This paper proposes a novel curve called the α-curve, which is specified by a variable shape parameter α and can be applied in curve reconstruction involving curve fairing, fitting, and segmentation. In contrast to classical log-aesthetic curves in which α is constant, this study contributes a novel formulation of α that varies monotonically within a specified segment. By introducing the variable α, the logarithmic curvature graph (LCG) becomes piecewise linear and parabolic, or any other function if necessary. Consequently, in comparison to conventional log-aesthetic curves that have limited field usage based on their rigidity, α-curves have various LCGs and flexible representations, thereby opening up many practical applications. Experimental results and comparative studies on curves created by CAD experts in the automotive industry demonstrate the theoretical and practical validity of α-curves.

For designing aesthetic surfaces, such as the car bodies, it is very important to use aesthetic curves as characteristic lines. In such curves, the curvature should be monotonically varying, since it dominates the distortion of reflected images on curved surfaces. In this paper, we present an interactive control method of log-aesthetic space curves. We define log-aesthetic space curves to be curves whose logarithmic curvature and torsion graphs are both linear. The linearity of these graphs constrains that the curvature and torsion are monotonically varying. We clarify the characteristics of log-aesthetic space curves and identify their family. Moreover, we present a novel method for drawing a log-aesthetic space curve segment by specifying two endpoints, their tangents, the slopes, α and β, of straight lines of the logarithmic curvature and torsion graphs, and the torsion parameter Ω. Our implementation shows that log-aesthetic curve segments can be controlled fully interactively.

Abstract The technique of solid modeling is essential in CAD/CAM applications, and is currently well established. However, problems remain, such as the lack of uniformity in geometric computations and the lack of stability of Boolean operations of two solids. In this paper, we introduce a theoretical solid modeling system that operates on boundary representations of polyhedral objects and is based on a new paradigm. The characteristics of the system are the following: (I) in Boolean Operations and modeling transformations, all geometric computations are performed by the 4 × 4 determinant method or the 4 × 4 matrix method in homogeneous space, which allows the system to avoid division operations, (2) all geometric computations are performed by the exact integer arithmetic, which makes the geometric algorithms stable and simple, and (3) primitive solids are constructed consistently in the integer domain, and the consistency is assured throughout Boolean operations and transformations.