48 COMPUTER AIDED ENGINEERING DESIGN
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0.9082 0.4083 1.9082 18165Previous sections discussed five kinds of transformations, namely, translation, rotation, reflection,
scaling and shear, in both two and three dimensions. Homogeneous coordinates were introduced to
unify all transformations into matrix multiplication operations. Of the five, the first three are rigid-
body transformations while the other two cause change in the shape of the object and/or size. It is
apparent from the examples that the mathematics of transformations at the back end of the CAD
software is quite involved. At the front end, however, a user barely feels the rigor as the operations
are hidden behind the graphical user interface that is designed to be very user-friendly. Transformations
are not only applied in Computer Aided Assembly of many engineering components, but are also,
significantly used in design operations. Interactively repositioning a data point in free-form curve/
surface design requires translation. In constructive solid geometry (Chapter 8), many primitives
(cylinders, blocks and others) require scaling and repositioning before they can be combined using
Boolean operations (cut, join and intersect) to obtain a desired solid model. Transformations form an
integral part of a CAD software.
2.7 Projections
Over a long period of time, designers and engineers have developed visualization techniques for
three-dimensional objects that have helped in their representation, comprehension, communication
and viewing. Pyramids, chariots, temples, canals, planned cities (Harappa-Mohen-jo-daro, for example),
cave paintings, all suggest that architects, city planners and designers may have used projections to
explain their ideas to the supervisors or artisans to execute the plan appropriately. A floor plan of a
building, belonging to 2150 B.C. has been found in Mesopotamia as a part of the statue of King
Gudea of the city of Lagash. Some temples and structures in South-East Asia have been carved out
of a single piece of rock, suggesting a remarkable sense of three-dimensional geometry and precision
in chipping off the stone pieces. Likewise, developments have also been observed in Roman and
Greek architectures. Some 15th century artists, namely, Brunellesci, Leone Alberti, and Leonardo da
Vinci, who were mathematicians as well, introduced perspective in their two-dimensional renderings.
In 17th century, Pascal, DesCarte and Kepler developed analytical tools for projective geometry. The
method of orthographic projections, as every engineer knows today, was developed by a French
engineer, Gaspard Monge (founding member of Ecole Polytechnique, 1746-1818). Engineering drawing
was further developed during the industrial revolution in 19th and 20th century, and since then, this
mode of representation for engineering components has been in wide use. The conventional paper
and pencil approach to represent engineering drawings is gradually paving way to computer graphics
that has been in use since the 1970s.
Visual communication has two aspects: (a) the information that a two-dimensional picture of a
three-dimensional component is trying to communicate and (b) how it communicates. Till recently,
two-dimensional drawings were the only means to reveal engineering ideas but now, with better
comprehension capabilities in three-dimensions, relatively cheap prototypes of machine parts designed
with intricate shapes can be manufactured with great precision. Numerically controlled manufacturing
machine tools can be programmed for a given geometry. Rapid prototyping machines can print
physical models after acquiring the instructions directly from the geometric model created using the