ETE {biogeom}

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

Description

ETE is used to calculate the y-value for an arbitrary point on the Troscianko curve that was generated by the explicit Troscianko equation.

Usage

ETE(P, x)

Arguments

P

the parameters of the explicit Troscianko equation, including a, \alpha_{0}, \alpha_{1}, and \alpha_{2}.

x

the x-value used in the explicit Troscianko equation.

Details

The explicit Troscianko equation is recommended as:

y = a\,\exp\left\{\alpha_{0}+\alpha_{1}\,\left(\frac{x}{a}\right)+\alpha_{2}\, \left(\frac{x}{a}\right)^2\right\}\sqrt{1-\left(\frac{x}{a}\right)^2},

where x and y represent the abscissa and ordinate of an arbitrary point on the explicit Troscianko curve; a, \alpha_{0}, \alpha_{1}, and \alpha_{2} are parameters to be estimated.

Value

The y values predicted by the explicit Troscianko 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 ETE is aligned to the x-axis. The argument, P, in the ETE function has the same parameters, \alpha_{0}, \alpha_{1}, and \alpha_{2}, as those in the TE function. However, the former has an additional parameter a than the latter, which represents half the egg's length. The lmTE function is based on the TE function, while the fitETE function is based on the ETE function here. 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

Biggins, J.D., Montgomeries, R.M., Thompson, J.E., Birkhead, T.R. (2022) Preston’s universal formula for avian egg shape. Ornithology 139, ukac028. doi:10.1093/ornithology/ukac028

Biggins, J.D., Thompson, J.E., Birkhead, T.R. (2018) Accurately quantifying the shape of birds' eggs. Ecology and Evolution 8, 9728-9738. doi:10.1002/ece3.4412

Shi, P., Gielis, J., Quinn, B.K., Niklas, K.J., Ratkowsky, D.A., Schrader, J., Ruan, H., Wang, L., Niinemets, Ü. (2022) 'biogeom': An R package for simulating and fitting natural shapes. Annals of the New York Academy of Sciences 1516, 123-134. doi:10.1111/nyas.14862

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

Troscianko, J. (2014). A simple tool for calculating egg shape, volume and surface area from digital images. Ibis, 156, 874-878. doi:10.1111/ibi.12177

See Also

curveETE, fitETE, SurfaceAreaETE, VolumeETE

Examples

  Par5 <- c(2.25, -0.38, -0.29, -0.16)
  xx2  <- seq(-2.25, 2.25, len=2000)
  yy3  <- ETE(P=Par5, x=xx2)
  yy4  <- -ETE(P=Par5, x=xx2)

  dev.new()
  plot(xx2, yy3, asp=1, type="l", col=4, cex.lab=1.5, cex.axis=1.5,
       xlim=c(-3, 3), ylim=c(-3, 3), 
       xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(xx2, yy4, col=2)  

  graphics.off()