AG1 / briefcourseinanal-geom-Yefimov

segments are laid off being determined by

the signs of the num-

bers x, y. Next, a line is drawn through

Mx

parallel

to the axis

Oy, and another line through My parallel

to

Ox; the

intersection

9. In Art.

4

it was explained how a

system of coordinates

is attached to

a

straight line. We shall

now attach a system

and choosing the point O as the origin of coordinates

on each

axis.

an arbitrary point M and its projection Mx on

Consider

the

axis Ox.

The coordinate of Mx on the axis Ox is equ^l to the

value

of

the segment

OMX , in Art. 7, this value was called the

abscissa

of the point M. Hence we conclude that the abscissa of the point

M is equal to the coordinate of the point Mx on the axis Ox.

Similarly, the ordinate of the

point M is

equal

to the coordinate

of the point Mv on the axis

Oy. For all

their

obviousness, these

apply Theorems 1 and 2 (Art. 5), which express

basic properties

of a coordinate system on a line, to all points in

a plane.

10. To facilitate subsequent formulations, let us now agree

about the use of certain terms.

The axis Oy divides the entire plane into

two half-planes;

termed

the right

half-plane, and the other, the

left

half-plane.

In like manner, the axis Ox divides the plane into two half-

planes,

of which

the one containing the positive

half

of the axis

the segment OMX is positively directed on the axis Ox and, con-

sequently, the abscissa x =

OMX

of the

point M is positive. On

the other hand, if M lies

in the

left

half-plane, then the seg-

ment OMX is negatively directed on the

axis Ox, and the number

x = OMX is negative. Finally, if the point

M lies on

the axis

Oyt

*) The adoption of these terms is justified

by

the usual

position of

the

Rectangular Cartesian Coordinates in a Plane

19

its

projection Mx on the

axis

Ox coincides

with the

point

and

x = OMX is zero.

Thus, all points of the right half-plane have positive abscissas

(x > 0) ; all points of the

left

half-plane have negative abscissas

(x < 0) ; the

abscissas

of

all

points

lying on the

axis Oy

are

equal to zero

(x = 0).

By similar reasoning, it can be established that

all

points of

the upper half-plane have positive

ordinates

(t/>0);

all points

of the lower half-plane have negative ordinates

(y < 0) ;

the

ordinates of

all

points

lying

on the axis

Ox

are equal to

zero

(y = o).

Note that the origin 0, as the point

of intersection of the axes,

has both coordinates equal to

zero: x

= 0,

y = 0, and

is charac-

terised by this property (that is, both

coordinates

are

zero

only

for

the

point 0).

H

12.

The two coordinate

axes jointly

divide the plane into four parts, called

quadrants and numbered according to

the

following rule:

The

first

quadrant

is the one lying simultaneously in the

^

IV

right and the upper half-planes; the

second quadrant lies in the left and the

upper

half-planes;

the

third

quadrant

lies in

the left and the lower half-planes;

p

ig>

7.

and,

finally,

the

fourth

quadrant

is

the

one

situated in

the

right

and

the

lower

half-planes.

(The order in which the quadrants are

numbered is illustrated in

Fig. 7.)

M be a point with

y. From the foregoing,

Let

coordinates

x,

it follows that,

M lies

if

x > 0,

y > 0,

in

the

first

quadrant;

if

x < 0,

y > 0,

M lies

in

the

second quadrant;

if

x < 0,

y < 0,

M lies

in

the

third

quadrant;

if

x > 0,

y < 0,

M lies

in

the

fourth

quadrant.

Such a system is determined by choosing

a

unit

of length

and two axes Ox, Oy, which intersect in the point

at

any angle

(other than

and 180).

Let M be an arbitrary point of the plane.

Draw two lines through

M parallel to the axes Ox, Oy and denote

their

respective

points

of

intersection

/y

with

Ox

and

Oy

by

Mx

and

My

M

In the chosen system, the coordi-

nates of the point M are defined as the

numbers

7

where OMX is the value of the segment

Fig.

8.

OMX

of

the axis Ox,

and

OMy is

the

polar axis, and a scale

for

measurement

of

lengths.

When

determining

a

polar

system, it must also be

specified

which

direction

of

rotation

about

the

point

is

to

be considered positive. Usually, counter-

clockwise rotation is taken as positive.

