Degree project

Projective Geometry

Author: Wu Wei Supervisor: Hans Frisk Examiner: Per Anders Svensson Course Code: 2MA41E Subject: Mathematics Level: Bachelor Department Of Mathematics

Projective Geometry

Wei Wu

June 13, 2019

Abstract

Projective geometry is a branch of mathematics which is founda-tionally based on an axiomatic system. In this thesis, six axioms fortwo-dimensional projective geometry are chosen to build the structurefor proving some further results like Pappus’ and Pascal’s theorems.This work is mainly in synthetic geometry.

Contents

1 Introduction 51.1 Informal description of projective geometry . . . . . . . . . . 51.2 What is projective geometry? . . . . . . . . . . . . . . . . . . 61.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 What you will find in this thesis . . . . . . . . . . . . . . . . 7

2 The axiomatic system and duality 82.1 First five axioms for the projective plane . . . . . . . . . . . . 92.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 An infinite model for projective plane . . . . . . . . . . . . . 13

3 Harmonic sets 153.1 Construction of the fourth point of a harmonic set . . . . . . 163.2 Construction of the fourth line of a harmonic set . . . . . . . 19

4 Perspectivities and Projectivities 214.1 Perspectivities . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Projectivities . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Construction of a projectivity between pencils of points . . . 26

5 Point and line conics 31

6 Conclusion 36

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List of Figures

1 Vanishing line BC and points B,C. . . . . . . . . . . . . . . 52 Finite model with 13 points. . . . . . . . . . . . . . . . . . . . 83 A triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 The diagonal points E, F , G from a complete quadrangle

ABCD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Illustration of Desargues’ theorem in 3D projective space. . . 116 The diagonal lines EF , BC, AD of a complete quadrilateral

abcd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4ABC and 4A′B′C ′ are perspective from the line l. Points

AC ·A′C ′ = R, BC ·B′C ′ = Q, AB ·A′B′ = P all lie on line l. 148 A projective plane model in 3D. . . . . . . . . . . . . . . . . . 149 Harmonic set H(AB,CD). . . . . . . . . . . . . . . . . . . . 1510 Five collinear points form a quadrangular set (AA′)(BB)(CD). 1611 Six collinear points form a quadrangular set (AA′)(BB′)(CD). 1612 Unique point D for harmonic set H(AB,CD). . . . . . . . . 1713 H(AB,CD)⇔ H(CD,AB). . . . . . . . . . . . . . . . . . . . 1814 H(ab, cd). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915 Pencil of lines. . . . . . . . . . . . . . . . . . . . . . . . . . . 2116 Pencil of points. . . . . . . . . . . . . . . . . . . . . . . . . . . 2117 Perspectivity between pencils of points. . . . . . . . . . . . . 2218 Perspectivity between pencils of lines. . . . . . . . . . . . . . 2319 Perspectivity between a pencil of points and a pencil of lines. 2320 A perspectivity between pencils of points. . . . . . . . . . . . 2421 A projectivity between pencils of points. . . . . . . . . . . . . 2422 ABC ∧A′′B′′C ′′. . . . . . . . . . . . . . . . . . . . . . . . . . 2423 abc ∧ a′′b′′c′′. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424 abc ∧A′′B′′C ′′. . . . . . . . . . . . . . . . . . . . . . . . . . . 2525 Axis of projectivity h for ABC ∧A′B′C ′. . . . . . . . . . . . 2726 A hexagon P1P2P3P4P5P6. . . . . . . . . . . . . . . . . . . . . 2827 ABC ∧BCA . . . . . . . . . . . . . . . . . . . . . . . . . . . 2928 Invariant points C,F . . . . . . . . . . . . . . . . . . . . . . . 2929 Only one invariant point C. . . . . . . . . . . . . . . . . . . . 3030 Five points U,U ′, A,B and C generate a point conic. . . . . 3231 Lines from S and R are projectively related. . . . . . . . . . . 3232 The Pascal’s line is outside of the ellipse. . . . . . . . . . . . 3333 Construction of a parabola. . . . . . . . . . . . . . . . . . . . 3434 Construction of a hyperbola. . . . . . . . . . . . . . . . . . . 3435 Five lines u, u′, a, b, and c generate a line conic. . . . . . . . 3436 The Brianchon’s point B is inside of the ellipse. . . . . . . . . 3537 The Brianchon’s point B is outside of the ellipse. . . . . . . . 35

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List of Tables

1 Point and line conic. . . . . . . . . . . . . . . . . . . . . . . . 31

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1 Introduction

1.1 Informal description of projective geometry

Projective geometry is a subject which originates from visual arts: usingfigures to record the shape by observation. The transformation that mapsobjects onto the plane is an example of a projective transformation. Wehave a lot of projective transformations in our daily life. One example isthat if we see a cylinder from above, it is a circle; but if we observe it fromthe side, it will be a rectangle.

