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2021年2月23日发(作者：巧点)

2.5

笛卡尔几何学的基本概念

（

basic concepts of Cartesian geometry

）

课文

5-A the coordinate system of Cartesian geometry

As mentioned earlier, one of the applications of the integral is

the

calculation

of

area.

Ordinarily

,

we

do

not

talk

about

area

by

itself ,instead, we talk about the area of something . This means that

we

have

certain

objects

(polygonal

regions,

circular

regions,

parabolic segments etc.) whose areas we wish to measure. If we hope

to arrive at a treatment of area that will enable us to deal with many

different

kinds

of

objects,

we

must

first

find

an

effective

way

to

describe these objects.

The most primitive way of doing this is by drawing figures, as

was done by the ancient Greeks. A much better way was suggested by

Rene

Descartes,

who

introduced

the

subject

of

analytic

geometry

(also

known

as

Cartesian

geometry).

Descartes’

idea

was

to

represent geometric points by numbers. The procedure for points in a

plane is this

：

Two perpendicular reference lines (called coordinate axes) are

chosen,

one

horizont

al

(called

the

“x

-

axis”),

the

other

vertical

(the

“y

-

axis”).

Their

point

of

intersection

denoted

by

O,

is

called

the

origin.

On

the x-axis

a

convenient

point

is

chosen

to

the

right

of

O

and its distance from O is called the unit distance. Vertical distances

along

the

Y-axis

are

usually

measured

with

the

same

unit

distance ,although sometimes it is convenient to use a different scale

on

the

y-axis.

Now

each

point

in

the

plane

(sometimes

called

the

xy- plane) is assigned a pair of numbers, called its coordinates. These

numbers tell us how to locate the points.

Figure

2-5-1

illustrates

some

point

with

coordinates

(3,2)

lies

three

units

to

the

right

of

the

y-axis

and

two

units above the number 3 is called the x-coordinate of the

point,2 its y-coordinate. Points to the left of the y-axis have a negative

x-coordinate; those below the x-axis have a negtive y-coordinate. The

x-coordinateof

a

point

is

sometimes

called

its

abscissa

and

the

y-coordinateis called its ordinate.

When

we

write

a

pair

of

numberssuch

as

(a,b)

to

represent

a

point, we agree that the abscissa or x-coordinate,a is written first. For

this reason, the pair(a,b) is often referred to as an ordered pair. It is

clear that two ordered pairs (a,b) and (c,d) represent the same point

if and only if we

have a=c

and

b=d.

Points

(a,b) with

both

a and

b

positiveare said to lie in the first quadrant ,those with a<0 and b>0

are in the second quadrant and those with a<0 and b<0 are in the

third

quadrant

;

and

those

with

a>0

and

b<0

are

in

the

fourth

quadrant. Figure 2-5-1 shows one point in each quadrant.

The

procedure

for

points

in

space

is

similar.

We

take

three

mutually

perpendicular

lines

in

space

intersecting

at

a

point

(the

origin)

.

These

lines

determine

three

mutually

perpendicular

planes

,and

each

point

in

space

can

be

completely

described

by

specifying , with appropriate regard for signs ,its distances from these

planes.

We

shall

discuee

three-dimensional

Cartesian

geometry

in

more

detail

later

on

;

for

the

present

we

confine

our

attention

to

plane analytic geometry.

课文

5-A

：笛卡尔几何坐标系

< p>

正如前面所提到的，

积分应用的一种是计算面积 。

通常我们不单

独讨论面积，

我们还讨 论其它物体的面积。

这意味着我们有特定的物

体

(

多边形

,

圆域

,

抛物弓形等

)

希望能测量 。如果我们希望获得面积的

计算方法以便能用它来处理多种不同类型的图形，

< p>
我们就必须首先找

出描述这些图形的有效方法。

做这件事的最原始的方法就是描绘出图形，

就如古希腊人所作的

那样。

笛卡尔

(1596-1650)

提出了一种好得多的方法并建立了解析几何

(

< br>也称为笛卡尔几何

)

这个学科。笛卡尔的思想就是用数字 来代表几何

中的点，对应点的过程如下：

选取两条相互垂直的线

(

称为坐标轴

)

，一条水平的

(

称为

x

轴

)

，

< br>另一条是垂直

(

称为

y

轴

)

，

它们的交点记为

O,

称为原点。

在

x

轴上

O

右边选定一个适当的点， 并把它到

O

的距离称为单位长度。沿着

y

轴的竖直距离通常也用相同的单位长度来测量，

不过有时采用 不同的

尺度较为方便。现在平面

(

有时 叫

xy

平面

)

上的每个点都对应一个数

对，称它为坐标。这些数对告诉我们如何定为一个点。

图

2-5-1

说明了 一些例子。坐标

(3,2)

的点位于

y

轴右边三个单位

且在

x

轴上方两个单位长度的地方。

3

称为该点的

< p>
x

坐标，

2

称为该

点的

y

坐标。

y

轴左边的点有负的

x

坐标，那些

< p>
x

轴下方的点有负的

y

坐 标。点的

x

坐标有时称为横坐标，

y< /p>

坐标称为纵坐标。

当我们用一对数如< /p>

(a,b)

代表一个点时，我们商定横坐标，

x

坐标

也就是

a

< br>写在第一位。

由于这个原因，

数对

(a,b)

是一个有序数对。

很

明显 ，当且仅当

a=c

，

b=d

< p>
时，两个有序数对

(a,b)

和

< br>(c,d)

代表同一个

点。点

( a,b)

当

a,b

同为正时，该点位于 第一象限，当

a<0,b>0

时位于

第 二象限，

当

a<0,b<0

时位于第三 象限，

当

a>0,b<0

时位于第四象 限。

图

2-5-1

画出了每个象限的一 个点。

在空间中点的表示方法是相似的。我们取空间中交于一 点

(

原点

)

的 三条相互垂直的线。

这些线决定了三个相互垂直的平面，

且空间 中

的每一个点通过它到三个平面的距离选取合适的记号，

都能完 全具体

的指定出来。

我们之后应该更细节的讨论三维笛卡尔几何 ，

目前我们

限制于关注平面解析几何。

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