I considerably improved the computation of the Gröbner bases in the qspray package, and I implemented something new: Gröbner implicitization. The Gröbner implicitization is able to transform a system of parametric equations to an implicit equation. Let’s see the example of the ellipse:

library(qspray)
# variables 
cost <- qlone(1)
sint <- qlone(2)
# parameters
a <- qlone(3)
b <- qlone(4)
#
nvariables <- 2
parameters <- c("a", "b")
equations <- list(
  "x" = a * cost,
  "y" = b * sint
)
relations <- list(
  cost^2 + sint^2 - 1 # = 0
)
# 
eqs <- 
  implicitization(nvariables, parameters, equations, relations)
## a^2*b^2 - b^2*x^2 - a^2*y^2

You see, a^2*b^2 - b^2*x^2 - a^2*y^2 = 0 is the implicit equation of the ellipse.

Gröbner implicitization is based on Gröbner bases. Unfortunately, while I considerably improved it, my implementation of the Gröbner bases can be slow, very slow. For the ellipse above, it is fast. But I tried for example to implicitize the parametric equations of the Enneper surface, and the computation was not terminated after 24 hours.

No worries. I have a new package coming to the rescue: giacR. This is an interface to the Giac computer algebra system, which powers the graphical interface Xcas. It is extremely efficient, and it is able to compute Gröbner bases.

Gröbner implicitization is not implemented in Giac. So I implemented it myself. Here is the implicit equation of the Enneper surface:

library(giacR)
giac <- Giac$new()

equations <-
  "x = 3*u + 3*u*v^2 - u^3, y = 3*v + 3*u^2*v - v^3, z = 3*u^2 - 3*v^2"
variables <- "u, v"

giac$implicitization(equations = equations, variables = variables)
## [1] "-19683*x^6+59049*x^4*y^2-10935*x^4*z^3-118098*x^4*z^2+59049*x^4*z-59049*x^2*y^4-56862*x^2*y^2*z^3-118098*x^2*y^2*z-1296*x^2*z^6-34992*x^2*z^5-174960*x^2*z^4+314928*x^2*z^3+19683*y^6-10935*y^4*z^3+118098*y^4*z^2+59049*y^4*z+1296*y^2*z^6-34992*y^2*z^5+174960*y^2*z^4+314928*y^2*z^3+64*z^9-10368*z^7+419904*z^5"

Finally we close the Giac session:

giac$close()
## [1] TRUE