Analytic Geometry - Mathematics Form and Function

gram of addition into a parallelogram). The same notion of linear transformation applies to higher dimensional vector spaces, and is the cornerstone of the study of linear algebra and matrix theory (Chapter VII).

The general idea that some geometrical facts can be handled well by appropriate algebraic operations has now appeared in the use of real numbers for one-dimensional geometry and of vectors for higher dimen

sions. The addition of vectors was possible only with the choice of an ori

gin

o.

Without a choice of origin one still has some algebraic operations such as the construction of the midpoint, written ( 1 /2 )P

+

( 1 /2 )Q, of two points P and Q or of the point ( 2/3 )P

+

^{( 1 /3 )Q}which is one third of the way from P to Q. The general such operation is the formation of the

weighted average

(4) of

n

₊

I

points with weights real numbers

Wi

with sum

Wo

₊ . ^. .

+ Wn

⁼

l .

A transformation preserving such averages is called an

affine

transformation; the resulting geometry (in any number of dimensions) is

affine geometry.

It is possible to write a complete set of axioms for the weighted average operations (4), but they are cumbersome

(A lgebra,

first edition). Even the simplest ternary operation P

o

- PI

+

2

of this type has not proven to be very useful. It seems more efficient to reduce to vector algebra by choosing an origin. Addition is a very convenient operation! Binary operations are much handier than ternary ones.

Analytic Geometry

Another and earlier reduction of plane geometry to algebra is provided by the familiar method of cartesian coordinates. Given an orientation, a choice of origin and unit, and two perpendicular coordinate axes, each point P is represented by its coordinates, a pair

(x,y)

of real numbers.

Each line in the plane may be described as the set of those points whose coordinates satisfy a linear equation, while the distance between two points is given by the familiar formula in coordinates, derived from the Pythagorean theorem. In this way, all sort of geometric facts about the plane are handled by algebraic. machinery; in effect, the plane is reduced to the cartesian product

R

⁼

R2

of two copies of the real line

R.

However, this reduction does depend on the choice of origin

and

of axes, and one must betimes verify that truly geometric facts are independent of the choice of coordinates.

Geometry and coordinates arise first in dimensions 2 and 3. The need for higher dimensional geometry is motivated by phenomena which need

1 1 0 IV. Real Numbers

specification of more than three coordinates: Events in space-time need four (position and time); in dynamics, the initial conditions for a particle need six, three for position and three for velocity. The use of such a six

dimensional

phase space

is a first example of the importance of mechanics as a motivation for mathematical developments (Chapter IX). To be sure, this example could be (but usually is not) described without explicit use of coordinates, as the product of a three-dimensional position space and another three-dimensional velocity space. A general n-dimensional Euclidean space may be constructed as the n-fold product Rn of real lines, with points the n-tuples of real numbers

(X I ,

^{. .} .

, xn ).

This is a vector space over

R,

with basis the n "unit vectors" (0 , . . . , 0, 1 ,0 , . . . , 0). Such use of coordinates allows for the efficient treatment of higher-dimensional phenomena in dimensions where the corresponding geometric intuitions are weak or non-existent. It serves to extend geometric ideas beyond the ordinary three dimensions.

Much of mathematical physics deals with phenomena in three dimen

sions and so often requires a formulation in triads of equations, one for each coordinate in

R3.

When

R3

is regarded as a vector space, many of these equations can be written as single vector equations. This accounts for the popularity of vector analysis in Physics. It includes the use of inner products, to which we now tum.

Trigonometry

Trigonometry is essentially a procedure for turning angular measures into linear measures. This appear directly in the definition of the two basic trigonometric functions sin

(J

and cos

(J

of the angle

(J.

In the oriented plane take a point P on the unit circle with center at the origin 0 so that the segment OP makes the given angle

(J

with the

X

coordinate axis. Then cos

(J

and sin

(J

are defined to be the

X

and

y

coordinates of P; since the circle has radius I, this immediately gives the identity

( 1 ) This defines sin

(J

and cos

(J

for angles

(J

of all sizes (not j ust for an angle

(J

in the first quadrant, as displayed in Figure 1 ).

When angles are measured (in radians) by numbers

t

we can also think of the sine and the cosine as functions of the

number t,

so that

Sin

t =

sin( angle of

t

radians ) . (2) Thus there are really two legally different functions: The Sine of a

number,

here with capital S, and the sine of an angle, with lower case s.

