Abstract
The curve is the most basic design element to determine shapes and silhouettes of industrial products and works for shape designers and it is inevitable for them to make it aesthetic and attractive to improve the total quality of the shape design. If we can find equations of the aesthetic curves, it is expected that the quality of the curve design improves drastically because we can use them as standards to generate, evaluate, and deform the curves. The authors have proposed the general equations of aesthetic curves as such a standard. However the aesthetic curves expressed by the general equations are limited to planar curves and it is necessary to convert the curves into B-spline forms to guarantee the compatibility of the curves on existing CAD systems. Hence in this paper, at first we show the necessary and sufficient condition for a given curve to have the self-affinity and then extend the aesthetic curves into 3-dimensional space. Furthermore we propose two methods to approximate them with B- pline curves by the least square to minimize the positional errors and the conjugate gradient to minimize the curvature errors as well as the positional ones.