We are all familiar with Euclidean geometry that has one character-istic of pairs of lines: parallel lines never meet. However, an exception isdiscovered by our eyes. The railway should be a pair of parallel lines butwhen we see the end of the railway it looks like they will meet somewherein the end. If the parallel lines will meet at infinity the space is transformedinto a new type of geometric object, the projective space. Projective spacecan thus be defined as an extension of Euclidean space in which two linesalways meet in an infinite point. In some way, Euclidean geometry is aboutthe object itself while projective geometry is about investigating the objectunder observation.

Take as an example the sides of a cube (see figure 1). Point B is theinfinite point of lines AB and DB. We call those infinite points the vanishingpoints and the line which goes through the vanishing points is the vanishingline [4].

Figure 1: Vanishing line BC and points B,C.

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1.2 What is projective geometry?

Let us now turn to projective geometry as a branch of mathematics. Theplane geometry of Euclid’s Elements is the same as the geometry of linesand circles: the tools are the ruler (the straight-edge or unmarked ruler) andthe compass. A Danish geometer Georg Mohr (1640-1697) and an ItalianLorenzo Mascheroni (1750-1800) have independently discovered that in thegeometrical constructions nothing is lost by only using the compass. Forexample for four points A, B, C, and D. We can find the intersection pointof lines AB and CD only by using compass although the actual process isquite complicated.

However, what’s going on if we only use rulers in geometry? It looksunacceptable since we can not even complete Euclid’s first proposition (Toconstruct an equilateral triangle on a given finite straight line) by only usinga ruler. Can we only use a ruler to develop a geometry which is not includingcircles, distance, angles, betweenness and parallelism? The answer is yes;this is projective geometry. It has not so much structure but anyhow full ofbeauty. Here, the basic concepts of projective geometry are listed here [2].

• Point, line and incident are taken as undefined terms

• Collinear: Any number of points that are incident with the same lineare said to be collinear.

• Concurrent: Any number of lines incident with a point are said to beconcurrent.

1.3 History

As previously stated in section 1.1, projective geometry is a sub-ject which originates from visual and it begins with work of an architect.In 1425, an Italian architect Brunelleschi started to discuss the geometri-cal theory of perspective which was summarized in a treatise by Albertia few years later. Although Menaechmus, Euclid, Archimedes and Apol-lonius studied conics in the fourth and third centuries B.C., the earliestprojective theorems were discovered by Pappus of Alexandria in the thirdcentury A.D. The French mathematician, J. V. Poncelet (1788-1867), wasthe first to prove such theorems by purely projective reasoning. The Ger-man astronomer Johann Kepler (1571-1630) and the French architect GirardDesargues (1591-1661) introduced however the concept of a point at infinitymuch earlier.

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Poncelet could then construct a projective space from ordinary spaceby introducing a line at infinity consisting of all the points at infinity. In1871, Felix Klein provided an algebraic foundation for projective geometryin terms of ”homogeneous coordinates”. This means projective geometrycould be analyzed with coordinates. However, in this thesis, we will notfocus on analytic projective geometry [1].

1.4 What you will find in this thesis

In this thesis, we select Judith N. Cederberg’s book as our main materialand take its six axioms. All definitions and theorems are built on the sixaxioms. Also, this thesis will using GeoGebra for visualization to help usunderstand the projective geometry. The paper will first introduce axiomaticsystem then find a particular set of points and lines, next talk about therelationships between the particular set of points and lines, finally describethe constructions which on the projective plane.

In section 2 the axioms are introduced and in section 3 a special setof four points, the harmonic set, is constructed. This relation between fourpoints is invariant under the projective transformations which is the topicof section 4. Finally, in section 5 the point and line conics are studied.

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2 The axiomatic system and duality

The statements that are accepted without proofs are known as ax-ioms. Other statements which can be proved by using the axioms are calledtheorems.

Mathematically, an axiomatic system is a collection of axioms, and oneor all of the axioms can be used to prove theorems logically. A mathematicaltheory consists of an axiomatic system and theorems which are derived bythe axiomatic system. In this thesis, we take the six axioms from our keymaterial [1] to develop the axiomatic system for the projective plane.

In this thesis only models with an infinite number of points will beconsidered. Finite models have also been studied [1], they use Axiom 1-3below but have another fourth axiom. Our first four axioms guarantee thatthere are at least four points on each line and in total 13 point (see figure2). The infinite model is presented at the end of this section.

If one includes axioms including the third dimension, like in [2], it ispossible to prove our Axiom 5 (Desargues’ theorem). This thesis is onlyabout 2D and then we have to take it as an axiom, see however figure 5 inpage 11 for a visualization in 3D.

Figure 2: Finite model with 13 points.

In this axiomatic system, the undefined concepts are point and line andincident. Points are said to be collinear if they are incident with the sameline and lines are said to be concurrent if they intersect at the same point.The first five axioms are presented below. The sixth axiom will be given insection 4.

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2.1 First five axioms for the projective plane

Axiom 1

Any two distinct points are incident with exactly one line.

