Abstract
The quality of a curve for industrial design and computer graphics can be interrogated using Logarithmic Curvature Graph (LCG) and Logarithmic Torsion Graph (LTG). A curve is said to be aesthetic if it depicts linear LCG and LTG function. The Log-aesthetic curve (LAC) was developed bearing this notion and it was later extended to a Generalized Log-aesthetic curve (GLAC) using the -shift and -shift approach. This paper reformulates GLAC by representing the Logarithmic Curvature and Torsion graph’s gradient function as a nonlinear ordinary differential equation (ODE) with boundary conditions. The outputs of solving the ODEs result in a well defined Cesaro equation in the form of curvature function that is able to produce both planar as well as spatial curves with promising entities for industrial product design, computer graphics and more.
Highlights
1.1 The Formulation of Logarithmic Curvature Graph That Leads to Log-Aesthetic CurvesDesigning visually pleasing industrial products is crucial since this feature dicates the success of a product (Pugh, 1991). Harada et al (1999) proposed a novel method to investigate curves used in automobile design called the Logarithmic Distribution Diagram of Curvature (LDDC)
In 2003, Kanaya et al simplified the formulation of LDDC to a simpler form, denoting it as Logarithmic Curvature Graph (LCG)
The Generalized Log-aesthetic curve (GLAC) comprises of high quality planar spirals such as Generalized Cornu Spiral, Log-Aesthetic Curve, clothoid, Nielsen’s spiral, logarithmic spiral, circle involute and etc
Summary
1.1 The Formulation of Logarithmic Curvature Graph That Leads to Log-Aesthetic Curves. Harada et al (1999) proposed a novel method to investigate curves used in automobile design called the Logarithmic Distribution Diagram of Curvature (LDDC). They proposed aesthetic curves as curves with a constant LDDC gradient where the gradient is denoted as. In 2003, Kanaya et al simplified the formulation of LDDC to a simpler form, denoting it as Logarithmic Curvature Graph (LCG). Yoshida and Saito (2006) further proposed a method to draw the Log-Aesthetic Curve segment interactively by using two endpoints and their respective tangent vectors, known as the G1 data. In 2012, Yoshida and Saito further derived a method to render the drawable boundary for Log-Aesthetic Curve segments to indicate whether a segment can be drawn from the given G1 data or otherwise
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