Chapter 7
Design of Surfaces
A closed, connected composite surface represents the shape of a solid. This surface, in turn, is
composed of surface patches, much like how a composite curve is a collection of juxtaposed segments.
Surface design may be influenced by a variety of factors such as aesthetics, aerodynamics, fluid flow
(for turbine blades, flow passages in a gas turbine, ship hull), ergonomics and many others. A free-
formed surface like the aircraft wings and fuselage, car body and its doors, seats and windshields are
all designed by combining surface patches at their boundaries.
Surface patches, similar to curve segments, can be modeled mathematically in parametric form
using two parameters u and v:
f (u,v) = [x(u,v)y(u,v)z(u,v)], u∈ [0, 1], v∈ [0, 1] (7.1)wherex(u,v),y(u,v) and z(u,v) are scalar polynomials in parameters (u,v). Note that a surface patch
is bounded by the curves f(u, 0), f(u, 1), f(0,v) and f(1,v).
With an implicit representation φ(x,y,z) = 0, many analytical surfaces such as a sphere, an
ellipsoid, a paraboloid and others can be represented accurately. In parametric form, surface patches can
be constructed (approximated) to closely represent the analytical counterparts. In a reverse engineering
application, a surface may also be required to fit a point cloud or a given set of large number of points
in space, usually obtained when scanning an existing surface by laser beam or the faro arm^1.
Though a set of data points and boundary curves may be known, the shape of the surface patch is
to be designed based on designers’ intuition and some qualitative data such as smoothness, flatness,
bumps and change in curvature. Design of the car rooftop as a surface and the manifold for coolant
circulation in the engine may require different methods of surface generation. We need to create a
surface without knowing a large number of points, because the analytic form of the desired surface
may not be known at the time of conception. A designer may often be required to change the shape
interactively to achieve the desired shape. In design, it is more convenient to deal with surface
patches and create a composite surface by stitching the patches ensuring Cn (n = 0 for position, 1 for
slope and 2 for curvature) continuity. Boundary surfaces (and hence solids) are to be eventually
manufactured using automated machine tools, press dies as in sheet metal forming, casting, molding
and other processes. The mathematical description of a surface must be eventually transformable via
the CAD/CAM interface to generate the manufacturing data and tool path generation codes.
(^1) A robot arm that has an end sensor to locate the coordinates of points on a physical surface.