Axiom 2

Any two distinct lines are incident with at least one point.

Notice that although Axiom 1 is the same as in Euclidean geometry,Axiom 2 is not. In projective geometry there are no parallel lines. Thereforewe have to introduce the concept of an ideal point which is the point at infin-ity added in projective geometry as the assumed intersection of two parallellines. Also, Axiom 1 and 2 do not show the existence of either points or lines.

Axiom 3

There exist at least four points, no three of which are collinear.

Axiom 3 shows us that points and lines exist in the projective plane sowe can consider a set of three noncollinear points and a set of four pointsno three collinear, the triangle and the quadrangle.

Definition 2.1

A set of three noncollinear points forms a triangle (4ABC), and the three

points determine three lines. The points A,B,C are called vertices and the

lines a, b, c are called sides of the triangle. (figure 3)

Figure 3: A triangle.

Definition 2.2

A set of four points A,B,C,D, no three collinear forms a (complete) quad-

rangle, and the four points determine six lines AB and CD, AC and BD,

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AD and BC. The points are called vertices and the lines are called sides

of the quadrangle. The points E, F , G at which pairs of opposite sides AB

and CD, AC and BD, AD and BC intersect are called diagonal points of

the quadrangle. (figure 4)

Figure 4: The diagonal points E, F , G from a complete quadrangle ABCD.

We see in figure 4 that the three diagonal points from a complete quad-rangle form a triangle and this observation is taken as the fourth axiom.

Axiom 4

The three diagonal points of a complete quadrangle are never collinear.

Axiom 1-4 describe the essential properties of the projective plane, allabout the points and lines themselves. The next two axioms are more aboutthe relation between a set of points and lines. In this section we only intro-duce the first five axioms; we put the sixth axiom in section 5.

Axiom 5 is about two relationships between pairs of triangles. Beforewe show the axiom, the properties of the relation will be given as a definition.

Definition 2.3

Triangles 4ABC and 4A′B′C ′ are said to be perspective from a point if the

three lines joining corresponding vertices, AA′, BB′ and CC ′ are concurrent.

The triangles are said to be perspective from a line if the three points of

intersection of corresponding sides, AB ·A′B′, AC ·A′C ′, and BC ·B′C ′ are

collinear. (figure 7, page 14)

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The corresponding axiom is given below.

Axiom 5 (Desargues’ theorem)

If two triangles are perspective from a point then they are perspective from

a line.

In plane projective geometry we have to set the Desargues’ theorem asan axiom since this statement does not hold for some geometries that alsosatisfy Axiom 1-4 [2].

For a visualization of Desargues’ theorem in 3D, see figure 5: A trian-gular pyramid OABC, points A′, B′ and C ′ are on the sides of the triangularpyramid. 4ABC and 4A′B′C ′ are perspective from the point O. We canobserve that point D, E and F lie on the line which is the same as themeeting line of plane ABC and plane A′B′C ′. Note that the lines A′B′ andAB must cross each other since they lie in the plane OAB.

Figure 5: Illustration of Desargues’ theorem in 3D projective space.

2.2 Duality

The dual of a statement is replacing each occurrence of the word”line” by the word ”point” and vice verse. Axioms 1 and 2 are nearly dualstatements.

If in an axiomatic system, the dual of any axiom can be proven as atheorem, it satisfies the duality principle. To ensure that the axiom system

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in this thesis satisfies the duality principle, we must prove the dual of eachaxiom. Some of the proofs will be given below.

Theorem 2.1 (Dual of Axiom 1)

Any two distinct lines are incident with exactly one point.

Proof. By Axiom 2, any two distinct lines are incident with at least one

point, assume there are two points incident with the two distinct lines. But

this assumption is a violation of Axiom 1. Hence this assumption is invalid.

Therefore any two distinct lines are incident with exactly one point.

Here we introduce the concept of a quadrilateral which is the dual ofa quadrangle, and figure 6 in page 13 will help us understand the conceptmore directly.

Definition 2.4

A (complete) quadrilateral is a set of four lines a, b, c and d, no three concur-

rent, and the four lines determine six points. The points are called vertices

and the lines are called sides of the quadrilateral. If a, b, c, d are four lines

of the quadrilateral, a · b and c · d, a · c and b · d, and a · d and b · c are said

to be pairs of opposite vertices. The lines joining pairs of opposite vertices

are called diagonal lines.

We see in figure 6 that the diagonal lines are not concurrent and thiscan be proven.

Theorem 2.2 (Dual of Axiom 4)

The three diagonal lines of a complete quadrilateral are never concurrent.

Proof. Let abcd be an arbitrary complete quadrilateral. See figure 6 in page

13, let points A = a · c, B = a ·d, C = c · b, D = b ·d, E = a · b, and F = c ·d.

Then the diagonal lines are EF , BC, AD and assume they are concurrent.

