Visual Introduction to Projective Geometry

Video Lectures

Displaying all 16 video lectures.
Lecture 1
Why Perspective Drawing Works
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Why Perspective Drawing Works
We begin our lecture series by describing the technique of perspective drawing which was responsible for popularizing projective geometry.

We use an image of railway tracks that appear to converge upon the horizon to introduce the idea of perspective drawing. We then describe the classic drawing technique used in the 1400's and investigate its consequences logically. In particular we show that parallel lines which are perpendicular to the picture plane will appear to meet at a special place within the horizon of the image (the principle vanishing point). We also give a proof of how parallel lines always meet somewhere upon the horizon within pictures produced via projective drawing. It is important to have a proof of this foundational fact about perspective drawing.

We also use Cartesian geometry to derive an expression for where the images of particular objects within the scene will get projected to within the picture plane. We use the formula we derive to examine the image a parabola in the scene. Remarkably we find the this parabola is projected onto a circle within the picture plane. By examining the underlying mathematics we obtain a rational parametrization of the circle which is rather similar to the one described within the video:

MathFoundations29: Parametrizing circles
By Norman Wildberger.
https://www.youtube.com/watch?v=xp0H3Aw0j6E


We are initiating our lecture series by explaining how the subject of projective geometry got popularized. Later we shall discuss how the study of projective geometry related to many other subjects including ancient Greek mathematics (e.g., Pappus's Theorem), conic sections, linear algebra, topology, and more specific types of geometry such as Euclidean, spherical and hyperbolic geometry.

My Website is

https://sites.google.com/site/richardsouthwell254/home
Lecture 2
Without Equations, Conics & Spirals
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Without Equations, Conics & Spirals
Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.

To get into this visual mindset we discus conics. We describe ellipses, hyperbola and parabola, and how they appear in astronomy, optics, and projectile movement. We show how these curves can be thought can thought of as conic sections, and how the Geogebra program called can be used to study these ideas.

We also discuss how logarithmic spirals occur in sunflowers, weather systems, and galaxy spirals, and we describe a simple geometric method for constructing them.

The subject matter in this introductory video is not exclusive to projective geometry. It was chosen to start the visual approach.

The Geogebra program can be found here:
http://www.geogebra.org/

I would like to thank Olive Whicher for his illuminating book on projective geometry, and Norman Wildberger for his insightful teachings on the subject.

I also cited a result from

Hilton, P.; Holton, D.; and Pedersen, J. Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, 1997.

I also use the following Wolfram Demonstration (by Phil Ramsden):

http://demonstrations.wolfram.com/ConicSectionsTheDoubleCone...

My website is:

https://sites.google.com/site/richardsouthwell254/
Lecture 3
Foundations & Tilings in Perspective
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Foundations & Tilings in Perspective
Protective geometry is more fundamental than euclidean geometry, with applications in physics, biology and perspective art. We shall introduce it visually, without equations.

This video begins by considering a hexagonal tiled hive from a bee's perspective. We then formally define projective geometry, discuss previous approach the study, like perspective art and geometry with only a straight-edge.

We discuss the famous hexagon theorem of Pappus. A foundation result of projective geometry. We also discuss the work of Descargues, and his brilliant idea about treating projective geometry as the study of constructions which can only be made using a straight-edge.

To contrast projective geometry with Euclidean geometry (which considers constructions using a straight edger `and' a compass), we compare how a hexagon is constructed in each geometry. In the projective case, we form the hexagon starting from three lines which pass through the vanishing line.

In the resulting projective construction of the hexagon we see many interesting qualities which are invariant of how the initial points are chosen. We point out how Pappus's theorem and Desgaurges' theorem can be seen working within this hexagon construction, and show how this construction can be extended to give a beautiful projective tiling of the plane with hexagons.

The Geogebra program can be found here:
http://www.geogebra.org/

I would like to thank Olive Whicher for his illuminating book on projective geometry, and Norman Wildberger for his insightful teachings on the subject.

