Graphics Reference
In-Depth Information
computation. For example, the expression
ax
+
by
d
equals zero for a straight
line. The variables
x
and
y
are the coordinates of any point on the line and
the values of
a, b, d
determine the position and orientation of the line. There
is an implied multiplication between
ax
and
by
, which would be expressed as
a
∗
x
and
b
∗
y
if we were using a programming language.
The = sign permits the line equation to be expressed as a self-evident
statement: 0 =
ax
+
by
−
d
. Such a statement implies that the expressions
on the left- and right-hand sides of the = sign are 'equal' or 'balanced'. So
whatever is done to one side must also be done to the other in order to
maintain equality or balance. For example, if we add
d
to both sides, the
straight-line equation becomes
d
=
ax
+
by
. Similarly, we could double or
treble both expressions, divide them by 4, or add 6, without disturbing the
underlying relationship.
Algebraic expressions also contain a wide variety of other notation, such as
√
x
−
square root of
x
√
x
n
th root of
x
x
n
x
to the power
n
sin(
x
) i e f
x
cos(
x
)
n
cosine of
x
tan(
x
)
tangent of
x
log(
x
)
logarithm of
x
ln(
x
)
natural logarithm of
x
Parentheses are used to isolate part of an expression in order to select
a sub-expression that is manipulated in a particular way. For example, the
parentheses in
c
(
a
+
b
)+
d
ensure that the variables a and
b
are added together
before being multiplied by
c
and finally added to d.
3.2 Algebraic Laws
There are three basic laws that are fundamental to manipulating algebraic
expressions: associative, commutative and distributive. In the following de-
scriptions, the term
binary operation
represents the arithmetic operations +
,
−
or
×
, which are always associated with a pair of numbers or variables.
3.2.1 Associative Law
The
associative law
in algebra states that when three or more elements are
linked together through a binary operation, the result is independent of how
each pair of elements is grouped. The associative law of addition is
a
+(
b
+
c
)=(
a
+
b
)+
c
(3.1)
e.g. 1 + (2 + 3) = (1 + 2) + 3