The diagonal points of quadrangle ABCD are E,F,G. Point G is on lines

BC and AD, these two lines cross on EF (the three lines where assumed

to be concurrent), so point G is on line EF . This is a contradiction since

according to axiom 4 the three diagonal points are not collinear. Therefore

the diagonal lines EF , BC, AG are never concurrent.

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Figure 6: The diagonal lines EF , BC, AD of a complete quadrilateral abcd.

Finally the dual of the Desargues’ theorem can be proved.

Theorem 2.3 (Dual of Axiom 5)

If two triangles are perspective from a line then they are perspective from a

point.

Proof. See figure 7 in page 14 to follow the proof. Assume 4ABC and

4A′B′C ′ are perspective from a line l, let AC · A′C ′ = R, BC · B′C ′ = Q,

AB ·A′B′ = P . To prove the two triangles are perspective form a point, we

need to show lines AA′, BB′, and CC ′ are concurrent. Let O = AA′ · BB′,then we have a look at 4RAA′ and 4QBB′. It is easy to see lines AB,

A′B′ and RQ are concurrent and intersect at the point P . So we can say

4RAA′ and 4QBB′ are perspective from a point. By using Axiom 5 we

should get 4RAA′ and 4QBB′ are perspective from a line. Hence we get

points AR ·BQ = C, A′R ·B′Q = C ′, AA′ ·BB′ = O and they are collinear.

So 4ABC and 4A′B′C ′ are indeed perspective from the point O.

2.3 An infinite model for projective plane

For a visualisation of the duality we need a model for the projectiveplane, see figure 8 in page 14. A point P in the plane α is represented by a

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Figure 7: 4ABC and 4A′B′C ′ are perspective from the line l. Points

AC ·A′C ′ = R, BC ·B′C ′ = Q, AB ·A′B′ = P all lie on line l.

ray a through the origin O. The ray a is the normal vector to a plane throughO which intersects the α-plane along a line l. To obtain the projective planewe add to α-plane the ideal points, represented by lines in the xz-plane. Theideal line corresponds then to the xz-plane. To each point in the projectiveplane corresponds a line and vice versa.

Figure 8: A projective plane model in 3D.

So far this section have focused on the axiomatic system, the followingsection will discuss a set of important points.

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3 Harmonic sets

This section is going to introduce a particular set of four collinearpoints (and a dual set of four concurrent lines) which is defined in terms ofa quadrangle. This four points have a special relation and it turns out thatthis relation remains after a so called projective transformation, which willbe the subject of section 4.

Definition 3.1

Four collinear points, A,B,C,D are said to form a harmonic set H(AB,CD)

if there is a complete quadrangle EFGH in which two opposite sides FE and

GH pass through point A, two other opposite sides EH and FG pass through

B, while the remaining side EG pass through point C and FH pass though

point D, respectively. C is called the harmonic conjugate of D (or D is the

harmonic conjugate of C) with respect to A and B. (figure 9)

Figure 9: Harmonic set H(AB,CD).

See figure 9, following the definition, with the quadrangle EFGH wecan have another harmonic set H(BA,CD) since points A and B bothare the intersection of two opposite sides of a quadrangle. Also, we haveH(BA,DC) from the same quadrangle since the two points C and D areboth the points of the remaining side.

So the quadrangle have at least four related harmonic sets which mean:H(AB,CD)⇔ H(BA,CD)⇔ H(AB,DC)⇔ H(BA,DC)Note: The symbol ⇔ means assume ... then we have ....

We have six sides of the quadrangle so normally we get six intersec-tions with an arbitrary line in the plane. If the line goes through one

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diagonal point we get five intersections (figure 10). We use the symbol(AA′)(BB′)(CD) to denote the six intersection points A,A′, B,B′, C,D. Inthis case, we call it a quadrangular set (figure 11). If the line goes throughtwo diagonal points it is a harmonic set and they are particular cases ofquadrangular sets. H(AB,CD) can also be written as (AA)(BB)(CD).

Figure 10: Five collinear points form a quadrangular set (AA′)(BB)(CD).

Figure 11: Six collinear points form a quadrangular set (AA′)(BB′)(CD).

3.1 Construction of the fourth point of a harmonic set

Focusing on the quadrangle AEBH (see figure 12 in page 17) we can findthe three diagonal points F , G and D and by Axiom 4 they are three distinctpoints which are never collinear. Points A and F are collinear, points B and

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G are collinear and points C,G, F are collinear, hence point D is distinctfrom point A,B and C. Is point D unique if we have points A,B and C?The answer is yes and we set the uniqueness of point D as a theorem.

Figure 12: Unique point D for harmonic set H(AB,CD).

Theorem 3.1

If A.B and C are three distinct, collinear points, then D, the harmonic con-

jugate of C with respect to A and B, is unique.

Proof. Let EFGH and E′F ′G′H ′ be two quadrangles which have the same

A, B, C then find their conjugate pointsD andD′ (see figure 12). This means

E′F ′ ·G′H ′ = A, F ′H ′ ·E′G′ = B, and F ′G′ ·AB = C. Let E′H ′ ·AB = D′.

