biogeom (version 1.4.3)

NRGE: 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)

Value

The \(y\) values predicted by the Narushin-Romanov-Griffin equation.

Arguments

P

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

x

the given \(x\) values.

Author

Peijian Shi , Johan Gielis , Brady K. Quinn .

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).

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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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()