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# Curves and surfaces in computer aided geometric design - Yamaguchi F.

Yamaguchi F. Curves and surfaces in computer aided geometric design - Tokyo, 1988. - 390 p.
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In actual design of a curve or surface, it is necessary to connect a number of curve segments or surface patches smoothly. In this case, in order to connect with continuity up to the curvature, troublesome restrictive conditions must be applied to the connecting sections in the vicinity of the polygon vertices. Bezier curves and surfaces have superior “controllability”, but have a problem with “connection”. Bezier curves and surfaces use Bernstein Basis functions as blending functions.
Gordon and Riesenfeld proposed curves and surfaces which use Basis splines as blending functions. These are called J3-spline curves and surfaces (refer to Chap. 6). B-spline curves, like Bezier curves, are defined by polygon
0. Mathematical Description of Shape Information
vertices and have properties similar to those of Bezier curves. That is, B-spline curves are expressed as a convex combination of polygon vertex position vectors, and also have the variation diminishing property. Curve shapes are smoothed versions of the polygon shapes and can be roughly predicted from the polygon shapes. In contrast to Bezier curve, which is a convex combination of all of the vertex position vectors, B-spline curve differs in that it is a convex combination of a number of vertex position vectors in their immediate vicinity. It follows from this that the shape variation properties of the polygon show up even more clearly in J3-spline curves than in Bezier curves. If n is the number of sides of the given polygon, the B-spline curve is formed by smoothly joining (n — M + 2) (M— l)-degree curve segments. A basis spline is determined by specifying the order M and the knot vector. The curve segments which make up a B-spline curve are defined by the M polygon vertices in the vicinity of each. Consequently, when one polygon vertex position is varied to control the curve shape, the effect of the change is locally confined. This property is very important in designing curves.
Ferguson curves and surfaces
(Problems in control and connection)
Coons surfaces
Bezier curves and surfaces
(improvement with respect to the connection problem)
I-------- t---------------------
I Interpolated spline curves and surfaces
(improvement with respect to the connection problem)
| g-splme curves and surfaces~j | Spline under tension
(Solves problems of control and connection)
(Attempt to improve controllability)
Fig. 0.2. Sequence of development of mathematical curve and surface description methods
0.4 The Development of Mathematical Descriptions of Free Form Curves and Surfaces 9
J3-spline curves are superior curves which combine the superior controllability of Bezier curves with the connection properties of spline curves. By specially specifying the knot vector in a B-spline curve segment, it can be made to agree with the Bezier curve segment.
The sequence of development of mathematical curve and surface description methods described above is shown schematically in Fig. 0.2.
B-spline curves can perhaps be thought of as possessing ideal “control” and “connection” properties. How well do J3-spline surfaces describe surface shapes? As shown in Fig.0.3, an ordinary, simple surface can be expressed by joining a number of surface patches together in a matrix. In the case of such a simple surface, the surface can be described and processed as a single J3-spline surface (a surface defined by mxn position vectors). When it comes to actual surfaces, it sometimes happens that it is difficult to describe and process the surface shape as such a simple surface. Figure 0.4 shows an example of the kind of surface shape that occurs in the vicinity of a corner of a solid object formed by the convergence of three ridges. In this case, there is a problem of how to describe a triangular area of a surface. Figure 0.5 shows another example, this time of a rounded concave corner formed by the convergence of 3 ridges. In this case the surface becomes pentagonal in shape. It is not impossible to express such special shapes other than standard quadrangular shapes by mathematical functions, but special, complicated treatment becomes necessary. Special considerations also become necessary when joining neighboring surface patches together.
Curves and surfaces which can be described as ordinary polynomials with respect to parameters have been described above. However, circles and circular arcs, which are very important in industry, cannot be rigorously expressed by ordinary polynomials. Rigorous mathematical expressions of conic section curves is possible in the form of rational polynomials with respect to a parameter (refer to Chap. 7). Consequently, rational polynomial descriptions of curves and surfaces are very interesting, but full-scale research and practical applications along this line remain as problems for the future.
Fig. 0.4. Example of a surface in which a triangular surface patch occurs
0. Mathematical Description of Shape Information
Fig. 0.5. Example of a surface in which a pentagonal surface patch occurs
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