Chapter 2
Transformations and Projections
Geometric transformations provide soul or lifetovirtual objects created through geometric modeling
discussed in later chapters. It is using transformations that one can manoeuver an object, view it from
different angles, create multiple copies, create its reflected image, re-shape or scale an object, position
an object with respect to the other, and much more. Projections, like orthographic and perspective on
the other hand, help comprehend an object for purpose of its fabrication. Transformations have many
uses, mainly pertaining motion, such as manipulating the relative positions of two objects in solid
modeling to create a complex entity, displaying motion of mechanisms, animating an assembly to
demonstrate its working or imparting motion to a virtual human for a walk through a virtual city or
a building. Motion simulators for aircrafts, tanks and motor vehicles extensively employ geometric
transformations.
Transformations may be employed to perform rigid-body motion wherein an object may be moved
from one position to another without altering its shape and size. Typical rigid body transformations
involvetranslation,rotation and reflection, the latter being a combination of the first two. Transformations
may also cause deformationslikeshear, scaling and morphingwherein the object is altered in size
and/or shape. For special effects, free-form deformation may be used where a geometric model is
embedded inside a grid of control points, and transformations are applied to these control points to
distort the object in a desired manner.
When dealing with transformations, an engineer would require a full description of the object, its
position relative to a fixed point called origin, and a specified set of coordinate axes. An object may
Figure 2.1 Use of transformation to view an object from different angles