We need to show D = D′. To do this, we need Axiom 5 and its dual.

By the dual of Axiom 5 4EFG and 4E′F ′G′ are perspective from a point

since they are perspective from a line AB. Also we can say EE′, FF ′ and

GG′ are concurrent. Similarly 4FGH and 4F ′G′H ′ are perspective from

AB, hence FF ′, HH ′ and GG′ are concurrent. Therefore EE′, FF ′, HH ′

and GG′ are all concurrent. By Axiom 5, 4EHG and 4E′H ′G′ are per-

spective from a line AB since they are perspective from a point. We get

the following relationship EH ·E′H ′, EG ·E′G′=B, and HG ·H ′G′=A are

collinear. Since EH ·AB=D and E′H ′ ·AB=D′, we get D=D′.

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If the point D is the harmonic conjugate to C with respect to A andB it turns out that the point A is conjugate to B with respect to C and D.This is the replaceability of a harmonic set.

Theorem 3.2

H(AB,CD)⇔ H(CD,AB).

Proof. Assume H(AB,CD), then there is a quadrangle EFGH. Set n=AB,

we get EG·n=C, FH ·n=D, EF ·GH=A and EH ·FG=B. Let DG·FC=S

and GE ·FH=T . Then consider the quadrangle TGSF . See figure 13. The

question is now if ST goes through point A? Think about 4THE and

4SGF , we can find points TE · SF=C, TH · SG=D and HE ·GF=B all

lie on line AB, so they are perspective from a line; by the dual of Axiom 5,

they should perspective from a point. We can find that the point is A. It

means A is incident with TS. Therefore H(CD,AB).

Figure 13: H(AB,CD)⇔ H(CD,AB).

Summarize it then we get a corollary.

Corollary 3.2.1

H(AB,CD)⇔ H(AB,DC)⇔ H(BA,CD)⇔ H(BA,DC)⇔ H(CD,AB)⇔H(CD,BA)⇔ H(DC,AB)⇔ H(DC,AB)

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3.2 Construction of the fourth line of a harmonic set

As we said in Section 2, the dual of a theorem is also a theorem. Wegive here the definition of the dual of fourth point of a harmonic set. Theproofs of the dual theorems will be similar to the previous proofs.

Definition 3.2

See figure 14. Four concurrent lines, a, b, c, d are said to form a harmonic set

H(ab, cd) if there is a complete quadrilateral in which two opposite vertices

A and B lie on a, two other opposite vertices C and E lie on b, while the

remaining two vertices D and F lie on lines c and d, respectively. The

quadrilateral AD,AE,BC,BD forms H(ab, cd).

Figure 14: H(ab, cd).

Then we would also have the uniqueness and replaceability of H(ab, cd).The two theorems are presented below, and the proofs are similar with theprevious theorems.

Theorem 3.3

If a, b and c are three distinct concurrent lines, then d, the harmonic con-

jugate of c with respect to a and b, is unique.

Theorem 3.4

H(ab, cd)⇔ H(cd, ab)

From the theorems in this section we got H(AB,CD) ⇔ H(CD,AB)andH(ab, cd)⇔ H(cd, ab). Are there any relationH(AB,CD)⇔ H(ab, cd)?

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Yes if we have a special relation, a so called perspectivity, between the linesand the points. This is the subject of the next section.

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4 Perspectivities and Projectivities

In this section we will talk about mappings from a set of points to aset of points, a set of lines to a set of lines, a set of points to a set of linesand vice versa. We learn how to use points and lines constructions to obtaincorrespondences synthetically.

There are two relationships to describe mappings. Before we learn therelationships, we need to know two basic concepts. They are presented asdefinitions with figures.

Definition 4.1

The set of all lines through a point P is called a pencil of lines with center

P.(figure 15)

Definition 4.2

The set of all points on a line p is called a pencil of points with axis p.(figure

16)

Figure 15: Pencil of lines. Figure 16: Pencil of points.

With the two basic definitions, one relationship of the mappings be-tween pencils (the pencil of points, the pencil of lines and the pencil ofpoints and lines) is known as a perspectivity. We show the definition ofperspectivity in the following and present some corresponding elementarymappings by figures.

We use the symbol ′′Z′′ to denote this mapping relationship which isdefined as perspectivity related.

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4.1 Perspectivities

Definition 4.3

A one-to-one mapping between two pencils is called a perspectivity.

Definition 4.3.1

A one-to-one mapping between two pencils of points with axis p and p′ is

called a perspectivity if each line joining the point A on p with the corre-

sponding point A′ on p′ is incident with a fixed point O. O is called the center

of the perspectivity. We can use AO∧A′ to denote this perspectivity.(figure 17)

Figure 17: Perspectivity between pencils of points.

