TSE {biogeom}

The Todd-Smart Equation (TSE)

Description

TSE is used to calculate y values at given x values using the Todd and Smart's re-expression of Preston's universal egg shape.

Usage

TSE(P, x, simpver = NULL)

Arguments

P

the parameters of the original Todd-Smart equation or one of its simplified versions.

x

the given x values ranging from -1 to 1.

simpver

an optional argument to use the simplified version of the original Todd-Smart equation.

Details

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

y = d_{0}z_{0} + d_{1}z_{1} + d_{2}z_{2} + d_{3}z_{3},

where

z_{0}=\sqrt{1-x^2},

z_{1}=x\sqrt{1-x^2},

z_{2}=x^{2}\sqrt{1-x^2},

z_{3}=x^{3}\sqrt{1-x^2}.

Here, x and y represent the abscissa and ordinate of an arbitrary point on the Todd-Smart curve; d_{0}, d_{1}, d_{2}, and d_{3} are parameters to be estimated.

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

y = d_{0}z_{0} + d_{1}z_{1} + d_{2}z_{2},

where x and y represent the abscissa and ordinate of an arbitrary point on the Todd-Smart curve; d_{0}, d_{1}, and d_{2} are parameters to be estimated.

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

y = d_{0}z_{0} + d_{1}z_{1},

where x and y represent the abscissa and ordinate of an arbitrary point on the Todd-Smart curve; d_{0}, and d_{1} are parameters to be estimated.

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

y = d_{0}z_{0} + d_{2}z_{2},

where x and y represent the abscissa and ordinate of an arbitrary point on the Todd-Smart curve; d_{0}, and d_{2} are parameters to be estimated.

Value

The y values predicted by the Todd-Smart equation.

Note

Here, x and y in the Todd-Smart equation are actually equal to y/a and x/a, respectively, in the Preston equation (See PE for details). Since a represents half the egg length, this means that the egg length is fixed to be 2, and the maximum egg width is correspondingly adjusted to keep the same scale.

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

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308-313. doi:10.1093/comjnl/7.4.308

Preston, F.W. (1953) The shapes of birds' eggs. The Auk 70, 160-182.

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

Todd, P.H., Smart, I.H.M. (1984) The shape of birds' eggs. Journal of Theoretical Biology 106, 239-243. doi:10.1016/0022-5193(84)90021-3

See Also

lmPE, PE

Examples

  Par <- c(0.695320398, -0.210538656, -0.070373518, 0.116839895)
  xb1 <- seq(-1, 1, len=20000)
  yb1 <- TSE(P=Par, x=xb1)
  xb2 <- seq(1, -1, len=20000)
  yb2 <- -TSE(P=Par, x=xb2)

  dev.new()
  plot(xb1, yb1, asp=1, type="l", col=2, ylim=c(-1, 1), cex.lab=1.5, cex.axis=1.5, 
    xlab=expression(italic(x)), ylab=expression(italic(y)))
  lines(xb2, yb2, col=4)

  graphics.off()