My website is:

https://sites.google.com/site/richardsouthwell254/
Lecture 4
When Does A Parabola Look Like An Ellipse?
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When Does A Parabola Look Like An Ellipse?
Curves can look very different when viewed from inside the plane they are drawn upon (using projective geometry), rather than being seen from above (using Cartesian geometry).

To explain, we describe many details of conic curves. In particular we show how these curves can be defined with respect to a focus points, and we show simple mechanical ways to draw conics.

In particular we describe how to use the `string and pins' method (also called the gardeners method) to draw an ellipse, and we show how to produce a very simple contraption for drawing Hyperbola, using household items.

We demonstrate how projective geometry can give powerful insights into the true nature of conics. Our main driving point is that a conic drawn on a plane stretching out ahead of one can look very different to a standard picture (much like rail road tracks appear to meet on the horizon). This could be viewed as a mere optical illusion, it seems to reveal deep truths about the underlying relationships between the conics.

We consider the issue of how conics appear in projective geometry. Using geogebra, we model how it might look to be standing on a grid
that has a giant parabola drawn upon it. The resulting perspective is surprising.

We also show how to construct ellipses manually using the string and pin method

The Wolfram demonstration I used for the conics can be found here

http://demonstrations.wolfram.com/LocusOfPointsDefinitionOfA...

It was written by Marc Brodie.

The Geogebra program can be found here:
http://www.geogebra.org/

I would like to thank Olive Whicher for his illuminating book on projective geometry, and Norman Wildberger for his insightful teachings on the subject.
Lecture 5
Desargues' Theorem Proof
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Desargues' Theorem Proof
Desargues' theorem is one of the most fundamental and beautiful results in projective geometry. Desargues' theorem states that if you have two triangles which are perspective to one another then the three points formed by the meets of the corresponding edges of the triangles will be colinear. We give an intuitive proof which is based at imagining this two dimensional situation from a three dimensional perspective.

Desargues' theorem is true on the projective plane. The projective plane can be thought of as the `extended' euclidean plane - i.e., the familiar 2D space, with extra `ideal' points at infinity, where parallel lines meet. The video's argument is not rigorous because we have not yet explained the axioms behind projective geometry. Our three dimensional argument quickly provides good evidence for Desargues' theorem, but if one wishes to prove it purely from within the two dimensional projective plane, one has to do quite some ground work.

Projective geometry can be set up using the following axioms

1: Two distinct points lie on a unique line.
2: Two lines meet at a unique point.
3: There exist three non-colinear points.
4: Every line contains at least three points.

(co-inear points are points that are on the same line.)

However the four axioms above are not actually enough to establish Desargues' theorem, working purely inside the projective plane. Adding extra axioms, such as the projectivity axiom

[Introduction To Projective Geometry, C.R. Wylie]

make it possible to prove Desargues' theorem from a two dimensional perspective. Coexter and Whitehead also set up different systems of axioms for projective geometry which allow the result to be proved. Some authors

[Projective Geometry, Finite and Infinite, Brendan Hassett,

just take Desargues' theorem as an axiom, and add it to axioms 1,2,3 & 4 above, to define projective geometry.

We take a simpler approach in our proof, and imagine that our projective plane is embedded in a three dimensional projective space. This allows us to make a very straight forward argument for the validity of Desargues' theorem, similar to the one found in

[Projective Geometry: Creative Polarities in Space and Time, Olive Whicher].

The converse of our stated result (i.e., that when the points formed by linking corresponding edges of a triangle lie upon a straight line, then the triangles are perspective to one another), is also true. And some consider this to be part of Desargues' theorem. Although we only prove that perspective triangles imply colinear meeting points of corresponding edges.
Lecture 6
Axioms, Duality and Projections
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Axioms, Duality and Projections
We discuss how projective geometry can be formalized in different ways, and then focus upon the ideas of perspective and projection. The interesting duality/polarity between points and lines also becomes apparent.
In particular, we approach the following issues:

The projective plane as an extension of the euclidean plane.

