DNRGE {biogeom}

Calculation of the First-Order Derivative of the Narushin-Romanov-Griffin Equation

Description

DNRGE is used to calculate the first-order derivative of the Narushin-Romanov-Griffin equation at a given x-value.

Usage

DNRGE(P, x)

Arguments

P

the parameters of the Narushin-Romanov-Griffin equation.

x

the x-value used in the Narushin-Romanov-Griffin equation.

Details

Let us define:

f_{1}(x) = \frac{B}{2}\sqrt{\frac{A^2-4x^2}{A^2+8Cx+4C^2}},

f_{2}(x) = \sqrt{\frac{A\left(A^{2}+8Cx+4C^{2}\right)}{2(A-2C)x^{2}+\left(A^{2}+8AC-4C^{2}\right)x+2AC^{2}+A^{2}C+A^{3}}},

f_{3}(x) = A^2 - 4x,

f_{4}(x) = A^2+8Cx+4C^2,

E = \frac{\sqrt{5.5A^{2}+11AC+4C^{2}} \cdot \left(\sqrt{3}AB-2D\sqrt{A^{2}+2AC+4C^{2 }}\right)}{\sqrt{3}AB\left(\sqrt{5.5A^{2}+11AC+4C^{2}}-2\sqrt{A^{2}+2AC+4C^{2}}\right)},

F = 2\left(A-2C\right),

G = A^{2}+8AC-4C^{2},

H = 2AC^{2}+A^{2}C+A^{3},

and then the first-order derivative of the Narushin-Romanov-Griffin equation at a given x-value is:

J(x) = -\frac{4\,f_{1}(x)\left[C\,f_{3}(x)+x\,f_{4}(x)\right]}{f_{3}(x) \cdot f_{4}(x)}\left\{1-E \cdot \left[1-f_{2}(x)\right]\right\}-\frac{AE}{2}\frac{f_{1}(x)}{f_{2}(x)}\frac{f_{4}(x) \cdot \left(2Fx+G\right)}{\left(Fx^2+Gx+H\right)^2},

where P has four parameters: A, B, C, and D.

Note

The argument P in the DNRGE function has the same parameters, as those in the NRGE function.

Author(s)

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

References

Narushin, V.G., Romanov, M.N., Griffin, D.K. (2021) Egg and math: introducing a universal formula for egg shape. Annals of the New York Academy of Sciences 1505, 169-177. doi:10.1111/nyas.14680

Narushin, V.G., Romanov, M.N., Mishra, B., Griffin, D.K. (2022) Mathematical progression of avian egg shape with associated area and volume determinations. Annals of the New York Academy of Sciences 1513, 65-78. doi:10.1111/nyas.14771

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

fitNRGE, NRGE, SurfaceAreaNRGE

Examples

  Par6 <- c(4.51, 3.18, 0.1227, 2.2284)
  xx3  <- seq(-4.51/2, 4.51/2, len=2000)
  J1   <- DNRGE(P=Par6, x=xx3)
  J2   <- -DNRGE(P=Par6, x=xx3)
  ind  <- which(is.na(J1) | is.na(J2))
  xx3  <- xx3[-ind]
  J1   <- J1[-ind]
  J2   <- J2[-ind]

  dev.new()
  plot(xx3, J1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-4.51/2, 4.51/2), ylim=c(-20, 20), xlab=expression(italic(x)), 
       ylab=expression(paste(italic(J), "(", italic(x), ")", sep="")))
  lines(xx3, J2, col=2)  

  graphics.off()