Graphics Reference
In-Depth Information
surfaces, Coons surfaces, and Bézier or B-spline surfaces are examples of these three
principles, respectively.
Lofting is sometimes considered as a special case of a more general surface con-
struction referred to as a
directrix-generator
representation. This is a representation
that describes a surface in terms of sweeping a generator curve along a set of guide-
lines. The three ingredients for such a representation are
(1) a set of longitudinal curves called
directrices
(also called
meridians
in
mathematics),
(2) a
correspondence rule
that relates each point of a directrix with a unique point
on every other directrix, and
(3) a
generator rule
that defines a curve through all the points on the directrices
that are related by the correspondence rule.
Figure 12.1 shows an example of how a surface can be defined by a directrix-
generator type of construction. It is a traditional conic lofting example of an airplane
fuselage. The directrices are conics and the generator consists of two conics, one
above the maximum width line and one below. The correspondence rule relates points
with equal x-values. Many surfaces can be defined by means of a directrix-generator
construction. [Sabi90] gives a number of explicit examples. In particular, ruled sur-
faces and surfaces of revolution clearly fall into this class of surfaces. The directrices
for the ruled surface are the bounding curves and the generators are the straight-line
segments that connect the corresponding points on these curves. A surface of revolu-
tion has a single directrix, namely, the curve being rotated, and the generators are
circles. The idea of a directrix-generator representation can be generalized to allow
directrices to act as control points rather than being interpolated.
One of the driving forces behind the existence of the many types of surfaces is to
make it easy for users (and programmers for that matter) to define the ones they need.
Providing interactive descriptions of sets of three-dimensional points is much harder
than for sets in the plane. One of the obvious reasons is that one is trying to describe
three dimensions on a two-dimensional medium, the computer screen. To make geo-
metric modeling programs “user friendly” one wants to avoid making a user enter
complicated mathematical equations. One common approach to accomplish this is
to let users define surfaces from one-dimensional sets using natural operations. For
example, to define a surface of revolution a user needs only specify the curve to revolve
and the axis of rotation.
Sections 12.2-12.12.1 describe some of the well-known basic surface types start-
ing with the simple and intuitively easy to understand surfaces and working up to the
more complicated ones. In Section 12.12.2 we show how the multiaffine approach to
Figure 12.1.
A fuselage as a directrix-
generator surface.
