Least Squares Fitting of Chacón-Gielis Curves by the Particle Swarm Method of Optimization.pdf

Least Squares Fitting of Chacón-Gielis Curves by the Particle Swarm Method of Optimization SK Mishra Dept. of Economics NEHU, Shillong (India) Introduction: Johan Gielis (2003) showed that his superformula 1 1 2 3 ( ) 1 1 4 4( ) ( ). cos( ) sin( ) ( ). ( ) ; 0 nn nm m a br f f g mθ θ θ θ θ θ ??
= + =    … (1) describes almost any closed curve in terms of the deformed circle, ( ),g θ and another function, ( ),f θ and their parameters. The function ( )f θ may be considered as a modifier of the Gielis function, ( )g θ . Ricardo Chacón (2004) pointed out that Gielis’ superformula [ ( )g θ in eqn-1 above] is inherently linear. Hence it can generate only idealized (Platonic) rather than real-world forms. However, natural shapes and patterns emerge as a result of nonlinear dynamic processes and should therefore be expressed in terms of related nonlinear functions. In view of this, Gielis’ superformula can be reformulated and generalized in which the trigonometric functions in ( )g θ would be replaced by the Jacobian elliptic functions. The use of elliptic parameters on the angle coordinate is the key feature providing diverse variations of a given initial shape. The rate and nature of such variations on an initial theme (pattern) can be controlled by changing the parameters. Thus, one can obtain sequences that mimic transformations of biological shapes, including growth processes. One of the generalizations of Gielis’ superformula, ( )g θ , suggested by Chacón is: 1 1 2 3 ( ) 1 1( ) ( ). cos( ( )) sin( ( )) ( ). ( ) nn n a br f fθ θ φ θ ψ θ θ γ θ ??
= + =   … (2) where, ( ) ( ) ( ) ( ); ( ) ( ); 2 2 K K am m and am m μ μφ θ θ ? μ ψ θ θ ? μ π π

′= + = +       …(3) In the expressions above, [ ; ]am u μ is the Jacobian elliptic function (JEF) of the parameter μ , ( )K μ is the complete elliptic integral of the first kind, and ( , )? ? ′ are additional parameters (Whittaker and Watson, 1996;