This pedantic (but real!) difference is usually ignored. It implicitly

9. Trigonometry

I I I

( 1 , 0) Figure 1

involves the wrapping function Or of §2 above. Recall that this function sends each real 1 to the point P

=

( cos Or, sin Or ) on the circle such that the length of the counterclockwise circular arc from ( 1 ,0) to P is congruent to I, modulo 2'11". Then the definition (2) reads

Sin 1

=

sin( Or )^, Cos 1

=

cos( Or ) : (3) it really amounts to a composite function I -+ Or -+ sin(Or ). Since the wrap

ping function has Or + 2'IT

=

Or we get

Sin( 1

+

2'11")

=

Sin( / ), COS( I

+

2'11")

=

Cos( t ) . (4) In other words, Sin and Cos are periodic functions, of period 2'11", as in the familiar graph of Figure 2. Now that this has been stated, we drop the S in Sin and the function Or ; they would just get in the way of trigonometric manipulations, but we emphasize that the wrapping function accounts for radian measure and the periodicity (4) of the trigonometric functions.

Indeed, the whole study of periodic functions, as carried on in Fourier analysis, concerns the expression of more or less arbitrary functions f(/) of period 2'11" in terms of the periodic functions sin

nl

and cos

nl

for the natural numbers

n.

A rotation o f the circle o f Figure 1 about the origin through the angle 0 will carry the point ( 1 ,0) to our point P and the point (0, 1 ) to a corresponding point p I ; thus this rotation has the effect

sin t

Figure 2

1 1 2 IV. Real Numbers

( 1 ,0 )

I->

( cos 8, sin 8), ( 0, 1 )

I->

( - sin 8,cos 8) (5) Since ( 1 ,0) and (0, I ) form a basis for the 2-dimensional vector space, any point Q with coordinates (x,y ) can be written as the linear combination (x,y )

=

x( 1 ,0 )

+

y( O, I ) of the two basis vectors. Now rotation is, as already noted, a linear transformation, so preserves linear combinations and thus by (5) has the effect

(x,y )

I->

(x cos 8 - Y sin 8, x sin 8

+

Y cos 8 ) .

When one writes (x ',y ') for the coordinates of the rotated point, this yields the customary equations in coordinates

x '

=

x cos 8 - Y sin 8 , (6) y '

=

x sin 8

+

Y cos 8

for a rotation (x,y )

I->

( x ',y ' ) of the plane about the origin. Also, if one takes Q

=

(x,y ) to be that point Q on the unit circle of Figure I for which OQ makes the angle cp with the positive x-axis, then x

=

cos cp, y

=

sin cpo Now the rotation by 8 clearly carries Q to a point Q ' where OQ ' makes the angle cp

+

8 with the positive x-axis. The coordinates (x ',y ' ) of Q ' are then cos( cp

+

8 ), sin( cp

+

8) and the equations (6) for rotation become

cos( cp

+

=

cos cp cos 8 - sin cp sin 8 , (7) sin(cp

+

=

cos cp sin 8

+

sin cp cos 8 .

These "addition formulas" for the trigonometric functions are thus another expression of the relation of angular to linear measure.

This relation also accounts for the use of trigonometric functions in cal

culating the connections between the sides and the angles of a triangle, as suggested by the Euclidean congruence theorems. In a triangle OA R with sides

a,

b and c , as illustrated in Figure 3, drop a perpendicular from A to the opposite side at M. If 8 and cp are the angles of the triangle at ° and

Figure 3

9. Trigonometry 1 1 3

B,

the definition of the sine function shows that the length

AM

can be expressed either as

b

sin () or as

c

sin cp ; this gives the

law of sines:

ble

⁼ sin cp Isin (). On the other hand, one may compute the length

c

of the side

AB

from the Pythagorean theorem for the right triangle

AMB

to be

c

2 ⁼(b sin (})2

+

(

a

b

cos (})2 ,

c

2 =

a

+ b

- 2ab

cos () .

(8)

This is the

law of cosines.

When

u

⁼

DB

and

v

⁼

OA

are regarded as vec

tors of the respective lengths

I DB I

⁼

a,

OA I

⁼

b,

the expression

ab

cos () which appears in (8) is called the

inner product u · v

of the two vectors; it is a scalar (a real number). If

A

has coordinates ( X bY I ) and

B

coordinates (X2,Y2 ), the lengths

a, b,

and

c

in (8) can be written in terms of these coordinates by the Pythagorean theorem; the formula (8) then becomes a formula for the inner product

u · v:

(9) This again is a linearization of angular phenomena.

With these ideas and formulas we have presented all of the essential concepts of trigonometry in less than 6 pages. It is curious (and trouble

some) that the standard elementary presentations of trigonometry have been inflated to inordinate lengths-an inflation hardly justified by the inevitable subtlety wrapped up in the comparison of linear and angular magnitude.

In (9), the inner product is expressed in terms of a choice of coordi

nates. However, as in other cases, the inner product can be described in a more invariant way by suitable axioms. It has three characteristic proper

ties:

Linearity, symmetry,

and

positive definiteness,

as expressed by the

In document Mathematics Form and Function (Page 120-124)