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0. Mathematical Description of Shape Information
Coons discussed the conditions which must be satisfied to join surface patches together to form a continuous surface, and gave a method for connecting them so as to be continuous up to the n-th order differential vector in the directions across the boundary curves.
Ferguson curve segments and surface patches and Coons surface patches share a number of problems in control and connection of segments and patches, such as the following.
1) It is difficult for a person to control the shape of a surface directly.
2) When curve segments or surface patches are connected, it is necessary to be concerned not only with local mathematical continuity at the contact points, but also with overall smooth connection of the whole curve or surface, but neither method specifies the conditions necessary for this.
Coons’ intention in conceiving the surface patch was to make it possible for a person to do design work interactively on a computer while watching a CRT screen connected to the computer. However, in practice Coons’ surface patch has found its greatest application in physical model surface fitting.
A technique for solving the “connection” problem is the spline (refer to Chap. 4). A spline is a flexible band made of wood, plastic or steel. In the design of products having free form surface shapes such as ships, airplanes and automobiles, splines held in shape by “weights” have long been used to obtain free form curve shapes. It is known from experience that curves produced by splines are fair. A curve produced by a spline is described by different cubic degree curves in different segments between “weight” and “weight”. At the positions of “weights”, that is, at the connecting points between curve segments, connections are continuous up to the curvature. Moreover, along the entire length of the spline, the integral of the square of the curvature must be a minimum among all possible curves passing through
the “weight” points. This means that the total bending energy stored in the
spline is a minimum.
Mathematical curves which approximate spline shapes by means of
parametric vector functions are very important in CAD. In particular, the natural spline (refer to Sect. 4.4) has the property that it gives the
interpolation that minimizes the integral of the square of the curvature. Ferguson’s and Coons’ conditions for connection of curve segments and surface patches insure local mathematical continuity. However, an infinite number of curves which are mathematically continuous at the contact points can be obtained by varying the magnitude of the tangent vector, resulting in an infinite number of possible curve shapes. The minimum interpolation property of the natural spline indicates what conditions should be satisfied in connecting curve segments and what shape the overall curve should have. In a spline.curve, the connection conditions are determined simultaneously at all connection points. By repeated application of a method similar to the spline curve method, spline surfaces can be created. For example, in a bi-cubic spline surface (refer to Sect. 4.11), continuity on the patch boundary curves is obtained up to the curvature in the direction across the boundary curves. Then the spline surface is uniquely determined. The spline method can be
0.4 The Development of Mathematical Descriptions of Free Form Curves and Surfaces
thought of as automatically solving the “connection” problem that exists with Ferguson and Coons curves and surfaces.
The shape of a spline curve is controlled by varying the positions of the connection points between curve segments (corresponding to “weight” positions) and by increasing the number of curve segments. Since the position of one point on a spline curve is determined by all data relating to the initially given points through which the curve must pass (which are also connecting points between curve segments), the effect of changing the position of one point through which the curve must pass extends throughout the whole curve. In addition, in some cases, curves are produced which vary in ways that are hard to predict from the given series of points through which the curve must pass. For example, there can be a loop in the curve that is produced even though there is no such a shape in the given series of points. As long as one wishes to use spline curves and surfaces as they are produced, there is a problem of “control”.
D. G. Schweikert proposed the use of a “spline under tension”2). This was an attempt to improve the controllability of the spline. By suitably adjusting the parameter that corresponds to tension, the production of loops in the spline can be prevented. The curve representation takes the form of a hyperbolic function.
P. Bezier of the Renault Company in France announced a curve representation that is defined by giving one polygon (refer to Chap. 5). This Bezier curve segment can be regarded as a curve obtained by smoothing the corners of the given polygon. Bezier curves have been put to practical use in Renault’s automobile body design system UNISURF3). Bezier curve segments and surface patches are defined only by the position vectors of polygon vertices; unlike the Ferguson and Coons methods, this method does not require analytical data that are hard to understand intuitively such as tangent vectors and twist vectors. Bezier curve segments are expressed as a convex combination of the polygon vertex position vectors which define the curve, and possess a variation diminishing property. Consequently the curve shape can be approximately anticipated from the polygon shape. In addition, it is also possible to increase the degree of the polynomial curve; for example, a curve segment can be split into two segments without changing the shape of the curve, or the degree of the curve can be formally increased also without changing its shape. That is to say, Bezier curves and surfaces are in a form that is easy for a person to control.