# Curve generation of implicit functions by incremental computers

**ABSTRACT** The conventional numerical solution of an implicit function f(x, y) = 0 is substantially complicated for calculating by any computer. We propose a new method representing the argument of the implicit function as a unary function of a parameter, t, if the continuous and unique solution of f(x, y) = 0 exists. The total differential constitutes simultaneous differential equations of which the solution about x and y is unique. The Newton-Raphson method must be used to calculate the values near singular points of an implicit function and then the sign of dt has to be decided according to four special cases. Incremental computers are suitable for curve generation of implicit functions by the new method, because the incremental computer can perform more complex algorithms than the analog computer and can calculate faster than the digital computer. This method is easily applicable to curve generation in three-dimensional space.

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**ABSTRACT:**Special purpose hardware devices devoted to display families of curves are very attractive when high performance graphic tools are to be designed. In this paper an architecture well suited for fast hardware curve generators is proposed, mainly based on the use of vector generators and ROMs. Curve graphs are approximated by polygonal lines, the extremes of which and a selected subset of vertices can be obtained with the required precision. Output rate is shown to be very close to available vector generators rate. As an example, a device adopting this architecture has been designed for the generation of conic and exponential curves. Precision figures have been obtained in the hypothesis that the generator hardware complexity allows a single chip implementation. The architecture is easily extensible to three-dimensional curves.Computers & Graphics 01/1986; 10(1):27-36. DOI:10.1016/0097-8493(86)90066-X · 0.91 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The paper presents an algorithm for general 3D and 2D parametric curve interpolation. This is an incremental step algorithm, and it uses two simple principles; firstly, each selected step has to follow a given direction along the curve, and, secondly, each selected step has to be at the closest distance from the curve. The mathematical formulation of the algorithm permits the interpolation of any kind of parametric curve, which is an important aspect of modern cad/cam systems. Its software implementation on risc workstations and the in-house academic tests to date have shown promising results, and it is hoped that it can be applied in the next generation of computerized numerical control systems.Computer-Aided Design 11/1994; 26(11-26):850-856. DOI:10.1016/0010-4485(94)90100-7 · 1.80 Impact Factor - Journal of Manufacturing Science and Engineering 01/1997; 119(4A). DOI:10.1115/1.2831191 · 1.02 Impact Factor