The projective plane, described by homogeneous coordinates.

Axioms for projective geometry
(here I refer to the document:

Projective Geometry, Finite and Infinite
Brendan Hassett
http://www.math.rice.edu/~hassett/RUSMP3.pdf
)

The definition of perspectives, with respect to points and lines.

The definition of a projection.

How to find a projection between any 3 co-linear points and any other 3 co-linear points.

How to find a projection between any 3 concurrent lines and any other 3 concurrent lines.

This gets us close to the point of being able to discuss the fundamental theorem of projective geometry.
Lecture 7
Conics Made Easily and Beautifully
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Conics Made Easily and Beautifully
Conics (ellipses, hyperbola, parabola) can be understood in terms of this simple construction method from projective geometry,
The idea is to create a projection from one line to another, and then draw (purple) lines joining points on the initial line to the images of such points under projection. This can be used to create ellipses, parabola and hyperbola.
The famous projective geometer Jacob Steiner described how each conic curve can be viewed as the envelope of lines connecting points which are related by a projection. The video also introduces the idea of a projection (as a sequence of perspectives). To make this I used the (free) geogebra software.

https://www.geogebra.org/

Although one could create such an image manually by drawing many lines joining points of the blue line to the position they get projected to. I also investigate how circles change with the movement of initial point, and obtain something that looks similar to the magnetic field lines of a small magnet.

Thanks to Norman Wildberger for his inspiring lectures on this subject, and Olive Whicher for her fascinating book on projective geometry.

We explain what perspectives are, what projections are, and how this method can be done in geogebra. We also show other interesting constructions which can be made along similar lines.

Also, I have made this video number 5 in my series (the previous video `Projective Geometry: Axioms, Duality and Projections' seemed premature).
Lecture 8
Harmonic Quadrangles & The 13 Configuration
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Harmonic Quadrangles & The 13 Configuration
We introduce harmonic ranges, harmonic conjugates and harmonic pencils. We discuss motivating examples from Apollonias and angle bisection. We also describe how harmonic ranges get induced by quadrangles, and how harmonic pencils get invoked by quadrilaterals. We also discuss the famous 13 configuration.

This videos uses some material from Wildberger's early lectures on universal hyperbolic geometry. The crucial point of our lectures is that harmonic conjugates are preserved by projection.

The cross ratio

http://en.wikipedia.org/wiki/Cross-ratio

is a more general projective invariant which takes a value of minus 1 when harmonic ranges occur.
Lecture 9
The Line Woven Net
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The Line Woven Net
The woven net (also known as the complete harmonic quadrangle-quadrilateral net) is a beautiful construction which begins with a quadrilateral (four points), and grows inwardly and outwardly as one repeatedly creates quadrilaterals (four lines) from quadrangles, which are dual to each other. Although many deep ideas can be seen by meditating on this construction, ultimately one just needs a pencil and a straight edge to make one. Essentially the woven net is a way to fill the projective plane by repeatedly creating `Thirteen Configurations'.
Lecture 10
Brianchon's Theorem (Pascal's Dual)
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Brianchon's Theorem (Pascal's Dual)
We briefly recap Pascal's fascinating `Hexagrammum Mysticum' Theorem, and then
introduce the important dual of this result, which is Brianchon's Theorem. Essentially
Brianchon's theorem says that if one circumscribes a hexagon on any circle (or, in fact, any conic section),
and then draws lines through opposite vertices of the hexagon, then these three lines meet at a unique point.
We also discuss relationships between Pascal's line and the Brianchon point. For example, it appears as though Pascal's line is the
polar of the Brianchon point (i.e., the Appolonian dual). The depth of these results is enough by itself, but these two theorems also
reveal important truths about the nature of conics in projective geometry.
Lecture 11
Five Points Define A Conic
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Five Points Define A Conic
We describe a remarkable implication of Pascal's theorem,
which is that given any 5 points in the plane (no 3 of which are co-linear), there
exists a single conic curve which passes through the 5 points. It also follows, by duality, that
a conic is uniquely defined by specifying 5 lines which are tangent to it.
Lecture 12
Projective Transformations Of Lines
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Projective Transformations Of Lines
Projective geometry may be defined as the study of features which do not change under projective transformations. This is one good reason to study such transforms. We begin by looking at simple cases where a projective transformation maps a line to itself. We focus upon how the number of fixed points depends upon the relative positions of
the primitives, and also have the novel idea of using a ghosted circle to illustrate the displacement between the image and the object at different points on the line.
Lecture 13
Involutions Of The Line
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Involutions Of The Line
We continue our study of projective transformations of the line, and study what configurations cause involutions. An involution is
a transformation which leaves positions unaltered, when applied twice. Interestingly, our questions about involutions quickly
return us to the consideration of harmonic quadrilaterals.
Lecture 14
Constructing The Dual Of A Quadrangle - The Thirteen Point Configuration
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Constructing The Dual Of A Quadrangle - The Thirteen Point Configuration
Here are the details regarding this classic construction from projective geometry:

