TSE function

The Todd-Smart Equation (TSE)

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. UTF-8

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.

Returns

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 pjshi@njfu.edu.cn , Johan Gielis johan.gielis@uantwerpen.be , Brady K. Quinn Brady.Quinn@dfo-mpo.gc.ca .

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")

Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal 7, 308$-$313. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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()