Definition 4.3.2

A one-to-one mapping between two pencils of lines with centers P and P ′ is

called a perspectivity if each point of intersection of the corresponding lines

a on P and a′ on P ′ lies on a fixed line o. o is called the axis of the per-

spectivity. We can use a o∧a′ to denote this perspectivity. (figure 18, page 23)

Definition 4.3.3

A one-to-one mapping between a pencil of points with axis p and a pencil of

lines with center P is called a perspectivity if each point A on p is incident

with the corresponding line a′ on P .

We can use AZ a′ or a′ ZA to denote the perspectivity. (figure 19, page 23)

Notice definition 4.3.3 includes two types of perspectivity, one is AZa′,another is a′ ZA. We use AZ a′ to denote the perspectivity with a pencil of

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Figure 18: Perspectivity between pencils of lines.

Figure 19: Perspectivity between a pencil of points and a pencil of lines.

points first, and then a pencil of lines; a′ ZA denotes the perspectivity witha pencil of lines first and a pencil of points second. Since perspectivities areone-to-one mappings, the inverses are also perspectivities.

4.2 Projectivities

Here we introduce another relationship of the mappings between thepencils.

Definition 4.4

A one-to-one mapping between the elements of two pencils is called a pro-

jectivity if it consists of a finite product of perspectivities.

When a projectivity exists between two pencils, the pencils are said tobe projectively related, we use the symbol ′′∧′′ to denote it.

In figure 20 and figure 21 in page 24 we show the mappings in 3D asan illustration of the difference between a perspectivity and a projectivity.So perspectivity is a special case of projectivity.

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Figure 20: A perspectivity be-

tween pencils of points.

Figure 21: A projectivity between

pencils of points.

Next we present figures of a projectivity between the pencils of points(figure 22), the pencils of lines (figure 23) and a pencil of points and a pencilof lines (figure 24 in page 25).

Figure 22: ABC ∧A′′B′′C ′′.

Figure 23: abc ∧ a′′b′′c′′.

If a projectivity maps either a pencil of points or a pencil of lines ontoitself, it is called a projectivity on the pencil. If a point (line) goes to thesame point (line), we say it is an invariant point (line). If the mappingrelates each point to itself, it is the identity mapping.

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Figure 24: abc ∧A′′B′′C ′′.

After the introduction of projectivity, we can give the final axiom whichdescribes an important property of projectivities on pencils. We formulateit for points, but it can also be formulated for a pencil of lines.

Axiom 6

If a projectivity leaves invariant each of three distinct points on a line, it

leaves invariant every point on the line.

With help of the Axiom 6, we can prove the fundamental theorem inprojective geometry.

Theorem 4.1 (fundamental theorem)

A projectivity between two pencils is uniquely determined by three pairs of

corresponding elements.

Proof. For the construction of the projectivity, see section 4.3.

The uniqueness follows from Axiom 6 in the following way. Assume there

exists a mapping T which we denote ABC ∧ A′B′C ′ and another mapping

S also denoted ABC ∧ A′B′C ′. Then TS−1 maps ABC ∧ ABC. Axiom 6

tells us it is the identity, TS−1 = I, so mapping S = T .

The proofs for the other two projectivities a done in the same way.

Corollary 4.1.1

If in a projectivity between two distinct pencils an element corresponds to

itself, then the projectivity is a perspectivity (i.e., the mapping requires only

one perspectivity).

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Proof. Take two distinct lines p and p′ with a common point P where points

A and B lie on p and points A′ and B′ lie on p′. If P is an invariant point

we have ABP ∧ A′B′P . We can take O = AB′ · A′B as the center of the

perspectivity.

The proofs for the other two projectivities a done in the same way.

After we have introduced the sixth axiom and the fundamental theorem,we now turn to constructions of projectivities.

4.3 Construction of a projectivity between pencils of points

Here we present a theorem which is about constructing the imagesunder a projectivity between pencils of points. This theorem is also almostshowing the Pappus’ theorem. We present the theorem with a definitionand a figure.

Definition 4.5

If A and A′, B and B′ are pairs of corresponding points, the cross joins of

those pairs are the lines AB′ and BA′.

The corresponding theorem is shown below.

Theorem 4.2

A projectivity between two distinct pencils of points determines a unique line

called the axis of projectivity, which contains the intersections of the cross

joints of all pairs of corresponding points.

Proof. Check figure 25 in page 27 to follow the proof.

Consider two distinct pencils of points with axes p and p′. Assume ABC ∧A′B′C ′. Set point P = p · p′ which is none of the six points.

Setting A′ as the center, we have a perspectivity between points A,B,C

and lines A′A,A′B,A′C; then change the center to A, we have another

perspectivity between points A′, B′, C ′ and lines AA′, AB′, AC ′. Hence we

have a projectivity between lines A′A,A′B,A′C and lines AA′, AB′, AC ′.

