The lecture on 22.10 began with a repetition of the “guided visualization” from the first meeting on 15.10.  Then Dr. Gunn demo-ed a couple of dynamic geometry applets demonstrating two theorems from the “early days” of projective geometry (circa 1640): Desargues Theorem and Pascal’s Theorem, emphasizing the qualities of mobility and variety which both theorems exhibit. Links can be found on the previous post.

Coordinates for projective geometry. When projective geometry was rediscovered in the 1820’s, the mathematics was available to introduce coordinates.  In modern form, this can be expressed as the projectivization of $\mathbb{R}^n$ introduced in the previous lecture:   \[\mathbf{P}^2(\mathbb{R}) := \mathbb{R}^3~/~\sim \text{  where } \mathbf{P} \sim \mathbf{Q} \iff \mathbf{P} = \lambda \mathbf{Q} \text{ for } \lambda \neq 0\].

Note: the following discussion focuses on the case of the projective plane $\mathbf{P}^2(\mathbb{R})$.  Everything said here generalizes easily to general dimension $n$, in particular to $n=3$ which is the case we are really interested in.

Projective transformations and the projective group.  A projective transformation, or projectivity for short, is defined as an invertible self-map of $\mathbf{P}^n(\mathbb{R})$ which preserves lines.  Here, a line is the set of points spanned by two points: $\alpha \mathbf{P} + \beta \mathbf{Q}$ for $\alpha, \beta \in \mathbb{R}$.  It’s not hard to see that this corresponds to a plane through the origin in $\mathbb{R}^{n+1}$.   An invertible self-map of $\mathbb{R}^{n+1}$ which maps planes to planes is what is called a linear map, and satisfies $f(\alpha \mathbf{P} + \beta \mathbf{Q}) = \alpha f(\mathbf{P})+\beta f(\mathbf{Q})$.  The set of all such self-maps forms a group, called the general linear group of dimension $n+1$, and written $GL(n, \mathbb{R})$.  Once a basis has been chosen one has a representation as matrices, hence $GL(n+1,\mathbb{R})$ is sometimes called a matrix group.

Exercise: Show that an element of $GL(n+1,\mathbb{R})$ induces a self-map of $\mathbf{P}^n(\mathbb{R})$ which maps lines to lines, that is, is projective.  And conversely, a projective transformation comes from some element of  $GL(n+1,\mathbb{R})$.  Two matrices give rise to the same projective transformation $\iff$ they are non-zero multiples of each other.

Hence the group of projectivities is the quotient of $GL(n+1)/ \mathbb{R}$.  It is written $PGL(n+1,\mathbb{R})$ to indicate that it results from projectivizing $GL(n+1,\mathbb{R})$. It has one less dimension than $GL(n+1)/ \mathbb{R}$.  Hence $PGL(3,\mathbb{R})$ has dimension $9-1=8$.

The group $$PGL(n+1,\mathbb{R})$ includes a huge range of possible transformations. A basic result which characterizes this group is:  a projectivity is uniquely determined by  choosing any $n+2$ points to be mapped to any other $n+2$ points (of course the points must be in general position).  The study of possible projective transformations is an interesting subject in its own right but one which we have to pass by right now.

Often one is interested only in the affine subgroup, that is, the transformations that map the ideal elements to themselves.  Exercise:  An affine transformation has last row $(0,0, ….,0,1$).

Among the affine transformations, one is interested particularly in isotropic scaling by the factor $\lambda$.  Such a transformation has a diagonal matrix all of whose diagonal entries are $\lambda \neq 0$, except the last one which is, as noted, 1.  For this course we will be particularly interested in the subgoup of the affine group that preserves distance and angle:

Euclidean Isometries. Recall that a euclidean transformation is a self-map of euclidean space that preserves euclidean distances and angles.   Exercise: A euclidean isometry is a projectivity.    Hence we can identify the euclidean group with a subgroup of $PGL(n+1, \mathbb{R})$.  Furthermore, the set of orientation-preserving isometries $\mathbf{E}^+(n)$ is a subgroup of the full euclidean group $\mathbf{E}(n)$.  Exercise: An isometry is orientation-preserving $\iff$ its determinant is positive.

