon Battista Alberti
• Filippo Brunelleschi
Omar Khayyam
Ibn al-Haytham (Alhazen)
Pappus of Alexandria

History of perspective (at Op.Art)
History of perspective (at Slideshare)
A Brief History of Perspective (at classicalart.org)
Brunelleschi’s Peepshow and the Origins of Perspective

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Representation: [ \, l_{ine} \, ]_{E_{uclidean \, Geometry}} \, = \, \left< \, a_{ffine} \, l_{inear} \, e_{quation} \, \right>_{E_{uclidean \, Geometry}}

Representation: [ \, p_{oint} \, ]_{P_{rojective \, Geometry}} \, = \, \left< \, h_{omogeneous} \, l_{inear} \,e_{quation} \, \right>_{P_{rojective \, Geometry}}

Representation: [ \, p_{oint} \, ]_{\mathbb{P}^2} \, = \, \left< \, l_{ine} \, o_{n} \, t_{he} \, p_{oint} \, O \, \right>_{{\mathbb{E}^3}_O}

Representation: [ \, p_{oint} \, ]_{\mathbb{P}^2} \, = \, \left< \, h_{omogeneous} \, l_{inear} \, e_{quation} \, \right>_{\mathbb{P}^2}

Representation: [ \, l_{ine} \, ]_{\mathbb{P}^2} \, = \, \left< \, p_{lane} \, o_{n} \, t_{he} \, p_{oint} \, O \, \right>_{{\mathbb{E}^3}_O}

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x^2 + y^2 = 0 \,\, [ \, \Rightarrow \, ]_{\mathbb{R}} \,\, x = 0, y = 0 \, .

x^2 + y^2 = 0 \,\, [ \, \Rightarrow \, ]_{\mathbb{C}} \,\, y = ix, y = -ix \, .

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Reconstruction: \left( \, \left< \, m_{odel} \, \right>_{P_{aradigm}} \, \right)_{P_{aradigm}} \, = \, r_{epresentee}

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General trilinear polarity in \, \mathbb{P}^2 \, :

General trilinear polarity in P^2

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Cartesian trilinear polarity in \, \mathbb{R}^2 \, :

Cartesian trilinear polarity in R^2

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Trilinear Polars and Harmonic Nets (Richard Southwell on YouTube):

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The projective point- and plane-coordinate systems in \, \mathbb{P}^3 \, :

The projective point- and plane-coordinate systems in P^3

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Dual unification in \, \mathbb{P}^3 \,
by the tetralinear pole-polar relationship
between the unit point and the unit plane:

Dual unification in P^3 by tetralinear pole-polar relationship of the unit point and the unit plane

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Plane projective CTU in \mathbb{P}^2 \, :

Plane projective CTU

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Line-projective CIU in \mathbb{P}^1 \, :

Line-projective CIU

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Plane euclidean CTU in \mathbb{R}^2 \, :

Plane euclidean CTU

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Trilinear euclidean pole-polar relation in \mathbb{R}^2 \, :

Trilinear pole-polar euclidean

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Trilinear projective pole-polar relation in \mathbb{P}^2 \, :

Trilinear pole-polar projective

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d-Grassmannians and their degrees of freedom in \mathbb{P}^n \, :

d-Grassmannians and their degrees of freedom

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Complete Quadrangle:

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This film shows the parallelogram (in white) that is formed by the midpoints of the four sides of a generic quadrangle in 3D.

The interactive simulation the created this movie.

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Projective Triangle – connected regions:

Projective Triangle - 4 regions

Interactive simulation of Projective Triangle – connected regions.

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Conic sections:

NOTE: Ellipses, parabolas and hyperbolas are not separate objects in projective geometry, since they can all be projected into each other. Hence, in projective geometry, we can only talk about “conics”.

The interactive simulation that created this movie.

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Perspective range-to-range transformation:

A perspective range-to-range transformation describes a point as an envelope of lines.

