ProjectiveTrig.pdf

Projective and spherical trigonometry

N J Wildberger

School of Maths UNSW

Sydney 2052 Australia

Introduction

Spherical trigonometry is historically one of the most important areas of mathematics, due to the

obvious applications to astronomy and navigation. Planar trigonometry, on the other hand, was for

ﬁ fteen centuries only of marginal interest, and the most important formulas were developed ﬁ rst in

the spherical case, even though they are more diﬃcult there. Only in the last four hundred years

or so, with the advent of modern technology and engineering, did planar trigonometry slowly begin

to gain the ascendancy. This reversal is now complete, and the modern curriculum largely ignores

spherical geometry. Perhaps the many complicated formulas present too much of a hurdle.

Recently a new framework for planar trigonometry has been proposed ([Wildberger]). Rational

trigonometry replaces distance and angle with quadratic concepts called quadrance and spread.

The usual laws are replaced by purely algebraic analogs, with the consequence that they hold in

much wider generality, allow more accurate calculations, and are much easier to learn. The usual

menagerie of transcendental circular functions and their inverses play no role.

Astronomy, however, requires understanding of how the earth’s rotation aﬀects our view of

the sky and the objects in it. As a consequence, longitudinal angle becomes an important and

unavoidable concept. But for many spherical geometrical applications, there is no uniform motion

around a ﬁ xed axis that plays such a distinguished role. For this kind of ‘stationary’ spherical

geometry, it turns out that there is a rational version of the classical theory which again is simpler,

more elegant and accurate. This theory is here developed in the more natural setting of projective

trigonometry.

The projective plane inherits a rich metrical structure which extends to higher dimensions and

arbitrary ﬁ elds, a fact which has major implications for algebraic geometry, and possibly also for

diﬀerential geometry. Thales’ theorem and Pythagoras’ theorem are particularly important, and

the wide variety of classical spherical formulas are replaced by simpler, polynomial relations. The

Platonic solids are seen in a new light. In a future article I will show that the formulas given here

hold in hyperbolic geometry too–essentially without any modi ﬁ cation.

Rational trigonometry

In this preliminary section we present some basic facts about (Euclidean) rational trigonometry,

taken from [Wildberger] and suitably modi ﬁ ed to hold in three dimensional space.

The quadrance Q(A 1 ,A 2 ) between the points A 1 ≡ [x 1 ,y 1 ,z 1 ] and A 2 ≡ [x 2 ,y 2 ,z 2 ] is the

number

Q(A 1 ,A 2 ) ≡ (x 2 −x 1 ) 2 +(y 2 −y 1 ) 2 +(z 2 −z 1 ) 2 .

Quadrance is just distance squared over the decimal numbers, although the algebraic de ﬁ nition

extends to arbitrary ﬁ elds. The points A 1 ,A 2 and A 3 are collinear precisely when the quadrances

Q 1 ≡Q(A 2 ,A 3 ) Q 2 ≡Q(A 1 ,A 3 ) Q 3 ≡Q(A 1 ,A 2 )

satisfy the Triple quad formula

(Q 1 +Q 2 +Q 3 ) 2 =2

Q 1 +Q 2 +Q 3

(1)

The points A 1 ,A 2 and A 3 form a right triangle with the lines A 1 A 3 and A 2 A 3 perpendicular

precisely when the quadrances satisfy Pythagoras’ theorem

Q 1 +Q 2 =Q 3 .

The notion of angle between two lines l 1 and l 2 is replaced by that of the spread s(l 1 ,l 2 ) which is a

(dimensionless) number between 0 and 1, and which is unchanged if either l 1 or l 2 is translated. To

de ﬁ ne it, suppose that l 1 and l 2 intersectatthepointA. Choose a point B 6= A on one of the lines,

say l 1 , and let C be the foot of the perpendicular from B to l 2 as in Figure 1. If Q(B,C)=Q and

Q(A,B)=R then de ﬁ ne

s =s(l 1 ,l 2 )= Q

R .

Figure 1: Spread as ratio

The spread between parallel lines is 0. The spread corresponding to 30 ◦ or 150 ◦ is s =1/4,

to 45 ◦ or 135 ◦ is 1/2,andto60 ◦ or 120 ◦ is 3/4. The spread between perpendicular lines is 1. It

is not hard to check that if O =[0,0,0], A 1 ≡ [x 1 ,y 1 ,z 1 ] and A 2 ≡ [x 2 ,y 2 ,z 2 ] then the spread

s =s(OA 1 ,OA 2 ) is

s = (y 1 z 2 − z 1 y 2 ) 2 +(z 1 x 2 − x 1 z 2 ) 2 +(x 1 y 2 − y 1 x 2 ) 2

(x 1 +y 1 +z 1 )(x 2 +y 2 +z 2 )

(2)

The spread may be measured by a spread protractor. The one in Figure 2 was created by

Michael Ossmann and may be downloaded at http://www.ossmann.com/protractor/.