Exercises.  All these exercises refer to the euclidean plane.

1. A euclidean translation by the vector $\mathbf{u} = (x_u, y_u)$ is given by the matrix\[ \mathbf{T}_{\mathbf{u}} = \left( \begin{array}{ccc} 1 & 0 & x_u \\ 0 & 1 & y_u \\ 0 & 0 & 1 \end{array} \right)\]
2. A euclidean rotation around the origin by angle $\alpha$ is given by the matrix \[ \mathbf{R}_{O,\alpha}  =  \left ( \begin{array}{ccc} \cos{\alpha} & -\sin{\alpha} & 0 \\ \sin{\alpha} & \cos{\alpha}  & 0 \\ 0 & 0 & 1 \end{array} \right)\]
3. Write the matrix form for a rotation $\mathbf{R}_{\mathbf{C},\alpha}$around the point $\mathbf{C} = (x_c, y_c)$ through angle $\alpha$.  [Hint: $\mathbf{R}_{\mathbf{C},\alpha} = \mathbf{T}_{\mathbf{C}} \circ \mathbf{R}_{O,\alpha} \circ \mathbf{T}_{\mathbf{-C}}$.]

How translations and really a kind of rotation [optional].  One of the unusual aspects of euclidean geometry is that translations and rotations seem so different, yet both are isometries.  Using the projective model we have developed, it’s possible to argue that a translation is really a kind of rotation.  Consider the translation $\mathbf{T}_{(1,0)}$ with translation vector $(1,0)$.  Imagine the series of rotations $R_n$ indexed by $n$ with center $(0,n)$ through an angle $\dfrac{1}{2 \pi n}$. Exercise: For any bounded region $D$ of the plane, I can find an $n_0$ such that $R_n -\mathbf{T}_{(1,0)}$ for $n> n_0$ is arbitrarily small for any point in $D$.  Taking this to the limit, one arrives at the conclusion that the translation $\mathbf{T}_{(1,0)}$ can be thought of as a rotation around the ideal point $(0,1,0)$ through a distance (not angle!) 1. Similar results apply for any translation; the “center point” is then the ideal point in the direction perpendicular to the translation.

Conclusion. This concludes the theoretical introduction to how euclidean isometries can be expressed as matrices. The next step is to study the jReality classes:

1. de.jreality.math.Rn  This class provides general report for the euclidean vector space $\mathbb{R}^n$.  This includes multiplying matrices times vectors and other operations from linear algebra.
2. de.jreality.math.MatrixBuilder  This is the primary class for constructing euclidean isometries.

Equipped with this knowledge, you should then be prepared to go to work on Assignment1.  For details consult the Javadoc for the Java class in eclipse.

The Renaissance was a time of great innovation in all areas of European life.  Science, art, and religion all received powerful new impulses.  I focused on the development of painting, where the invention of perspective painting had important consequences for mathematics. The first perspective paintings appeared around 1420 (Masaccio).  The slides containing the paintings discussed in the lecture can be found here.

From Art to Geometry. Around 100 years later the process had been mechanized (Dürer), and finally, after another 100 years, the French architect and mathematician Rene Desargues published a book (1642) which provided a new geometry to represent perspective painting.  This new geometry is called projective geometry.  The name comes directly from its roots in perspective painting, as follows.

Central Projection. The essential operation of perspective painting was identified as a central projection, in which points in the world are projected onto points of an image plane by means of lines passing through the center point of the projection (think of the eye of the painter, and the image plane as a plane standing between the painter and the world — which he transfers to his canvas). The essential problem in terms of describing such a projection geometrically was the existence of vanishing points: points in the image where parallel lines in the world appear to meet.  It was awkward to have a pairing of points in which some points did not have a partner: the vanishing points on the image had no real partner point in the world since parallel lines, by definition, do not meet.

Ideal elements. Desargues innovation was to suppose that parallel lines in fact do meet, in a special sort of point, an ideal point.  All lines parallel to a given line share the same ideal point. All the ideal points of all the lines in a plane go together to form a line, the ideal line of the plane. (You might think they would form a circle, but since it’s “infinitely” far away, it can’t be curved, so is a line.)  And all the ideal points of all the lines in space form a plane, the ideal plane.  (These ideal points bear a strong resemblance to free vectors, see Exercise 3 below.)

