Third Century Pappas of Alexandria Discovered Some Properties

Chapter 6Projective Geometry

Homework:

6.11, 2 prism only

6.24a

6.33, 4

History:

Third century Pappas of Alexandria discovered some properties

and constructions.

1425Brunelleschi started investigating the geometry of perspectives with respect to art. Alberti, an architect, develops the method of perspective drawing. (1435 – Della Pittura, Alberti’s seminal

1600ishKepler and Desargues independently worked out the “points at infinity” – a key concept in this type of geometry

1800ishPacal became interested and published some work

1822Poncelet made huge strides – declaring projective geometry to be metric-free

1892Fano’s geometry

1972Dirac gave a talk using Projective Geometry to discuss his revolutionary work in Physics

Overview:

A geometry using the Euclidean plane and transformations that are Projective

(a special subset of Affine). Projective transfomations preserve the degree of the

curve being mapped. They do not preserve angle measure or size. They do

preserve the cross-ratio and incidence relations. We use most often the projections

that map lines to lines. A projective transformation describes what happens to the

perceived position of objects when the point of view of the observer changes.

Every projective transformation has an inverse.

Think of a cube. If you are facing a side, it looks like a square. If you point a

vertex at your eye, you have a very different view of it. If you look straight at

an edge, it looks different again. Imagine the 8 vertices and their point values…a

projective transformation will “rotate” the cube so you can see it from any point of

view at all.

Look at a cube from various points of view here!

To see videos of the various views of a cube, check out:

In Projective Geometry the set of Euclidean points is augmented by a set of “points

AtInfinity”. The set of Euclidean lines is augmented by a set of “lines at

infinity”…so there is a feeling that there are more points and one more line than

usual…thoughinfinity plus infinity is still infinity, it feels like more to most people.

Axioms:

There are several sets of axioms floating around. Here are a couple of them.

From Wolfram Mathematics:

Undefined Terms:point, line, on

Axioms:

1 If A and B are distinct points on a plane, there is at least one line

containing both of them.

2 If A and B are distinct points on a plane, there is not more than one line

containing both of them.

3 Any two lines in a plane have at least one point of the plane in common.

(note: this may be a point at infinity)

4 There is at least one line on a plane

5 Every line contains at least three points of the plane.

6 All the points of the plane do not belong to the same plane.

Let’s look at A3 closely…

Where have we seen this axiom before?

Alfred Whitehead’s Axioms:

(English geometer early 20th century)

Undefined terms:point, line, on

Axioms:

1 Every line contains at least 3 points.

2 Every two points, A and B, lie on a unique line AB.

3 If lines AB and CD intersect, then so do lines AC and BD.

A, B, C, and D are distinct.

What does A3 give us?

What’s different between these 2 sets of axioms?

Here is a third set from our author Dave Thomas, page 283

1 There exists at least one line.

2 Each line contains at least 3 points.

3 Not all the points are on the same line.

4 Two distinct points determine a distinct line.

5 Two distinct lines determine a distinct point.

Note: An early off-shoot from Projective Geometries were the Finite Geometries

of which Fano’s Geometry is the most famous

7 lines, 7 points, no parallel lines – Where did you see this last?

Axioms for Fano's Geometry

Undefined Terms.point, line, and incident.

Axiom 1.There exists at least one line.

Axiom 2.Every line has exactly three points incident to it.

Axiom 3.Not all points are incident to the same line.

Axiom 4.There is exactly one line incident with any two distinct points.

Axiom 5.There is at least one point incident with any two distinct lines.

Let’s compare Fano’s with Thomas’s axioms.

There is a set representation of Fano’s that avoids the use of Euclidean lines

Points:{A, B, C, D, E, F, G}

Lines:{ADG} {BEG} {FCG} {DEF}

{BDC} {AEC} {ABF}

Here lines are just sets of 3 points that meet the axioms.

Here are some facts worth noting:

The number of points on a line is 2 + 1 = 3

The number of lines on each point is2 + 1 = 3

The number of points is

The number of lines is

This is an order 2 finite projective plane!

By convention a finite projective geometry is called:PG(a,b)

wherea is the projective dimension and b is one less than the number of points on a line (aka the order of the Geometry).

Fano’s Geometry is PG (2, 2).

Later on we’ll look at an order 3 projective plane (in 6.2)

4.1Alberti’s Method of Perspective Drawingpage 268

You have one rather like this in your homework, so let’s run through the steps.

You may sketch it by hand (clearly!) or use Sketchpad.

Position the object in a rectangular window:

Draw a line (M) parallel to and to the left of the window rectangle:

Put in a grid of parallel lines that are perpendicular to M

Quadrille graph paper is perfect for this!

Put in a line (L) perpendicular to M at the top of the window rectangle

Put in vertical spacers perpendicular to L and the grid lines from step 3

Reproduce the spacing of the grid lines from step 3 on line L…slanting up to reach line L

Construct a vertical line to the left of the leftmost transferred point on L, call this line N. It is perpendicular to L and to the left.

To the left of Line N, construct point P. Sketch in the lines connecting P to each intersection of the horizontal grid lines from step 3 and line M.

Now mark the intersection of these lines with line N.

10.

Construct lines through the interection points in step 9 parallel to line L

Construct a line through P parallel to L (this is called the Horizon Line).

Position the Vanishing Point and construct segments from the points along line L back to the this point.

Transfer the salient features of the original object to this new grid.

Do the steps here and toggle:

Let’s look at a vanishing point with a dilation and a triangle:

Imagine this is a PRISM and you get the picture with respect to parallel lines and ideal points

There is a nice exploration of multiple vanishing points in the book, 4.1

4.2 Introduction to Projective Geometry

We’ve already seen three versions of axioms for this space. Let’s work on the

visualizations.

