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1.2 Analytic representation of surfaces

Similar to the curve case there are mainly three ways to represent surfaces, namely parametric, implicit and explicit methods. In parametric representation the coordinates of a point $ (x, y, z)$ of the surface patch are expressed as functions of the parameters $ u$ and $ v$ in a closed rectangle:
$\displaystyle x=x(u,v),\;\;y=y(u,v),\;\;z=z(u,v),\;\;u_1\leq u \leq u_2,\;\;v_1\leq v \leq v_2\;.\;\;$     (1.12)

The functions $ x(u,v)$ , $ y(u,v)$ and $ z(u,v)$ are continuous and possess a sufficient number of continuous partial derivatives. The parametric surface is said to be of class $ r$ , if the functions have continuous (partial) derivatives up to the order $ r$ , inclusively. In case the class is not explicitly given, it is assumed that the functions have infinitely many derivatives. In vector notation the parametric surface can be specified by a vector-valued function
$\displaystyle {\bf r} = {\bf r}(u,v)\;.$     (1.13)

An implicit surface is defined as the locus of points whose coordinates $ (x, y, z)$ satisfy an equation of the form

$\displaystyle f(x,y,z)=0\;.$     (1.14)

When (1.14) is linear in variables $ x$ , $ y$ and $ z$ , it represents a plane. If (1.14) is of second degree in the variables $ x$ , $ y$ , $ z$ , it represents quadrics [79]
$\displaystyle ax^2 + by^2 + cz^2 + dxy + eyz + hxz +kx + ly + mz + n =0\;.$     (1.15)

Some of the quadric surfaces such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder have explicit forms (see Fig. 8.9). Paraboloid of revolution is a special case of elliptic paraboloid where the major and minor axes are the same. The rest of the quadrics have implicit forms including ellipsoid, elliptic cone, elliptic cylinder, hyperbolic cylinder, hyperboloid of one sheet and two sheets, where the hyperboloid of revolution is a special form. The natural quadrics, sphere, circular cone and circular cylinder, which are special cases of ellipsoid, elliptic cone and elliptic cylinder, are widely used in mechanical design and CAD/CAM systems. Also they result from standard manufacturing operations such as rolling, turning, filleting, drilling and milling [149]. According to a survey conducted by the Production Automation Project group at the University of Rochester in the mid 1970's, 80-85% of mechanical parts were adequately represented by planes and cylinders, while 90-95% were modeled with the addition of cones [434,363,149].

If the implicit equation (1.14) can be solved for one of the variables as a function of the other two, say $ z$ is solved in terms of $ x$ and $ y$ , we obtain an explicit surface

$\displaystyle z=F(x,y)\;.$     (1.16)

This is always possible at least locally when $ \frac{\partial f}{\partial z}\neq 0$ [166]. And if the two variables $ u$ , $ v$ of the parametric form can be solved in terms of $ x$ and $ y$ , we can substitute $ u=u(x,y)$ and $ v=v(x,y)$ into $ z=z(u,v)$ which yields an explicit form. This is possible when $ \frac{\partial x}{\partial u} \frac{\partial y}{\partial v}- \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}\neq0$ [76]. Conversely when the explicit form $ z=F(x,y)$ is given, the parametric form is derived by setting $ x=u$ , $ y=v$ , $ z=F(u,v)$ . Thus, the explicit form can be considered as a special case of implicit and parametric forms.

Example 1.2.1. Let us consider a hyperbolic paraboloid surface patch in the parametric form:

$\displaystyle x=u+v,\;\;\;y=u-v,\;\;\;z=u^2 - v^2, \;\;\;0\leq u,v \leq 1\;.$     (1.17)

Since we can easily solve for $ u$ and $ v$ in terms of $ x$ and $ y$ as $ u=\frac{x+y}{2}$ and $ v=\frac{x-y}{2}$ , the explicit form is obtained as
$\displaystyle z=xy, \;\;\; 0\leq x+y \leq 2, \; 0\leq x-y \leq 2\;.$     (1.18)


Table 1.1: Representations of curves and surfaces

Geometry		 Parametric Implicit Explicit

Plane curves		 $ x=x(t)$ , $ y=y(t)$ $ t_1\leq t \leq t_2$ $ f(x,y)=0\;\;\;$ or $ {\bf r}={\bf r}(u,v)$ $ \cap$ plane $ y=F(x)$

Space curves		 $ x=x(t)$ , $ y=y(t)$ , $ z=z(t)$ , $ t_1\leq t \leq t_2$ $ f(x,y,z)=0 \cap g(x,y,z)=0$ or $ {\bf r}={\bf r}(u,v)$ $ \cap$ $ f(x,y,z)=0$ or $ {\bf r}={\bf p}(\sigma,t)$ $ \cap$ $ {\bf r}={\bf q}(u, v)$ $ y=Y(x)\;\cap$ $ z=Z(x)$

Surfaces		 $ x=x(u,v)$ , $ y=y(u,v)$ , $ z=z(u,v)$ , $ u_1\leq u\leq u_2$ , $ v_1\leq v\leq v_2$ $ f(x,y,z)=0$ $ z=F(x,y)$


Table 1.2: Comparison of different methods of curve and surface representation

Disadvantages

Explicit		 Implicit Parametric

Infinite slopes are impossible if $ f(x) $ is a polynomial.		 Difficult to fit and manipulate free form shapes. High flexibility complicates intersections and point classification.
Axis dependent (difficult to transform). Axis dependent.  
Closed and multivalued curves are difficult to represent. Complex to trace.  

Advantages

Explicit		 Implicit Parametric

Easy to trace.		 Closed and multivalued curves and infinite slopes can be represented. Closed and multivalued curves and infinite slopes can be represented.
  Point classification (solid modeling, interference check) is easy. Axis independent (easy to transform).
  Intersections/offsets can be represented. Easy to generate composite curves.
    Easy to trace.
    Easy in fitting and manipulating free-form shapes.

Table 1.1 summarizes the three representation forms for plane curves, space curves and surfaces. Table 1.2 compares the three representations [119,116]. It is clear from the tables that the parametric form is the most versatile method among the three and the explicit is the least. Furthermore, the explicit form can always be easily converted to parametric form. Therefore we will mainly focus on the parametric and implicit forms throughout this book. Methods to fit and manipulate free-form shapes in implicit form are more complex than those for the parametric form both with respect to computation and geometric intuition. However, a considerable body of research aimed at alleviating precisely this obstacle has been published over the last fifteen years, see for example [373,299,16]. In this book we do not cover implicit surface fitting and design methods.

next up previous contents index
Next: 1.3 Bézier curves and Up: 1. Representation of Curves Previous: 1.1.2 Space curves   Contents   Index
December 2009