The first flower. Desargues developed his new geometry, based on the addition of these ideal elements to the ordinary elements, purely synthetically, that is, without coordinates. He discovered beautiful theorems (for example, Desargues Theorem).  His younger colleague, Blaise Pascal, contributed other remarkable theorems, such as Pascal’s Theorem. But the new geometry was too far ahead of its time.  The geometry of Desargues’ friend Rene Descartes was better suited to  new impulses in natural science which remained within the framework of euclidean geometry.  Cartesian geometry spread rapidly, while Desargues’ work was forgotten and disappeared.  It was rediscovered, independently, in 1822 by another Frenchman, Jean-Victor Poncelet, and has since then developed into a broad and deep branch of mathematics.  There is reason to believe that it may yet prove useful to the further development of natural science just as Cartesian geometry has been useful during the past 400 years.

When it was rediscovered, projective geometry was outfitted with coordinates, it is these coordinates which I want now to describe, since they are important to the field of mathematical visualization.

Mathematical definitions. The real projective plane $\mathbf{P}^2(\mathbb{R})$ is defined by applying an equivalence relation to the euclidean vector space $\mathbb{R}^3$.  Two points with coordinates $(x,y,z)$ and $(x’,y’,z’)$ are defined to be equivalent $\iff$ $(x’,y’,z’)=(\lambda x, \lambda y, \lambda z)$ for some $\lambda \neq 0$.  This equivalence relation can be expressed geometrically by saying, “the points of the projective plane are the lines of $\mathbb{R}^3$}”.   Each projective points has many possible representatives $(\lambda x, \lambda y, \lambda z)$ from $\mathbb{R}^3$.  We write $(x,y,z)$ (a bit sloppily) to mean the equivalence class of $(x,y,z)$.

The connection to Desargues. In this definition of the projective plane, where are the “ordinary” points and where are the “ideal” points discussed above?  The simplest answer is to define a point $(x,y,z)$ to be ordinary if $z \neq 0$, and to be ideal if $z = 0$.  In the former case, we can choose $\lambda = \dfrac{1}{z}$ to obtain an equivalent representative point $(\dfrac{x}{z}, \dfrac{y}{z}, 1)$.  Hence we can assume the ordinary points are of the form $(x,y.1)$ while, as we show below, the ideal points appear in the form $(x,y,0)$.  We call these coordinates affine coordinates for the projective plane.

Intersection of parallel lines. Let’s apply this framework to the problem of representing the intersection of parallel lines.  If we represent a line by its line equation in ordinary coordinates, it has the form $ax+by+c=0$ where not both $a$ and $b$ are zero.  Using affine coordinates, this takes the form $ax+by+cz=0$.  In this form the expression is symmetric in $(a,b,c)$ and $(x,y,z)$ hence $(x,y,z)$ coordinates for the projective plane are sometimes called homogeneous coordinates. It’s clear that if $(x,y,z)$ satisfies the line equation, then so does $(\lambda x, \lambda y, \lambda z)$ for $\lambda \neq 0$, hence the line equation is well-defined for points of the projective plane, and our choice of $z=1$ is not essential.

