NRGE {biogeom}

The Narushin-Romanov-Griffin Equation (NRGE)

Description

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

Usage

NRGE(P, x)

Arguments

P

the four parameters (i.e., A, B, C, and D) of the Narushin-Romanov-Griffin equation.

x

the given x values.

Details

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

f_{1}(x)=\frac{B}{2}\sqrt{\frac{A^2-4x^2}{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}(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)=\pm f_{1}(x) \cdot \left\{1-E \cdot \left[1-f_{2}(x)\right]\right\}.

Here, f(x) is the Narushin-Romanov-Griffin equation, which is used to predict the y coordinates at the given x coordinates; A represents the egg's length; B represents the egg's maximum breadth; C is a parameter to be estimated, and it can be expressed as \left(A-B\right)/(2q), where q is a parameter to be estimated; D represents the egg's breadth associated with (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 y 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 [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

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()