R.U. Gobithaasan: The Approximation of Generalized Log-Aesthetic
Curves with Cubic Trigonometric Bézier Functions
Abstract:
Curves with monotonic curvature profiles are coined as fair curves and
are essential for aesthetic product design. However, the de facto
flexible curves available in CAD systems are
Bézier, B-Spline and NURBS
which have a complex curvature function; thus the practitioners need to go
through a fairing procedure during the design process. In 2011, Gobithaasan &
Miura introduced Generalized Log-Aesthetic curves (GLAC), which are
a family of aesthetic curves which possess a monotonic curvature
profile. Even though they can be directly used for design tasks without
a fairing process, they are in the form of transcendentals, so a numerical
method is necessary for GLAC rendering. Recently, Trigonometric
Bézier
(T-Bézier) has been introduced by Han (2009). The advantage of
T-Bézier
over traditional Bézier
curves are twofold; firstly, they can precisely
represent circular arcs, cylinders, cones, tori etc., making them suitable for
CAD implementation. Secondly, T-Bézier curves are located closer
to the control
polygon as compared to traditional Bézier curves which is of use during
the design process. In this paper, we propose an algorithm employing
cubic T- Bézier curves with two shape parameters to approximate GLAC
with G2 continuity by means of curvature error measure. The
approximation formula inherits the shape parameters of GLAC which can be
used for freeform design, whereas T- Bézier shape parameters
are
utilized to satisfy terminal G2 constraints. The final approximated GLAC
is represented in the form of trigonometric basis functions, thus
enabling us to preserve essential freeform curve design entities, e.g.,
the convex hull property, partition of unity, symmetric, non-negativity and
monotonicity properties. Numerical results indicate that the proposed
algorithm capable of approximating GLAC within given tolerance in two
iterations, indicating straightforward implementation for CAD systems.