# Research

### Interactive Control of 3D Class A Bezier Curves

R. Fukuda, N. Yoshida and T. Saito, Interactive Control of 3D Class A Bezier Curves, SIGGRAPH ASIA Sketch, to appear.

### Interactive Control of 2D Class A Bezier Curves

N. Yoshida , T. Hiraiwa, and T. Saito, Interactive Control of Planar Class A Bezier Curves using Logarithmic Curvature Graphs, Computer-Aided Design & Applications, Vol.5, Nos.1-4, pp.121-130, 2008. [PDF]

Abstract: We present a method for interactively drawing a planar class A Bzier curve segment. First, wepresent a method for interactively drawing a typical class A Bzier curve segment by specifying threepoints like a quadratic Bzier curve segment. We show that as the degree of a typical class A Bziercurve segment is elevated, the curve converges to a logarithmic spiral segment. At the limit ofinfinite degree, the curve segment becomes a logarithmic spiral segment. We also present a methodfor drawing a general class A Bzier curve segment by perturbing the elements of the typical class Amatrix so that the endpoint constraints are satisfied. To see the characteristics of the generatedcurves, we propose to use logarithmic curvature graphs.

### Quasi-Log-Aesthetic Curves

N. Yoshida and T. Saito, Quasi-Aesthetic Curves in Rational Cubic Bezier Forms, Computer-Aided Design & Applications, Vol. 4, Nos. 1-4, pp.477-486, 2007. [PDF]

Abstract: Designing aesthetically appealing models is vital for the marketing success of industrial products. In this paper, we propose quasi-Aesthetic Curves that can be used in CAD systems for aesthetic shape design. Quasi-Aesthetic Curves represented in rational cubic Bezier Forms are curves whose logarithmic curvature histograms (LCHs) become nearly straight lines. The monotonicity of curvature of quasi-Aesthetic Curves is checked by the proposed method. We generate quasi-Aesthetic Curves by approximating the Aesthetic Curves whose LCHs are strictly represented by straight lines. We show that one Aesthetic Curve segment whose change of tangential angle is less than 90 deg. can be replaced by one quasi-Aesthetic Curve segment guaranteeing the monotonicity of the curvature in most of practical situations.

*Now, aesthetic curves are called log-aesthetic curves

### Log-Aesthetic Curves

Norimasa Yoshida and Takafumi Saito, Interactive Aesthetic Curve Segments, The Visual Computer (Pacific Graphics), Vol. 22, No.9-11, pp.896-905, 2006.[Video(WindowsMedia)] [PDF] (Homepage)

Abstract: To meet highly aesthetic requirements in industrial design and styling, we propose a new category of aesthetic curve segments. To achieve these aesthetic requirements, we use curves whose logarithmic curvature histograms(LCH) are represented by straight lines. We call such curves aesthetic curves. We identify the overall shapes of aesthetic curves depending on the slope of LCH $\alpha$, by imposing specific constraints to the general formula of aesthetic curves. For interactive control, we propose a novel method for drawing an aesthetic curve segment by specifying two endpoints and their tangent vectors. We clarify several characteristics of aesthetic curve segments.

*Now, aesthetic curves are called log-aesthetic curves