biogeom (version 1.4.3)

DSGE: Calculation of the First-Order Derivative of the Simplified Gielis Equation

Description

DSGE is used to calculate the first-order derivative of the simplified Gielis equation at a given \(\varphi\)-value.

Usage

DSGE(P, phi)

Arguments

P

the parameters of the simplified Gielis equation, including \(a\), \(n_{1}\), and \(n_{2}\).

phi

the \(\varphi\)-value used in the simplified Gielis equation.

Author

Peijian Shi , Johan Gielis , Brady K. Quinn .

Details

The first-order derivative of the simplified Gielis equation with arguments simpver = 1 and m = 1 at a given \(\varphi\)-value is: $$g(x) = \frac{a}{4}\,\frac{n_{2}}{n_{1}}\,\left[\left(\cos{\frac{\varphi}{4}}\right)^{n_{2}-1}\left(\sin{\frac{\varphi}{4}}\right)-\left(\sin{\frac{\varphi}{4}}\right)^{n_{2}-1}\left(\cos{\frac{\varphi}{4}}\right)\right]\ \left[\left(\cos{\frac{\varphi}{4}}\right)^{n_{2}}+\left(\sin{\frac{\varphi}{4}}\right)^{n_{2}}\right]^{-\frac{1}{n_{1}}-1}, $$

where P has three parameters: \(a\), \(n_{1}\), and \(n_{2}\).

References

Chen, Z. (2012) Volume and area of revolution under polar coordinate system. Studies in College Mathematics 15(6), 9\(-\)11.

Shi, P., Chen, L., Quinn, B.K., Yu, K., Miao, Q., Guo, X., Lian, M., Gielis, J., Niklas, K.J. (2023) A simple way to calculate the volume and surface area of avian eggs. Annals of the New York Academy of Sciences 1524, 118\(-\)131. tools:::Rd_expr_doi("10.1111/nyas.15000")

See Also

GE, fitGE, SurfaceAreaSGE

Examples

  Par7 <- c(1.124, 14.86, 49.43)
  phi1 <- seq(0, pi, len=2000)
  g1   <- DSGE(P=Par7, phi=phi1)

  dev.new()
  plot(phi1, g1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlab=expression(italic(varphi)), 
       ylab=expression(paste(italic(g), "(", italic(varphi), ")", sep="")))

  graphics.off()