According to fundamental theorem (theorem 4.1) and corollary 4.1.1, there

is a perspectivity between lines AA′, AB′, AC ′ and lines A′A,A′B,A′C with

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a axis of the perspectivity. Let point F = AB′ · BA′ and G = AC ′ · CA′

then we have the axis h which is determined by points F and G.

Thus, by axis h we can find the mapping of another distinct point D. Let

E = A′D ·h then the mapping of point D is D′ = AE ·p′. Next we are going

to prove that the line is unique.

To prove that the line h is unique, we need to show that the line h is

independent of the choices for the centers of pencils. Set R = p · h and

Q = p′ · h, the image of P is Q since P A′

∧ Q and QA∧Q; the pre image of

P is R since RA′

∧ R and RA∧P . According to theorem 4.1, the image is

unique. Then we can say points R and Q are independent of the choices for

the centers of pencils, and the two points determine the axis h. Therefore,

points R and Q determine a unique line h.

Figure 25: Axis of projectivity h for ABC ∧A′B′C ′.

From theorem 4.2, we can prove one of the most well-known theoremwhich is Pappus’s theorem. However, first we need a new definition.

Definition 4.6

A hexagon is given by a set of six distinct points called vertices, they are P1,

P2, P3, P4, P5 and P6. Lines P1P2, P2P3, P3P4, P4P5, P5P6 and P6P1 are

called the sides of the hexagon P1P2P3P4P5P6. Points P1 and P4, P2 and

P5, P3 and P6 are pairs of opposite vertices. Lines P1P2 and P4P5, P2P3

and P5P6, P3P4 and P6P1 are pairs of opposite sides. The three points A,B

and C which are the intersection of opposite sides are called diagonal points.

(figure 26, page 28)

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Figure 26: A hexagon P1P2P3P4P5P6.

Theorem 4.3 (Pappus’s theorem)

If alternate vertices of a hexagon lie on two lines, then the diagonal points

are collinear.

The proof of Pappus’s theorem is similar to the proof of theorem 4.2which refer to the hexagon AB′CA′BC ′A.

Axiom 6 and the fundamental theorem also show that a projective trans-formation of a line into itself can not have more than two invariant pointswithout being the identity. The elliptic projectivity have no invariant point,the parabolic projectivity have one invariant point and the hyperbolic pro-jectivity have two invariant points. We will show these three constructionsnow.

Question 1

How is the elliptic projectivity looking like?

Answer 1

For an elliptic projectivity three perspectivities are needed. An example,

ABC ∧BCA, is given in figure 27 in page 29. First we do the perspectivity

ABC O∧A′B′C ′, then we put B = A1, C = B1 and A = C1. The second

perspectivity has A1 as the center and maps the points A′, B′, C ′ on the line

h. Finally we take A′ as the center and maps the line h to line AC.

Question 2

How do we construct projectivities with one or two invariant points?

Answer 2

Assume we have a quadrangular set (AD)(BE)(FC), see figure 28 in page

28

Figure 27: ABC ∧BCA

29, there exists a quadrangle QRSP such that A = p · SP , B = p · SQ,

C = p · SR. D = p · QR, E = p · RP , F = p · QP where p = AB. The

projectivity AEC P∧SRC

Q∧BDC leaves C invariant. Let G = SR · PQ. It is

clear that point F is also an invariant point since F P∧G

Q∧F .

The projectivity AECF ZBDCF is hyperbolic if C and F are distinct.

Figure 28: Invariant points C,F .

The projectivity AEC Z BDC is parabolic if C and F are coincident,

see figure 29 in page 30.

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Figure 29: Only one invariant point C.

30

5 Point and line conics

The following part of this thesis describes the construction of conics ingreater detail by using a different style, compared to previous sections, norigorous style with definitions, theorems and proofs anymore. A table willpresent the concepts and the point and line constructions will be exhibited byfigures [3]. According to the axiomatic system: two points determine a line,three points determine a triangle and four points determine a quadrangle.Five points determine a point conic by a projectivity of pencils of lines.What can be clearly seen in table 1 is the characteristics of a point conicand its dual.

Point conic Line conic

A point conic is the set of points of

intersection of corresponding lines

of two projectively, but not per-

spectively, related pencils of lines

with distinct centers U and U ′.

These points and the lines p and

q and the center F determine the

projectivity.

A line conic is the set of lines

joining corresponding points of two

projectively, but not perspectively,

related pencils of points with dis-

tinct axes u and u′. These lines

and the points P and Q and the

line f determine the projectivity.

A tangent to a point conic is a line

that has exactly one point in com-

mon with the point conic.

A point of contact of a line conic

is a point that lies on exactly one

line of the line conic.

Table 1: Point and line conic.

The figure 30 in page 32 shows that a point conic is determined by fivepoints. Points U and U ′ are taken as the centers for the pencil of lines, takealso three other points A, B and C in the plane, no three of the five chosenpoints can be collinear. We can now construct the two lines p = AB andq = AC; and the point F = UB · U ′C. The two lines p and q and the pointF are now used for the projectivity of the pencils in the following way. Aray from U will meet the line q at a point K, this point is mapped by aperspectivity with centre F to a point L on p. The final perspectivity givesthe line U ′L. The intersection UK ·U ′L gives a point X on the point conic.

