Computer Aided Engineering Design

SOLID MODELING 257

Cylindrical half-space with radius R: H = {(x,y,z)|x^2 + y^2 < R^2 }

Spherical half-space with radius R: H= {(x,y,z) | x^2 + y^2 + z^2 <R^2 }

Conical half-space with cone angle α: H = ( , , )| + <^22 tan 2

2

xyz x y z

⎛ α

⎝

⎞

⎠

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎪

⎩⎪

⎫

⎬

⎪

⎭⎪

Toroidal half-space with radii R 1 and R 2

H = {( , , ) | (xyz x^222 + + – y z R 22 – R 122 ) < 4R R 22 ( 122 – z)}

Intricate solids can be represented using half spaces by treating them as lower level primitives and
combining them using set operations like those in Constructive Solid Geometry (Section 8.9). For
example, a block B with a cylindrical void C
shown in Figure 8.13 is represented using seven
half spaces. Six of those are planar half spaces
with their material sides pointing into the solid.
The block is the union of six intersecting half
planes. The 7th half space is cylindrical with its
material side pointing towards the axis. The
complement of the cylindrical half space has the
material direction pointing away from the axis. If
this half plane is intersected with the block and
then a union is taken, Figure 8.13 results, that is,
in general, any solid may be considered as the
union of intersections of the half planes or their
complements.
Any representation scheme for computer
modeling of solids should: (a) be versatile and
capable of modeling a generic solid, (b) generate
valid solids having characteristics described in

Figure 8.13 A block with a cylindrical void

requiring seven half spaces for its

representation

Section 8.1, (c) be complete such that every valid representation (solid) produced is unambiguous, (d)
generate unique solids in that no two different representations should generate the same object, (e)
have closure implying that permitted transformation operations on valid solids would always yield
valid solids and (f) be compact and efficient in matters of data storage and retrieval. This chapter
discusses three solid modeling techniques, namely, wireframe modeling, boundary representation
method and constructive solid geometry. The schemes, by themselves, may not have all the features
described above and thus most commercially existing solid modelers employ them in combination as
required by the application.

8.7 Wireframe Modeling

This method is perhaps one of the oldest to represent solids. The representation is essentially through
a set of key vertices connected by key edges. Consequently, two tables are generated for data storage,
one storing the topology (connectivity) and other the geometry. The edges may be straight or curved.
In former, the coordinates of the end points are stored. For curved edges, the control points, slopes
and knot vector may be stored depending on the Ferguson, Bézier or B-spline segments modeled. For
example, the data tables for a tetrahedron are given in Figure 8.14 with the edges numbered within
parenthesis.