Table 1 Details of the mathematical... | Emerald Publishing

Table 1

Details of the mathematical functions used in bead modeling. The height and width of the bead are reported in terms of the coefficients α and β appearing in the analytical expressions. The area A_p of the bead is formulated in terms of w and h

Model	Expression	Height (h)	Width (w)	Predicted area (A_p)
Parabola	$y = α x^{2} + β$	β	$2 \sqrt{- β / α}$	$(2 / 3) w h ≅ 0.667 w h$
Cosine	$y = α \cos (β x)$	$α$	$π / β$	$(2 / π) w h ≅ 0.637 w h$
Half ellipse	$y^{2} / α^{2} + x^{2} / β^{2} = 1$	$α$	$2 β$	$(π / 4) w h ≅ 0.785 w h$
Circular arc	$y = \sqrt{α^{2} - x^{2}} + β$	$α - β$	$2 \sqrt{α^{2} - β^{2}}$	$f w h : f \in [0.667, 0.785]$

Source(s): Table by authors