Control-Line Curves: The formulations for parametric curves defined by control points is revised to improve their design capabilities. By simply replacing the control points with control lines in the formulations of these curves, powerful tools will be obtained that enable creation of complex geometries more precisely and in a shorter amount of time. Curves designed by control lines will be called control-line curves or plus curves. The plus implying that in addition to the control points, tangent vectors at the control points are used to design a curve. A control-line curve with Bernstein-Bézier basis functions is called a Bézier+ curve.

Examples:

Fig. 1. (a)–(c) Different control-line Bézier curves of order 3 obtained by varying the tangent direction at the middle point.

Fig. 2. (a) A traditional Bézier curve of order 5. The curve does not resemble its control polygon. (b) The curve obtained using the revised formulation.

Fig. 3. (a) A Bézier curve of order 3. (b) - (d) Bézier+ curves obtained from the same control points and the same tangent directions, but with different tangent magnitudes.

Fig. 4. Comparison of Bézier curves (yellow) and Bézier+ curves (blue). The control points are shown in white, the control lines are shown in green, and the control polygons are shown in purple. (a) Bézier and Bézier+ curves are tangent to the control polygon at start and end points. (b) Closed Bézier and Bézier+ curves obtained from the same control points. (c) A Bézier+ curve may have one more inflection point than a Bézier curve of the same order. (d) Bézier+ curves do not always provide linear precision.

Control-Plane Surfaces: The same idea can be extended to parametric surfaces also. By replacing the control points with control planes, it becomes possible to revise the local shape of a surfaces by changing the orientation of the control points (or normals to the control planes). An Example is demonstrated below.

Fig. 5. (a) A Bézier patch defined by nine control points. Once the control points are selected, the surface becomes fixed. Note that the patch is very far from the center control point. (b) A Bézier+ patch obtained from the same control points. The normal vector magnitudes are interactively increased until the surface gets sufficiently close to the control point. (c) Overlaying of (a) and (b) for comparison purposes. (d)-(h) Different surfaces obtained by changing the orientation of the center control-plane (normal).

Use of Control-Plane Surfaces in Smooth Parametric Representation of Polygon Meshes

To show the power of control-plane surfaces, suppose a polygon mesh is given and we want to represent the mesh by a smooth surface. If we use the polygon faces as the control planes in the equation of rational Gaussian surfaces [1], we will obtain a smooth parametric surface approximating the mesh. An example of this is shown in Fig. 6. Note that the mesh is not required to be made of a homogenous mesh elements, such as triangles. A mixture of different polygon types may be used. In this example, a mesh made up of triangluar and quadrilateral mesh elements is shown. Fig. 6a shows the mesh, and Figs. 6b - f show surfaces obtained with different smoothness levels of the rational Gaussian.

Fig. 6. Approximation of a polygon mesh by a control-plane surface. The polygon faces are used as the control planes in the equation of a rational Gaussian surface [1]. (a) The given mesh. (b) - (f) Control-plane surfaces approximating the mesh at increasing smoothness levels.

Examples of larger data sets are shown in Fig. 7. Figs. 7a, 7c, and 7e show three polygon meshes with thousands of faces and Figs. 7b, 7d, and 7f, show approximation of the meshes by control-plane surfaces. These data sets are courtesy of the U.S. Geological Survey, Stanford University, and SGI.

Fig. 7. Smooth parametric representation of irregular polygon meshes by control-plane surfaces. (a) A digital elevation map courtesy of the U.S. Geological Survey and (b) its smooth representation. (c) The Stanford bunny and (d) its smooth representation. (e) A vase courtesy of SGI and (f) its smooth representation.

[1] A. Goshtasby, "Geometric Modeling Using Rational Gaussian Curves and Surfaces," Computer-Aided Design, May 1995, pp. 363-375.

For more information contact A. Goshtasby (agoshtas@cs.wright.edu).

Last modified: 1/31/03.