# Projective Geometry

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

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Related KMR-pages:

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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:

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:

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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 $\, \mathbb{P}^2 \,$:

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

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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 $\, \mathbb{P}^3 \,$:

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

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

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

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

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

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

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

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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.

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Projective Triangle – connected regions:

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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”.

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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.

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Projective range-to-range transformation:

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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):

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

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

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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:

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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.

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

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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.

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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.

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

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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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(changing the scale):

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

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

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

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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]:

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Hyperbolic yardsticks in action:

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Pascal-Brianchon’s theorem:

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Polar reciprocity:

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The pencil on two conics:

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Projective involutions:

Projective involution (reflection in line and point) 1:

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Projective involution (reflection in line and point) 2:

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