Exercises

1. Intersection of two lines.
   1. Show that the intersection of the two lines $l : ax+by+cz=0$ and $m : dx+ey+fz=0$ is the projective point $(bf-ce, -af+cd, ae-bd)$. [Hint: show this expression satisfies both equations.]
   2. Show that $l$ and $m$ are parallel $\iff$ $ae-bc = 0$, hence the intersection point $(x_i, y_i, z_i)$ is ideal $\iff$ $z_i = 0$. Hence the ideal line of the plane is given by the equation $z = 0$.
   3. The ideal point of the line with equation $ax+by+cz=0$ is $(b, -a, 0)$. [Hint: apply A + B above.] Conclude that all lines of the form $ax+by+cz=0$ for fixed $a$ and $b$ have the same ideal point $(b,-a,0)$.
2. Joining line of two points.
   1. Show that the joining line of  two points $P : (x_0,y_0,z_0)$ and $Q: (x_1,y_1,z_1)$  is the line with line equation $ax + by + cz = 0$ where $(a,b,c) = (y_0 z_1 – y_1 z_0, x_1z_0 – x_0 z_1, x_0 y_1 – x_1 y_0)$. [Hint: show that both points satisfy this equation.]
   2. In high school, using traditional Cartesian coordinates, perhaps you learned that the line through the point $P := (x_0,y_0)$ with direction vector $V := (u,v)$ has line equation $-vx +uy + x_0v-uy_0 = 0$.  Show that you obtain the same line equation in the projective setting by joining the point $(x,y,1)$ with the ideal point $(u,v,0)$ using the formula in part A above.
   3. Result B) above implies that a free vector in Cartesian geometry is similar to an ideal point in projective geometry.  Consider other properties of free vectors and test whether and how they carry over to ideal points.  For example, a vector is often defined as the difference of two points.  (For this exercise, assume that we are using affine coordinates for the ordinary points in $\mathbf{P}^2(\mathbb{R})$, i.e., points have the form $(x,y,1)$.)  On the basis of this, discuss the following statement:  “An ideal point is a direction.”

Hopefully these exercises have convinced you that projective geometry in fact solves the problem posed by vanishing points, and provides a  representation with which we can compute practical problems.

Eye and hand:   There are in good reasons to deny the existence of the ideal points of projective geometry, as well as to assert their existence.  How can one resolve this contradiction? One way is to understand the two positions as representing two different human sense organs.  The euclidean view, that parallel lines do not meet,  is derived from the evidence provided by the sense of touch.  I can never hope to touch an ideal point since if I try to move towards it, the parallel lines remain the same distance apart.  On the other hand, the sense of sight sees the parallel sides of a train track intersect at the horizon as clearly as the sense of touch knows they stay the same distance apart forever.  The different senses give rise to different geometries.  Neither one is right or wrong, just different.

Exercise. It’s only been around 600 years, since the early Renaissance, that human beings (first just a few, now everyone) have been conscious of seeing parallel lines intersect in a “realistic” perspective picture. Why do you think this development took so long to occur?

Resources: For more detailed treatments of projective geometry see the resources listed on the webpage for the course Geometrie I, such as these lecture notes by Boris Springborn.

Coming next: Using projective geometry to represent geometric transformations, such as translations, rotations, scaling, and more.

We start out on our expedition.  We began by doing some “guided mathematical visualization” with our eyes closed, led by Dr. Gunn.  At the other extreme Dr. Gunn asked whether a crystal he brought with him represents “mathematical visualization”. We chose to leave unanswered the question of whether some being could have thought the crystal into existence as a kind of “mathematical visualization”.  In any case we decided that neither example qualified as a mathematical visualization (MV) in the sense of this course.  To be a MV in the sense of this course means to combine something of the thought quality of pure mathematics with the sense-perceptible, material nature of the crystal.  It must be an “embodiment” of mathematics.

Next, Dr. Gunn introduced 3 types of “product” that qualify as mathematical visualization:

1. 3D prints (the ones he passed around he had made in the last 6 years at the 3D Lab at TU Berlin.) Please also check out this blog post from Franziska Lippoldt — a student in last year’s course — describing her experience making a 3D print from her semester project.
2. Mathematical movies, where the content was generated directly from mathematical models in the computer.
   1. Outside In: A movie from the Geometry Center on turning the sphere inside out.
   2. The Borromean Rings: A movie from TU Berlin introducing the new logo (new in 2006!) of the International Mathematical Union.
3. Interactive applications which allow the user to explore and experience a mathematical domain.  Example of the Archimedean solids app.

We noted that these three categories fit in sequence between the crystal and pure math, as they progress from more “thing-like” to more “mind-like”, the latter being characterized by pure freedom, the former by pure necessity — in philosophical terms.

We established that most if not all of the course participants plan to go into some “applied” area when they graduate from the TU.  Hence they will not be working in “pure math” mode but in a mode in which mathematics is brought into some “embodied” form in the real world.  Hopefully the visualization course can be a preparation for this direction of work and will develop relevant non-mathematical skills  such as design, communication, and teamwork.