Points:

We have two flavors of points:

Regular, ordinary Euclidean points in the Cartesian Plane and

Ideal points at “infinity”….math folks really just call these Ideal…

these correspond todifferent directions in the Euclidean plane and are at the ends of

lines that areEuclidean-parallel.These are the collection of vanishing points. The

point at whichrailroad tracksmeet at the horizon is one of these.

A set of collinear points is called a pencil of points. The line containing these

points is called the axis of the pencil.

Lines:

Ordinary Euclidean lines and we have

lines that contain both regular points and ideal points.

The set of all ideal points is called the ideal line. It corresponds to the set of all

possible directions in the Euclidean plane.

A set of concurrent lines is called a pencil of lines. The point of concurrency is

called the center of the pencil of lines.

Parallel Lines:

In Euclidean Geometry we have the Playfair set up…

In Projective Geometry, Euclidean parallel lines meet at an ideal point…in other

words we don’t have any parallel lines any more.

In fact, sets of parallel lines pointing in different directions have different ideal

points. The set of all ideal points is called the ideal line.

Page 281

Projective transformations:

A PERSPECTIVITY is a one-to-one mapping from a pencil of lines to a pencil of points if each line in the pencil of lines is incident with exactly one point in the pencil of points.

A PERSPECTIVITY is a one-to-one mapping from a pencil of points to a pencil of points if each line joining a point in the domain to a range point is incident with a fixed point O, called the center of perspectivity.

A PERSPECTIVITY is a one-to-one mapping from a pencil of lines to a pencil of lines if the intersection of each line in the domain with its corresponding range line lines on a fixed line called the axis of perspectivity.

Complete Quadrangle and Complete Quadrilateral

A complete quadrangle is a set of 4 points (no 3 collinear), and the 6 lines determined by those points.

{A, B, E, D} for example…list the 6 lines!

A complete quadrilateral is a set of 4 LINES, no 3 concurrent, and the 6 points determined by those lines. {ABF, DEF, BEG, ADG} …list the points!

Page 285

An order 3 projective plane

Points: {A, B, C, …, M}

Lines{ABCD} {AEFG} {AHIJ} {AKLM} {BEHK} {BFIL} {BGJM}

{CEIM} {CFJK} {CGHL} {DEJL} {DFHM} {DGIK}

Number of points on each line3 + 1 = 4

Number of lines on each point3 + 1 = 4

Number of points

Number of lines

These are heavily pattern related. The author notes some of the patterns to this type of representation on pages 284 and 285.

Duality

One of the most fun aspects of projective geometry is a property called duality.

You replace the words

“point” with “line” and vice versa

Collinear and concurrent

Intersect and join

Lie on and pass through

in the axioms and the theorems and you find that you have a perfectly respectable new geometry that works in a “truth value” sense.

This was discovered in about 1825 by Gergonne. And independently by Poncelet.

This allows a dual correspondence between two geometries and receiprocities between figures.

We won’t spend any time on it, of course. It is interesting if you want to follow up on your own.

The author has an interesting section of Error-Correcting Codes and how it ties in to projective geometry.

Here is another interesting use for projective geometry for in geo-referencing and literal mapping of spaces.

1.2. Georeferencing

In this chapter, you will learn about the process of georeferencing, what it means and how it works.

1.2.1. Spatial adjustment methods

Often, different sets of spatial data don't have the same coordinate system, even if they cover the same area. Therefore, it is not possible to combine these datasets directly. To solve this problem, one dataset has to be adapted to the other by a spatial adjustment what means that the coordinate system of one dataset needs to be transformed into the other coordinate system. In modern GIS software, this process is quite easy for the user. By selecting identical points in both the correct dataset and the one that needs to be fitted, a displacement link is being created and then, an algorithm calculates automatically the transformation process. Depending on the transformation, two or more points need to be evaluated in every dataset. After that, the second dataset is adapted to the first one. In contrast to a datum transformation, this transformation relies just on a local area .

Displacement links connect the two datasets

A dataset can be transformed in the following way: rotate, scale, skew and translate. They look like that:

Possible transformations (ArcGIS)

Not every transformation process contains all of these elements. The most important transformation methods are the following ones:

Similarity Transformation

The Similarity Transformation is the simplest one. It rotates, scale (same scale in x and y direction) and translates the object, but does not skew it. The following four parameters need to be calculated: rotate angle, scale factor, translation along x-axis and translation along y-axis. Since four unknown parameters have to be evaluated, the similarity transformation requires a minimum of two displacement links. Every displacement link results in a movement of the x and y coordinate, each coordinate gives one known parameter, and therefore, two displacement links give four parameters. Consequently, the transformation can be calculated. This transformation is congruent what means that it preserves the proportion of the objects; a square or (circle) will still be a square (circle) after the transformation.

Affine Transformation

Unlike the Similarity Transformation, the Affine Transformation includes an independent scale factor for the x and y direction and additionally, a skew factor. The process of rotating and skewing the coordinate system can also be imagined as two independent rotations of the x and y axis. Therefore, the axes aren't orthogonal any longer. Three displacement links are necessary for evaluating the six unknown parameters. This transformation is not congruent, but preserves parallelism what means that parallel lines will be still parallel. A square will be a rhomboid, a circle will be an ellipse.

Projective Transformation

The most complex one of the habitually used transformations is the Projective Transformation. It has eight unknown parameters what means a minimum of four displacement links. This transformation does not preserve parallelism. Squares will be transformed into a quadliteral.

Transformation methods