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Figure 30: Five points U,U ′, A,B and C generate a point conic.

Using this construction we realize that the five points U,U ′, A,B, andC also belong to the point conic. For example, sending a ray from point Utowards point U ′ will intersect with the corresponding line at point U ′. Thename point conic reminds us about the well known conics, the ellipse, theparabola and the hyperbola and it turns out it is exactly these three curvesthat can be generated by this construction. To show this, coordinates mustbe used. From now on we often drop the name point conic and simplify sayconic.

It is obvious from the construction in figure 30 that the three pointsK,F,L lie on a line. This observation was first made by Blaise Pascal (1623-1662) and he formulated Pascal’s theorem. It tells us that the diagonalpoints of a hexagon which is inscribed in a point conic are collinear. Thetheorem is almost directly seen from the figure. The line which is determinedby three diagonal points is called Pascal’s line.

Figure 31: Lines from S and R are projectively related.

For full understanding of Pascal’s theorem let us take three new points,S, Y and R on the point conic constructed with centers U and U ′ above.The three points U , U ′ and X are the same as in figure 30. Let us now

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investigate if the diagonal points W , F , Z of the hexagon UXU ′SY RU arecollinear?

Let M = UR ·SY and N = U ′S ·Y R, if we set U as the center, we havea perspectivity between points S,W,M, Y and lines US,UX,UR,UY . Thenchange the center to U ′, we have a projectivity between lines US,UX,UR,UYand lines U ′S,U ′X,U ′R,U ′Y . Next we set line Y R as the axis and we havea perspectivity between lines U ′S,U ′X,U ′R,U ′Y and points N,Z,R, Y .Hence we get a projectivity between points S,W,M, Y and pointsN,Z,R, Y .Since point Y goes to itself, Y is an invariant point. According to corol-lary 4.1.1, SWM ∧NZR is a perspectivity! Then we can define the centerF = SN ·MR. Therefore the diagonal points W,F,Z lie also in this caseon a straight line. This shows that there is nothing special with centersU and U ′, we can equally well use S and R as centers and we still have aprojectivity between the pencils of lines.

Sometimes the Pascal’s line can lie outside the conic as in figure 32.

Figure 32: The Pascal’s line is outside of the ellipse.

As indicated previously, the parabola and the hyperbola can also begenerated by the same construction as in figure 30. Figure 33 and figure 34in page 34 are two illustrations for parabola and hyperbola, line DFE is thePascal’s line.

After the introduction of the point conic construction, the dual con-struction will be presented. Figure 35 in page 34 reveals that lines u and u′

are taken as the lines for the pencil of points. Take also three other lines a,b and c in the plane, no three of the five chosen lines can be concurrent. Wecan now construct the two points P = a ·c, Q = a ·b and line f is determinedby points b · u′ and c · u. The two points P and Q and the line f are nowused for the projectivity of the pencils in the following way. Take a pointA on u′, the ray from point P through A will meet the line f at a point F ,

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Figure 33: Construction of a

parabola.

Figure 34: Construction of a hy-

perbola.

the intersection point of QF and u gives a point B on the line u. In thisway we get a new line x = AB of the line conic. From this construction wecan realize that the joins between opposite vertices of the hexagon u′xucabu′

goes through a point F in Figure 35. This point is called Brianchon’s point.

Figure 35: Five lines u, u′, a, b, and c generate a line conic.

In figure 36 and figure 37 in page 35, a hexagon 1234561 is circumscribedaround a conic section, those lines connecting opposite vertices (12 and 45,23 and 56, 34 and 61) meet in a single point (Brianchon’s point). Thisobservation was first made by Charles Julien Brianchon (1783-1864) and heformulated it as a theorem: Brianchon’s theorem.

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Figure 36: The Brianchon’s point

B is inside of the ellipse.

Figure 37: The Brianchon’s point

B is outside of the ellipse.

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6 Conclusion

Overall, this study explained two famous theorems in the axiomaticsystem, Pappus’ theorem and Pascal’s theorem. Also a lot of points andlines constructions were shown. The aim of the present research was tounderstand the basics of projective geometry. A limitation in this thesisis that only 2D is considered and focus on the constructions alone withoutcoordinates.

The study of projective geometry could now go in another three direc-tions: finite geometry, analytical geometry (with coordinates) or computergraphics. And one modern application of projection is motion pictures dis-played on a screen.

References

[1] Judith N. Cederberg, 2001, A Course in Modern Geometries, Springer.

[2] H.S.M, Coxeter, 1991, Projective Geometry, Springer.

[3] Bengt Ulin, 2000, Projective Geometry, Ekelunds forlag.

[4] Dan Pedoe, 1976, Geometry and the Visual Arts, Dover edition.

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