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Proceedings of CAD'15, 2015, 42-46
Reformulation of Generalized Log-aesthetic Curves with Bernoulli Equations

Kenjiro T. Miura, Sho Suzuki, Shin Usuki, Shizuoka University
R. U. Gobithaasan, Universiti Malaysia Terengganu

Abstract. A Generalized Log-Aesthetic Curve (GLAC) is the general formulation of emerging the Log-Aesthetic (LA) curves for aesthetic industrial design. GLAC has an extra degree of freedom compared to LA curve which makes it versatile for design. There are two approaches employed to develop GLACs namely ρ-shift and κ-shift [2]. To note, κ-shift GLAC is a better formulation of GLAC since its directional angle can be obtained analytically as compared to ρ-shift GLAC. Recently, Sato and Shimizu reported the relationship between the fundamental equation of Log-aesthetic curve and Riccati differential equations. They considered the case of ρ-shift GLAC and reported its representation in the form of Riccati equation. It is well known that solving Riccati equation involves reduction of order which is a painstaking trial and error approach to find for a solution. This paper completes the investigation by analyzing κ-shift GLAC. We derived the formula of the κ-shift GLAC as a solution of a Bernoulli equation which can be solved with various approaches. 

Keywords. Log-aesthetic Curve, Generalized Log-aesthetic Curve, Bernoulli Differential Equations, Riccati Differential Equations

DOI: 10.14733/cadconfP.2015.42-46