504 transformation of coordinates

The angles associated with a transversal

has the effect of, respectively, reflecting points across

the x-axis, rotating points about the origin through an

angle θ, and projecting points onto the x-axis.

In algebra, a change in the form of a mathematical

expression, without changing its validity, is called a

transformation. For example, the action of

EXPANDING

BRACKETS

transforms the expression y= (x+ 1)2into

the equivalent expression y= x2+ 2x+ 1.

See also

AFFINE TRANSFORMATION

;

LINEAR TRANS

-

FORMATION

;

PROJECTION

.

transformation of coordinates (transformation of

axes) In coordinate geometry, it is often convenient to

change the position of a set of coordinate axes, either by

a

ROTATION

or a

TRANSLATION

, to simplify the expres-

sion of a given curve under study. For example, consider

the curve given by the equation x2– 6x+ y2+ 4y= 3. By

COMPLETING THE SQUARE

, this equation can be rewritten:

(x– 3)2+ (y+ 2)2= 16

Setting X= x– 3 and Y= y+ 2 this reads:

X2+ Y2= 16

identifying the curve as a

CIRCLE

of radius four. In this

process we have, in effect, introduced two new coordi-

nate axes, Xand Y, each a translation of one of the

original x- and y-axes. The origin of the new coordi-

nate system lies at the location where X= 0 and Y= 0,

that is, at the point (3,–2).

Changing from the use of C

ARTESIAN COORDI

-

NATES

to

POLAR COORDINATES

, or, in three-dimensional

geometry, to

SPHERICAL COORDINATES

or

CYLINDRI

-

CAL COORDINATES

, is also deemed a transformation of

coordinates.

See also

PRINCIPAL AXES

;

TRANSFORMATION

.

translation See

GEOMETRIC TRANSFORMATION

.

transversal (traverse) A line cutting two or more

other lines is called a transversal. When a transversal

cuts just two other lines, eight angles are formed. The

four angles lying between the two lines are called inte-

rior angles, and the four lying outside are called exte-

rior angles. Special names are given to pairs of angles,

as shown in the following diagram:

According to the

PARALLEL POSTULATE

, two lines

cut by a transversal are parallel if, and only if, two

alternate interior angles are equal (or, equivalently, if

any two corresponding angles are equal).

trapezoid/trapezium A

QUADRILATERAL

with two

sides parallel is called a trapezoid in the United States

and a trapezium in the United Kingdom. Matters are

confusing, for a four-sided figure with no two sides

parallel is called a trapezium in the United States and a

trapezoid in the United Kingdom. We follow U.S. usage

of the terms here.

The area of a trapezoid is given by the formula

where b1and b2are the lengths of the two parallel

edges, and his the distance between them. (See

AREA

.)

The midline or median of a trapezoid is the straight-

line segment that joins the

MIDPOINT

s of the nonparal-

lel sides. An exercise in geometry shows it has length

m= (b1+ b2). (Use similar triangles.) Thus the area

of a trapezoid is the same as that of a

RECTANGLE

of

length mand width h.

If the two nonparallel sides of a trapezoid are equal

in length (and not themselves parallel), we then call the

figure an isosceles trapezoid.

See also

SIMILAR FIGURES

.

trapezoidal rule See

NUMERICAL INTEGRATION

.

1

–

2