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In 1963, Ivan E. Sutherland of M.I.T. announced a revolutionary system called “Sketchpad”. It created the possibility that basic design and detailed
0. Mathematical Description of Shape Information
design could be done under computer control.1} Even if a shape is described mathematically, it is hard for a person to understand what kind of shape is being described just by looking at the mathematical description. The mathematical description of the shape is stored in a computer memory device. If it can be displayed on a CRT screen, then it becomes visible to people and can be understood easily by a human being. In this case, the shape description and transmission can be regarded as being carried out objectively. Since it is easy to create a drawing of the shape from its mathematical model, by creating various drawings of the shape as seen from various locations and directions, a person’s understanding of the shape can be deepened. The person can draw a freehand sketch by stylus of the desired shape on the CRT screen, and can instruct the computer to perform specialized processing of the shape by pointing to a particular point on the CRT screen image with the stylus. Consequently, if a mathematical model can be created based on the picture drawn by a person and on information indicated by a stylus, it becomes possible to do even basic design and detailed design on the CRT screen through conversational interaction with the computer (Fig. 0.1). That is to say, in place of the conventional design process based on drawings, a new design process can be conceived which is based on a mathematical model of a shape stored in a computer. In this method, in contrast to the method described above in which a physical model is converted to a mathematical model, mathematical model is created in the very beginning stage of design; if necessary a physical model can then be created from the mathematical model.
0.4 The Development of Mathematical Descriptions of Free Form Curves and Surfaces
At the start of the 1960s, J. C. Ferguson of Boeing Aircraft Company in the U.S.A. announced a method of describing curve segments as vectors, using parameters (refer to Chap. 3). A Ferguson curve segment is a cubic vector function with respect to a parameter obtained by specifying the position
0.4 The Development of Mathematical Descriptions of Free Form Curves and Surfaces 5
vectors and tangent vectors of the starting and end points of the curve segment. An actual curve is created by joining these curve segments smoothly. In addition, Ferguson, using these curve segments, proposed a method of creating a portion of a surface (called a surface patch) that satisfies the conditions imposed by specifying position vectors and tangent vectors at 4 points, and put this method to practical use in Boeing’s surface creation program FMILL. FMILL is a system intended to create NC tapes. Before this work by Ferguson, mathematical representation descriptions of curves had been the form y=f(x) or F(x, y) = 0. In contrast, Ferguson curves have the following advantages.
1) Not only curves in a plane, but curves in space can be expressed by simple functions.
2) The part of a curve that is needed can easily be specified by a parameter range.
3) Since a slope parallel to the y-axis can be expressed by dx/dt = 0, it is not necessary to use dy/dx= oo.
4) Transformations of a curve such as translation and rotation can be carried out simply by multiplying by a transformation matrix.
Subsequently, parametric description of curves and surfaces became the standard method of mathematical representation.
It can be seen from NC-processed surface shapes that when Ferguson surface patches are taken to be relatively large, the surface tends to be flattened in the vicinity of the 4 corners of the patch.
Ferguson curves and surfaces are an example of the use of Hermite interpolation functions.
In 1964, S. A. Coons of M.I.T. announced a surface description method in which one considers the position vectors of the 4 corner points of a surface patch and the 4 boundary curves, and derives a mathematical description which satisfies those boundary conditions (refer to Sect. 3.3.2). A generalized version of this concept was announced in 1967 (refer to Sect. 3.3.3). Coon’s surface patch is defined not only by the position vectors and higher order differential vectors with respect to the 4 corner points of the patch, but also the position vectors related to the 4 boundary curves and the higher order differential vectors with respect to the directions across the boundary curves. In practice, it is sufficient to give positions with respect to the 4 corner points of the surface patch, tangent vectors in 2 directions, mutual partial differential vectors (twist vectors) and positions with respect to the 4 boundary curves and tangent vector functions in the directions across the boundary curves. In this method, if the 4 boundary curves and tangent vector functions are expressed by Hermite interpolation formulas, a representation resembling the Ferguson patch is obtained. This is called a bi-cubic Coons surface patch; it is expressed in a simple form and is widely used. If the twist vector is set to zero, the bicubic Coons surface patch agrees with the Ferguson surface patch. That is, the Ferguson surface patch is a special case of the Coons surface patch. The reason that the Ferguson surface patch flattens the surface in the vicinity of the 4 corners is that the twist vector has been taken to be zero. In his paper,