Figure 2: A spread protractor

The notion of spread may be extended to include planes in three dimensions. The spread

between a line l and a plane Π is de ﬁ ned to be 0 if they are parallel, 1 if they are perpendicular,

and otherwise is de ﬁ ned to be the spread between l and the unique line m which is the intersection

of Π and the plane perpendicular to Π passing through l.

The spread between two planes Π 1 and Π 2 in three dimensional space is de ﬁ ned to be 0 if they

are parallel, 1 if they are perpendicular, and otherwise is de ﬁ nedtobethespreadbetweenthetwo

lines formed by intersecting Π 1 and Π 2 with a plane Π perpendicular to them both, that is a plane

perpendicular t o the line of intersection of Π 1 and Π 2 .

A triangle A 1 A 2 A 3 is a set of three non-collinear points. Given three distinct points A 1 ,A 2

and A 3 ,de ﬁ ne the quadrances Q 1 ,Q 2 and Q 3 as above and the spreads

s 1 ≡s(A 1 A 2 ,A 1 A 3 ) s 2 ≡s(A 2 A 1 ,A 2 A 3 ) s 3 ≡s(A 3 A 1 ,A 3 A 2 ).

A 2

Q 3 s 2

Q 1

A 1

s 1

s 3

A 3

Q 2

Figure 3: Quadrances and spreads

Here are the other main laws of rational trigonometry.

Spread law

Q 1 = s 2

s 1

Q 2 = s 3

Q 3 .

Cross law

(Q 1 +Q 2 −Q 3 ) 2 =4Q 1 Q 2 (1−s 3 ).

Triple spread formula

s 1 +s 2 +s 3

(s 1 +s 2 +s 3 ) 2 =2

+4s 1 s 2 s 3 .

This last law replaces the rule that the sum of a triangle’s angles is π,andisamodi ﬁ cation of

the Triple quad formula. One of its important consequences is that if l 0 ,l 1 ,l 2 ,l 3 ,··· are equally

spaced intersecting lines, with common spread s= s(l 0 ,l 1 )=s(l 1 ,l 2 )=··· then

s(l 0 ,l 2 )=4s(1−s)=S 2 (s)

s(l 0 ,l 3 )=s(3−4s) 2 =S 3 (s)

s(l 0 ,l 4 )=16s(1−s)(1−2s) 2 = S 4 (s)

s(l 0 ,l 5 )=s

5−20s+16s 2

Figure 4: Multiple spreads

The ﬁ fthspreadpolynomialS 5 (s)=s

5−20s+16s 2

2 controls ﬁ ve-fold symmetry, and over

the decimal ﬁ eld has non-trivial zeroes

α =

5− √

/8 ≈ 0.345491503... and β =

√

/8 ≈ 0.904508497...

which are spreads in the regular pentagon of Figure 5. Note that S 2 (α)=β and S 2 (β)=α.

Figure 5: A pentagon

=S 5 (s)

and so on. The ﬁ rstofthesestatementsiscalledtheEqual spreads theorem and is very useful.

Note that S 2 (s) is the logistic map of chaos theory fame.

An interesting example

Here is an example that is surprisingly involved considering its simpli city (a nd has a spherical analog

of importance to the Platonic solids). Take an equilateral triangle ABC and draw lines AD, B E

and CF as in Figure 6, all making the same spreads with the sides of the triangle, so that DEF is

equilateral, and so that

Q(A,D)=Q(D,E)=Q(B,E)=Q(E,F)=Q(C,F)=Q(F,D).

Figure 6: An equilateral arrangement

The spread s =s(AC,CD) turns out to be 3/28, and the corresponding angle is about 19.11 ◦ .

If we complete the con ﬁ guration to make it more symmetrical, then numerous other spreads and

quadrances may be calculated using the basic laws above, resulting in Figur e 7. The qu adra nces

have been normalized so they are all commensurable. Note that the segment AX trisects AB, and

that Q(A,X):Q(D,X)=7:1. Checking the numbers in this Figure is an excellent exercise.

2800

3600

27/28

400

3/4

225

2800

576

75/196

441

576

441

225

2800

48/49

3600

3/28

27/196

3/28

15/28

225

3/4

3600

75/196

400

27/28

2800

Figure 7: More quadrances and spreads

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