Advantage and disadvantage of synthetic curve over analytic one
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Last updated: Yeh-Liang Hsu (2010-10-05).
Note: This is the course material for “ME550 Geometric modeling and computer graphics,” Yuan Ze University. Part of this material is adapted from CAD/CAM Theory and Practice, by Ibrahim Zeid, McGraw-Hill, 1991. This material is be used strictly for teaching and learning of this course.
Curve representation
1. Wireframe models
There are three types of geometric models, wireframes, surfaces and solids. Typically, a wireframe model consists of a finite set of points (vertices), connected in pairs by straight lines (edges), or arcs, circles, conics, and curves, so that the three- dimensional form of a solid object can be visualized.
The major advantage of wireframe modeling is its simplicity to construct. It does not require as much computer time and memory as does surface or solid modeling. Wireframe modeling is considered a natural extension of traditional methods of drafting. Consequently, it does not require extensive training of users; nor does it demand the use of unusual terminology as surfaces and solids. Wireframe models form the basis for surface models. Most existing surface algorithms require wireframe entities to generate surfaces.
The disadvantages of wireframe models are manifolds. Primarily, wireframe models are usually ambiguous representations of real objects and rely heavily on human interpretation. A wireframe model of a box offers a typical example where the model may represent more than one object depending on which face(s) is assumed to exist. Models of complex designs having many edges become very confusing and perhaps even impossible to interpret. Moreover, as shown in Figure 1, it is often difficult to display objects with curve surfaces using wireframe.

Figure 1. Displaying holes and curved ends in wireframe models
◇Assignment 1
Construct an object with several curve surfaces, the object in Figure 1 for example, using your CAD software. Display the object as a wireframe model. Change the views of the display. Do you find anything interesting on the curve surfaces? ◇
2. Parametric curves
2.1 Curve representation
Curves can be described mathematically by nonparametric or parametric equations. Nonparametric equations can be explicit or implicit. For a nonparametric curve, the coordinates y and z of a point on the curve are expressed as two separate functions of the third coordinate x as the independent variable [see Equation (1)]. This curve representation is known as the nonparametric explicit form. If the coordinates x, y and z are related together by two functions [see Equation (2)], a nonparametric implicit form results.
 (1)
 (2)
Note: This is the course material for “ME550 Geometric modeling and computer graphics,” Yuan Ze University. Part of this material is adapted from CAD/CAM Theory and Practice, by Ibrahim Zeid, McGraw-Hill, 1991. This material is be used strictly for teaching and learning of this course.
Curve representation
1. Wireframe models
There are three types of geometric models, wireframes, surfaces and solids. Typically, a wireframe model consists of a finite set of points (vertices), connected in pairs by straight lines (edges), or arcs, circles, conics, and curves, so that the three- dimensional form of a solid object can be visualized.
The major advantage of wireframe modeling is its simplicity to construct. It does not require as much computer time and memory as does surface or solid modeling. Wireframe modeling is considered a natural extension of traditional methods of drafting. Consequently, it does not require extensive training of users; nor does it demand the use of unusual terminology as surfaces and solids. Wireframe models form the basis for surface models. Most existing surface algorithms require wireframe entities to generate surfaces.
The disadvantages of wireframe models are manifolds. Primarily, wireframe models are usually ambiguous representations of real objects and rely heavily on human interpretation. A wireframe model of a box offers a typical example where the model may represent more than one object depending on which face(s) is assumed to exist. Models of complex designs having many edges become very confusing and perhaps even impossible to interpret. Moreover, as shown in Figure 1, it is often difficult to display objects with curve surfaces using wireframe.

Figure 1. Displaying holes and curved ends in wireframe models
◇Assignment 1
Construct an object with several curve surfaces, the object in Figure 1 for example, using your CAD software. Display the object as a wireframe model. Change the views of the display. Do you find anything interesting on the curve surfaces? ◇
2. Parametric curves
2.1 Curve representation
Curves can be described mathematically by nonparametric or parametric equations. Nonparametric equations can be explicit or implicit. For a nonparametric curve, the coordinates y and z of a point on the curve are expressed as two separate functions of the third coordinate x as the independent variable [see Equation (1)]. This curve representation is known as the nonparametric explicit form. If the coordinates x, y and z are related together by two functions [see Equation (2)], a nonparametric implicit form results.
 (1)
 (2)
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