- The logarithmic curve and aestheticー
Norimasa Yoshida (norimasa@acm.org) (Nihon University)
Takafumi Saito (txsaito@cc.tuat.ac.jp) (Tokyo University of Agriculture and Technology)
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Be able to appeal the "beauty" is very important for industrial products to be successful in the market. For example, the characteristic line of the body of the car, you will greatly affect their impression of the beauty of the car. Many of the traditional curves and surfaces as represented by NURBS, but has been represented by a polynomial or rational expression, and meets the requirements for a high level of beauty are as hard to say. The main reason lies in the difficulty of controlling the change increases or decreases in addition to the continuity of the curvature. Curvature, so that closely related to the strain of reflection lines strongly influence the beauty of the curved surface, control of the change in curvature is very important.
Would say that if, it is necessary to give the knot vector weights and unnatural, and a typical example of that it is difficult to control the change in curvature (representing the arc that is) to maintain a constant curvature in the NURBS . The figure below shows the different reasons. (A) is an example of a curve with a desirable change in curvature, (b) Bezier curve is approximated by the following three cases of that. Has become undesirable curve shape is simple, the curvature of the Bezier curve is repeatedly increased or decreased.
Wakayama University of declared interest Harada teacher, "Many of the curve in a variety of beautiful nature and artifacts, that can be approximated by a straight line logarithmic distribution diagram of curvature" was pointed out various. Aesthetic and logarithmic curve is logarithmic curvature graph curve becomes a straight line. Mr. Harada was examined in a beautiful curve, which contains the key line of the body of the car, such as butterfly wings or. As a special case of the logarithmic curve aesthetic, in the case of α = -1 the slope of the straight line in logarithmic distribution diagram of curvature in the case clothoid curve, and α = 1 will be a logarithmic spiral has been pointed out. In this study, referred to as planar curves aesthetic curves monotonic change in curvature of the logarithmic distribution diagram of curvature is represented by a straight line.
* Note: change in curvature for the curve is not monotone, can draw a logarithmic distribution diagram of curvature, and that figure does not call distribution, we have to be referred to as logarithmic curvature graph.
Of Shizuoka University Kenjiro Miura teacher, has been derived a general formula of aesthetic curves. This study is one of the most important and the basis for our research, or draw, such as whether to change how by α the overall picture of the aesthetic curve, nor been elucidated, and how the curve segment also not clear that.
We are, by considering the standard form was added to the constraint of some sort against the logarithmic curve aesthetic, to clarify the overall picture of the aesthetic curve log (a), when (2) α is specified, has devised a technique to draw the aesthetic curve segment by three control points as well as the Bezier curve of order 2. The main contribution of our research is located in the following points.
Formulated as a standard form imposes certain constraints, the elucidation of the overall picture of the logarithmic curve aesthetic aesthetic logarithmic curve (1). We derive the relationship between the radius of curvature of the curve, direction angle, arc length, to clarify whether you want to change how the overall picture of the curve by the slope of the straight line R of LCH. Also shown that the aesthetic of Λ = 2 curve is a circle involute curve.
Also (by the trajectory of the center of curvature curve) of the curve evolute evolute of the curve aesthetic aesthetic understanding of the logarithm log (2), indicate that it is aesthetic curve.
Interactive drawing of the curve when α aesthetic logarithm (3) is specified, as the next Bezier curve 2, by specifying three points, to control the aesthetic curve segment interactively in real-time log We propose a method.
To clarify how it changes for the placement of control points α and elucidation of the nature of the various logarithmic aesthetic curve segment (4), is it aesthetic logarithmic curve segment and its evolute. In addition, the placement of control points from restraint and strong for the change in curvature, that is specified by α if there is a curve segment is not drawn. experimentally to clarify the relationship between the propriety of drawing the curve segment and control point placement and α
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