NRGE {biogeom}R Documentation

The Narushin-Romanov-Griffin Equation (NRGE)

Description

NRGE is used to calculate yy values at given xx values using the Narushin-Romanov-Griffin equation (NRGE).

Usage

NRGE(P, x)

Arguments

P

the four parameters (i.e., AA, BB, CC, and DD) of the Narushin-Romanov-Griffin equation.

x

the given xx values.

Details

The Narushin-Romanov-Griffin equation (Narushin et al., 2021) has four parameters in total, among which three parameters have clear geometric meanings.

f1(x)=B2A24x2A2+8Cx+4C2,f_{1}(x)=\frac{B}{2}\sqrt{\frac{A^2-4x^2}{A^2+8Cx+4C^2}},

E=5.5A2+11AC+4C2(3AB2DA2+2AC+4C2)3AB(5.5A2+11AC+4C22A2+2AC+4C2),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)},

f2(x)=A(A2+8Cx+4C2)2(A2C)x2+(A2+8AC4C2)x+2AC2+A2C+A3,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(x)=±f1(x){1E[1f2(x)]}.f(x)=\pm f_{1}(x) \cdot \left\{1-E \cdot \left[1-f_{2}(x)\right]\right\}.

Here, f(x)f(x) is the Narushin-Romanov-Griffin equation, which is used to predict the yy coordinates at the given xx coordinates; AA represents the egg's length; BB represents the egg's maximum breadth; CC is a parameter to be estimated, and it can be expressed as (AB)/(2q)\left(A-B\right)/(2q), where qq is a parameter to be estimated; DD represents the egg's breadth associated with (3/4)L(3/4)L from the egg base (to the egg tip) on the egg length axis (which can be regarded as the major axis of the egg shape).

Value

The yy values predicted by the Narushin-Romanov-Griffin equation.

Note

Here, parameter C is a parameter to be estimated, which can be directly calculated numerically based on the egg-shape data.

Author(s)

Peijian Shi pjshi@njfu.edu.cn, Johan Gielis johan.gielis@uantwerpen.be, Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca.

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

Shi, P., Gielis, J., Niklas, K.J. (2022) Comparison of a universal (but complex) model for avian egg shape with a simpler model. Annals of the New York Academy of Sciences 1514, 34-42. doi:10.1111/nyas.14799

Tian, F., Wang, Y., Sandhu, H.S., Gielis, J., Shi, P. (2020) Comparison of seed morphology of two ginkgo cultivars. Journal of Forestry Research 31, 751-758. doi:10.1007/s11676-018-0770-y

See Also

curveNRGE, fitNRGE, SurfaceAreaNRGE, VolumeNRGE

Examples


P0 <- c(11.5, 7.8, 1.1, 5.6)
x  <- seq(-11.5/2, 11.5/2, len=2000)
y1 <- NRGE(P=P0, x=x)
y2 <- -NRGE(P=P0, x=x)

dev.new()
plot(x, y1, cex.lab=1.5, cex.axis=1.5, type="l", 
     col=4, ylim=c(-4, 4), asp=1, 
     xlab=expression(italic(x)), ylab=expression(italic(y)) )
lines(x, y2, col=2)

graphics.off()

[Package biogeom version 1.4.3 Index]