biogeom (version 1.4.3)

ETE: 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)

Value

The \(y\) values predicted by the explicit Troscianko equation.

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.

Author

Peijian Shi , Johan Gielis , Brady K. Quinn .

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.

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