DNRGE {biogeom} | R Documentation |
DNRGE
is used to calculate the first-order derivative of the Narushin-Romanov-Griffin equation at a given x-value.
DNRGE(P, x)
P |
the parameters of the Narushin-Romanov-Griffin equation. |
x |
the x-value used in the Narushin-Romanov-Griffin equation. |
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
.
The argument P
in the DNRGE
function has the same parameters, as those in the
NRGE
function.
Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.
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
fitNRGE
, NRGE
, SurfaceAreaNRGE
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()