EPE {biogeom}

Calculation of the Ordinate For an Arbitrary Point on the Preston Curve in the Plane

Description

EPE is used to calculate the y-value for an arbitrary point on the Preston curve that was generated by the explicit Preston equation or one of its simplified versions for a given x-value.

Usage

EPE(P, x, simpver = NULL)

Arguments

P

the parameters of the explicit Preston equation or one of its simplified versions.

x

the x-value used in the explicit Preston equation.

simpver

an optional argument to use the simplified version of the explicit Preston equation.

Details

When simpver = NULL, the explicit Preston equation is selected:

y = b\ \sqrt{1-\left(\frac{x}{a}\right)^2}\left(1+c_{1}\ \frac{x}{a}+c_{2}\left(\frac{x}{a}\right)^2+c_{3}\left(\frac{x}{a}\right)^3\right),

where P has five parameters: a, b, c_{1}, c_{2}, and c_{3}.

\quad When simpver = 1, the simplified version 1 is selected:

y = b\ \sqrt{1-\left(\frac{x}{a}\right)^2}\left(1+c_{1}\ \frac{x}{a}+c_{2}\left(\frac{x}{a}\right)^2\right),

where P has four parameters: a, b, c_{1}, and c_{2}.

\quad When simpver = 2, the simplified version 2 is selected:

y = b\ \sqrt{1-\left(\frac{x}{a}\right)^2}\left(1+c_{1}\ \frac{x}{a}\right),

where P has three parameters: a, b, and c_{1}.

\quad When simpver = 3, the simplified version 3 is selected:

y = b\ \sqrt{1-\left(\frac{x}{a}\right)^2}\left(1+c_{2}\left(\frac{x}{a}\right)^2\right),

where P has three parameters: a, b, and c_{2}.

Value

The y values predicted by the explicit Preston equation.

Note

We only considered the upper part of the egg-shape curve in the above expressions because the lower part is symmetrical to the upper part around the x-axis. The mid-line of an egg's profile in EPE is aligned to the x-axis, while the mid-line of an egg's profile in PE is aligned to the y-axis. The EPE function has the same parameters, P, as those in the PE function. The explicit Preston equation is used for calculating an egg's volume and surface area, when the parameters, which are saved in the P vector, are obtained using the fitEPE function or the lmPE function based on the TSE function. In addition, the values in x > a (i.e., the first element in P) are forced to be a, and the values in x < -a will be forced to be -a.

Author(s)

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

References

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

Shi, P., Wang, L., Quinn, B.K., Gielis, J. (2023) A new program to estimate the parameters of Preston's equation, a general formula for describing the egg shape of birds. Symmetry 15, 231. doi:10.3390/sym15010231

See Also

curveEPE, fitEPE, PE, SurfaceAreaEPE, VolumeEPE

Examples

  Par3 <- c(4.27, 2.90, 0.0868, 0.0224, -0.0287)
  xx1  <- seq(-4.27, 4.27, by=0.001)
  yy1  <- EPE(P=Par3, x=xx1, simpver=NULL)
  yy2  <- -EPE(P=Par3, x=xx1, simpver=NULL)

  dev.new()
  plot(xx1, yy1, asp=1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-5, 5), ylim=c(-5, 5), 
       xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(xx1, yy2, col=2) 

  graphics.off()