SurfaceAreaSGE {biogeom}

Calculation of the Surface Area of An Egg Based on the Simplified Gielis Equation

Description

SurfaceAreaSGE is used to calculate the surface area of an egg that follows the simplified Gielis equation.

Usage

SurfaceAreaSGE(P, subdivisions = 100L,
          rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol,
          stop.on.error = TRUE, keep.xy = FALSE, aux = NULL)

Arguments

P

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

subdivisions

please see the arguments for the integrate function in package stats.

rel.tol

please see the arguments for the integrate function in package stats.

abs.tol

please see the arguments for the integrate function in package stats.

stop.on.error

please see the arguments for the integrate function in package stats.

keep.xy

please see the arguments for the integrate function in package stats.

aux

please see the arguments for the integrate function in package stats.

Details

The formula of the surface area (S) of an egg based on the simplified Gielis equation is:

S(\varphi)=2\,\pi\int_{0}^{\pi}\sin{\left(\varphi\right)}\ r\,\sqrt{r^2+\left(\frac{dr}{d\varphi}\right)^2}\,d\varphi,

where the polar raidus (r) is the function of the polar angle (\varphi):

r\left(\varphi\right) = a\left(\left|\mathrm{cos}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}+ \left|\mathrm{sin}\left(\frac{m}{4}\varphi\right)\right|^{n_{2}}\right)^{-\frac{1}{n_{1}}},

namely the simplified Gielis equation (i.e., GE) with arguments simpver = 1 and m = 1.

Note

The argument P in the SurfaceAreaSGE function only has the three parameters: a, n_{1}, and n_{2}.

Author(s)

Peijian Shi [e-mail], Johan Gielis [e-mail], Brady K. Quinn [e-mail].

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. doi:10.1111/nyas.15000

See Also

DSGE, fitGE, GE, VolumeSGE

Examples

  Par7 <- c(1.124, 14.86, 49.43)
  SurfaceAreaSGE(P = Par7)