Take a quadrangle (4 blue points in our case).

Make this into a `complete quadrangle' by drawing a line through every pair of points (so 6 blue lines are added).

The newly added lines meet at three extra points -the diagonal triangle of the quadrangle (so 3 yellow points, and three yellow lines are added).

Mark where edges of the diagonal triangle intersect with complete quadrangle edges (so 6 green points are added).

Draw all lines linking the six newly generated points (so 4 green lines are added).

The final four lines create a quadrilateral which is the dual of the initial quadrangle.
Also the initial quadrangle can be considered to be the dual of the quadrilateral.
Lecture 15
Pascal's Hexagrammum Mysticum Theorem
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Pascal's Hexagrammum Mysticum Theorem
Take a circle, and add six points to make a hexagon

Find the three points where opposite lines of the hexagon meet

The result claims these points always line up

Even if the order of the points are changed, or if the circle
is replaced by another conic (e.g., a hyperbola).

It only takes a minute to understand what is says, but why is it true ??

See my other videos on projective geometry for more details
Lecture 16
Non Euclidean Geometry & Hyperbolic Social Networks
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Non Euclidean Geometry & Hyperbolic Social Networks
My Website
https://sites.google.com/site/richardsouthwell254/

I begin by introducing non-euclidean geometry by discussing Euclid's postulates. I then discuss spherical geometry, great circles, warped triangles and map making. Then I discuss Beltrami's hyperboloid model, and its relationship to the Poincare disk. Then I discuss regular tiling of the plane, platonic solids and regular tilings of the hyperbolic plane. Then I discuss how to make hyperbolic paper, how triangles on saddles are distorted, and how to model hyperbolic space using the pseudosphere. The next topics are the Euler number and genus of a graph, and combinatorial curvature. I also discuss graph scaling dimension and the notion of scaled Gromov hyperboic graphs. Using this idea I discuss how real complex networks such as the LiveJournal can have negative curvature on a large scale.

Wolfram Demonstrations Used:

Lines through Points in the Poincaré Disk
http://demonstrations.wolfram.com/LinesThroughPointsInThePoi...

Tiling the Hyperbolic Plane with Regular Polygons
http://demonstrations.wolfram.com/TilingTheHyperbolicPlaneWi...

Hyperbolic paper from `The Shape of Space' by Jeffrey R. Weeks. Original credit for the Hyperbolic paper idea is given to Bill Thurston.

I reccommend Wildberger's course:

Universal Hyperbolic Geometry 0: Introduction
http://www.youtube.com/watch?v=N23vXA-ai5M&list=PLC37ED4C488...