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

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0. Mathematical Description of Shape Information

What, then, is the relation between shape and drawings for Type 2 shapes? Since Type 2 shapes include curved surfaces which vary in a complicated manner, in general there are many cases in which a small number of curves drawn on paper are not in themselves sufficient to describe the shape. In the case of a Type 2 shape, it is not possible to rely on the ability of the person reading the drawings to infer the necessary shape information from it. If not enough curves are given, the shape between one curve and the next will vary depending on the subjective judgment of the person reading the drawings. Often the model builder’s interpretation of the drawings differs somewhat from what the designer intended. When the designer sees the model, he tries to tell the model builder what shape it is that he really intended, but in the absence of adequate tools for this purpose there is no way to give a really accurate explanation.

0.2 Processing and Analysis of Shapes

In the design of an industrial product, various types of processing and analysis are performed with respect to the shape to make sure that it not only has a beautiful exterior appearance, but also satisfies a number of necessary technical conditions. For example, calculation of the surface area, volume, weight and moment of inertia of a shape, structural strength analysis, vibration analysis, fluid flow analysis, thermal conductivity analysis and NC tape preparation also may become necessary. All of these processings and analyses are performed by having a person read shape information from drawings and then using the resulting data. Recently, through technological advances such as the Finite Element Method (FEM) and the Finite Boundary Method (FBM), it has become possible to perform these analyses even on complicated shapes. In the case of a Type 1 shape, it is possible to assume that there will be no problem of ambiguity in the representation of a shape by drawings, so there are few problems in shape processing and analysis. Meanwhile, in the case of a Type 2 shape, there are many ambiguities in the drawing representations, so it is very difficult for a person to read the data necessary for shape processing and analysis.

In addition, viewing this problem from the point of view of amount of information, in general a Type 2 shape involves a much greater amount of information than a Type 1 shape. It is very troublesome for a person to read such a large amount of information and then perform calculations using that large amount of information, and it is easy for mistakes to occur. Let us consider the case of a container as an example. The internal volume of the container must be rigorously adjusted to a specified value. If the container that is designed has an actual volume that is smaller than the nominal volume, the customers will probably complain. If the volume is too large, the company that bottles drinks will suffer a loss. Since volume calculation for a complicated shape cannot be carried out easily, normally it is done by the end summing rule. Based on curve information on the drawings, the cross-

0.3 Mathematical Description of Free Form Shapes

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sectional shape which the container is expected to have at a certain height is drawn on graph paper, and the number of squares enclosed by the curve is counted to give the area of the cross-section. The shape of the container is approximated as a collection of a large number of thin slices, the cross-sectional area of each of which is calculated by the above method, and then the volumes of the slices are added up to give the total volume. Calculating the volume in this manner takes a great deal of time and is very troublesome, and then the shape must be adjusted to bring the actual volume into agreement with the specified volume, which is also very difficult.

From the point of view of description and transmission of shape information and processing and analysis with respect to that shape, many problems lurk in a design and production system based on drawings, particularly in the case of Type 2 shapes.

0.3 Mathematical Description of Free Form Shapes

One method of solving this kind of problem is to describe shapes mathematically. When this is done shape description becomes objective rather than subjective, and it becomes possible to use the power of computers to do the various processings and analyses with respect to the shape.

Technology for describing shapes mathematically was first developed in connection with numerical control processing technology. Since the world’s first NC 3-dimensional milling machine was perfected at M.I.T. in 1951, concentrated research, mainly at M.I.T., began on software technology for describing shapes mathematically and preparing NC tapes using a computer. The large-scale software system called “APT III” was perfected in the early 1960s. In the APT system, a program reads shape information off of drawings. When the shape and the movement of a tool that processes that shape are programmed, the computer creates a numerical model of the shape; then, based on that model, it calculates accurate coordinates of the tool path and outputs an NC tape. At the stage at which the APT system is used, design work related to the shape has been completed. To put it another way, the APT system deals with the last stage of design; it uses a computer to create an NC tape which contains what might be called “production command information”. Use of the APT system still requires drawings; consequently the problem of “ambiguity” of drawings in shape design remains. In this design method, shape creation and modeling and in fact all decision-making operations in the basic design and detailed design stages are done completely by human beings, and a finished physical model is produced. Using measurement of the model a mathematical model is created on the computer to produce a N/C machining tape for the part. In other words, in this method a copy of the physical model is produced mathematically; that is, this is a design method “from a physical model to mathematical model”.

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