Given

the

pole

and

the

polar

axis

A

(Fig.

9),

let

us consider

an arbitrary

point

M.

Let

p

denote the

distance

of

M

Fig. 9.

from

the

point

(p = |OM|),

and

6

the

angle through which the

ray OA must be rotated to reach coinci-

dence with

the

ray

OM

(6

= Z AOM). This

angle 8 will be un-

lar angle may be defined

verbally as the angle through which

the ray OA must be rotated

(no matter which direction) to reach

Note 2. If the point M coincides

with 0, then p = \OM\ = 0.

This means that the first coordinate

of the pole is equal to zero;

its second coordinate obviously has no definite value.

15. Sometimes, the cartesian and

the polar systems are to be

to

find

its

cartesian coordinates;

and,

conversely, given the cartesian coordi-

nates

of

a

point,

to

find its

polar

coordinates.

We

shall

now derive the

formulas of this coordinate transfor-

mation

(that

is,

the formulas for tran-

sition from polar to rectangular coor-

dinates, and vice versa) for the partic-

ular

case

when

the pole of the polar

system coincides with the origin of rec-

tangular cartesian coordinates, and the

polar axis coincides with the positive

semi-axis of abscissas

(Fig 10). Also, when determining the po-

lar

angle,

we

shall

regard as

positive

the direction of

the

shortest

rotation

of

the positive

semi-axis

Ox into

the

positive

semi-axis Oy.

Let

M be

an

arbitrary point in the plane, and let

(x,

y)

and

(p,

8)

be its

cartesian and polar coordinates, respectively. Describe

diameter. From the point Af, drop perpendiculars to the axes Ox

and Oy\ denote their feet by Mx and My respectively

(see Fig.

10).

The segment OMX

is

the cosine

line

of

the

angle

Q;

therefore,

OMX = \OM\ cos 8.

The

segment

OMy

is

the

sine

line of

the

angle

6; therefore

OMy =* \OM\ sin 8.

But OMX = x,

OMy

= y,

\OM\

= p; hence, the above relations give

,

y = psin9.

(1)

tan6 =

1.

According to the

second

of

these

relations,

6 =

3

or 6 =

1

Since the

-jU

-J-TC.

given point lies

in the

second quadrant,

the first of

the two

indicated values

of 8 must be chosen as

the principal value.

Thus,

p = 2

Distance Between Two Points

25

OM extending from the origin

to the given

point M is called

the radius vector of that point.

Formulas (!')

express the obvious

Theorem 4. For any position of points

MI (x\, y\)

and Af 2 (x2,

y%) in the plane, the distance

d between

them is determined by

the formula

2

yi) -

(2)

Proof. We shall keep the notation used in the preceding theo-

rem, and, in addition, denote

by N the

intersection

of the lines

Example. Find the distance

between

the

points M\ (2, 3)

and Af2 (5, 4).

Solution. By formula (2),

^3p = )/~50 = 5

19. Let us once more

consider

the segment MiMz. Through

its initial point A4, draw

a

ray

u

parallel

to the

axis Ox and

having the same direction

as

Ox (Fig. 14).

Denote by 8 the angle

stood as in trigonometry

(that is, as a signed quantity deter-

mined to within the term

2nji).

system

whose

pole is the point MI and whose polar axis is the

ray w,

in this

polar system, the length d of the given segment

sian coordinate system, whose axes are

directed similar to

the

axes of

the

original

cartesian

system

(in

Fig. 14,

the

new

M\M2

on

the

corresponding

axes of

the

old

and

the new

system being identical, we shall

denote

them,

as

before, by

X,

Y. The

numbers

X, Y

are

the

cartesian

coordinates

of

the

point M2 with respect to the new

system. Applying to them for-

mulas

(1) of

Art. 15, we find

Fig. 14,

X=

,

K=rfsin8. (3)

From these formulas

and in virtue of Theorem

3,

y2

y 1 ==

16,

(4)

or

cos 9 =

=

~

sin

'

< 5)

Formulas (5) make it

possible to

determine the

polar angle of

tan = --=-

(6)

can

also

be

conveniently

used; this

formula is

readily derived

from

(4).

Example

1.

Given the length

d = 2 Y"2

and the polar

angle 9 = 135 of a

X = 2 J/l cos 135

= 2

/

2

=* - 2,

(-

)

135 = 2

=- = 2.