Transformations - Transformations and Projections in Computer Graphics - page 2

mapping is not onto because no real number maps to zero. Requiring a transformation
to be onto makes sense since it guarantees that there will not be any special points P ∗
that cannot be reached by the transformation.

An arbitrary function may map two distinct points x and y into the same point.
Function f ( x ) above maps the two distinct numbers 9.2 and 9.9 into the integer 9. A
one-to-one function satisfies x

= f ( y ). Function g ( x ) above is one-to-
one. Requiring a transformation to be one-to-one makes sense because it implies that a
given point P ∗ is the transformed image of one point only, thereby making it possible
to reconstruct the inverse transformation.
Definition . A geometric transformation is a function that is both onto and one-
to-one, and whose range and domain are points.

= y → f ( x )

Exercise 1.1: Do either of the two real functions f 1 ( x, y )=( x 2 ,y )and f 2 ( x, y )=
( x 3 ,y ) satisfy the definition above?

There are two ways to look at geometric transformations. We can interpret them
as either moving the points to new locations or as moving the entire coordinate system
while leaving the points alone. The latter interpretation is discussed in Section 1.5, but
the reader should realize that whatever interpretation is used, the movement caused by
a geometric transformation is instantaneous . We should not think of a point as moving
along a path from its original location to a new location, but rather as being grabbed
and immediately planted in its new location.

The description of right lines and circles, upon which geometry is founded, belongs to
mechanics. Geometry does not teach us to draw these lines, but requires them to be
drawn.

—Isaac Newton (1687)

Combining transformations is an important operation that is discussed in detail
in Section 1.2.2. This paragraph intends to make it clear that such a combination
(sometimes called a product ) amounts to a composition of functions. If functions f and
g represent two transformations, then the composition g

f represents the product of the
two transformations. Such a composition is often written as P ∗ = g ( f ( P )). It can be
shown that combining transformations is associative (i.e., g

◦

h ). This
fact, together with a few other basic properties of transformations, makes it possible
to identify groups of transformations. A discussion of mathematical groups is beyond
the scope of this topic but can be found in many texts on linear algebra. A set of
transformations constitutes a group if it includes the identity transformation, if it is
closed, and if every transformation in the set has an inverse that is also included in the
set.

◦ ( f ◦ h )=( g ◦ f ) ◦

An example of a group of transformations is the set of two-dimensional rotations
about the origin through angles of 0 ◦ and 180 ◦ . This two-element set is a group since
a zero-degree rotation is an identity transformation and since a 180 ◦ rotation is the
inverse of itself.