Two ranges can only be perspective from a point
if this point does not belong to anyone of the two ranges.

The interactive simulation that created this movie.

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Projective range-to-range transformation:

The interactive simulation that created this movie.

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The fundamental theorem of projective geometry:

Two projectively related ranges
determine the line-locus (= the envelope) of a unique conic.

In general, a conic is determined by five lines in generic position.

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Conics via projective geometry (N. J. Wildberger on YouTube):

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Projective range-to-range transformation (elliptic envelope 1):

Projective range-to-range transformation (elliptic envelope 1) (lettered)

The interactive simulation that created this movie.

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A non-degenerated projective range-to-range transformation
determines a conic envelope (an ellipse in the depicted case):

Projective range-to-range transformation (elliptic envelope 2) (lettered)

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A non-degenerated projective range-to-range transformation
determines a conic envelope (a hyperbola in the depicted case):

Projective range-to-range transformation (hyperbolic envelope 1) (lettered)

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The interactive simulation that created this movie.

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A degenerated projective range-to-range transformation
is called a perspective range-to-range transformation
and it determines a point as an envelope of lines.

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A perspective range-to-range transformation
describes a point as an envelope of lines:

The interactive simulation that created this movie.

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DUALITY

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The dual version of the fundamental theorem:

A projective pencil-to-pencil transformation determines a conic
as a point-locus (= an ellipse in the depicted case):

Two projectively related pencils determine the point-locus of a unique conic.
This conic is often referred to as the Steiner conic.

In general, a conic is determined by five points in generic position.

The interactive simulation that created this movie.

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A projective pencil-to-pencil transformation determines a conic
as a point-locus (= a hyperbola in the depicted case):

The interactive simulation that created this movie.

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A non-degenerated projective transformation is called a projectivity. It can be shown that any projectivity can be expressed as the product of at most two different perspective transformations. If the projectivity degenerates, it becomes a perspective transformation, and in this case it is called a perspectivity.

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A perspective pencil-to-pencil transformation determines a line as a locus of points:

Two pencils can only be perspective from a line
that does not belong to anyone of the two pencils.

The interactive simulation that created this movie.

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The dual configuration:

A perspective range-to-range transformation describes a point as an envelope of lines:

Two ranges can only be perspective from a point
that does not belong to anyone of the two ranges.

The interactive simulation that created this movie.

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Compare this video (presented elsewhere)
which is repeated here for comparison with the preceding video:

The interactive simulation that created this movie.

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Constructing the harmonic conjugate of a point on a range
with respect to two other points on that range:

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Projecting a quadratic grid
(changing the scale):

The interactive simulation that created this movie.

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Changing the projection of a square grid with diagonals 1
[a1 = 3n , b2 = 3(1-n)]:

The interactive simulation that created this movie.

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Changing the projection of a square grid with diagonals 2
[b1 = 3n , b2 = 3(1-n)]:

The interactive simulation that created this movie.

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Changing the projection of a square grid with diagonals 3
[c3 = -n , 0 ≤ n ≤ 20]:

The interactive simulation that created this movie.

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Projection of a square grid with a point moving on the line at infinity:
[a1=3, b1=0, c1=5, a2=0, b2=3, c2=10, a3=1, b3=1, c3=1]:

The interactive simulation that created this movie.

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Hyperbolic yardsticks in action:

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Desargue’s theorem:

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PappusPascal’s theorem:

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Pascal’s theorem:

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Pascal-Brianchon’s theorem:

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Polar reciprocity:

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Reciprocal conics:

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The pencil on two conics:

The interactive simulation that created this movie.

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Projective involutions:

Projective involution (reflection in line and point) 1:

Projective involution (reflection in line and point) 1

The interactive simulation that created this movie.

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Projective involution (reflection in line and point) 2:

Projective involution (reflection in line and point) 2

The interactive simulation that created this movie.

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