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In the previous post, I showed how to expand a polynomial with symbolic parameters with the help of the spray package. As I said, it has one problem: it doesn’t preserve the rational numbers in the polynomial expression.

I’m going to provide a solution here which overcomes this problem, with the help of the Ryacas package. In fact I provide a solution with the packages Ryacas and partitions, and then I improve it with the help of the spray package.

Here is the polynomial expression I use for the illustration:

expr <- "((x*x+y*y+1)*(a*x*x+b*y*y)+z*z*(b*x*x+a*y*y)-2*(a-b)*x*y*z-a*b*(x*x+y*y))^2-4*(x*x+y*y)*(a*x*x+b*y*y-x*y*z*(a-b))^2"

In fact, the equation expr(x,y,z) = 0 defines a solid Möbius strip. That is why I was interested in this expression, because I wanted to draw the solid Möbius strip with POV-Ray. It is nice, in spite of a sad artifact (please leave a comment if you know how to get rid of it):

Let’s assign this expression to Yacas and let’s have a look at the degrees of the three variables x, y and z:

library(Ryacas)
yac_assign(expr, "POLY")
yac_str("Degree(POLY, x)")
## [1] "8"
yac_str("Degree(POLY, y)")
## [1] "8"
yac_str("Degree(POLY, z)")
## [1] "4"

The degrees are 8, 8 and 4 respectively. So we can get all possible combinations $$(i,j,k)$$ of $$x^iy^jz^k$$ with the blockparts function of the partitions package:

library(partitions)
combins <- blockparts(c(8L, 8L, 4L))
dim(combins)
## [1]   3 405

There are $$405$$ such combinations. Of course they don’t all appear in the polynomial, and that is the point we will improve later. For the moment we will iterate over all these combinations. Here is the function which takes one combination and returns the corresponding coefficient of $$x^iy^jz^k$$ in terms of $$a$$ and $$b$$, written in POV-Ray syntax:

f <- function(combin) {
xyz <- sprintf("xyz(%s): ", toString(combin))
coef <- yac_str(sprintf(
"ExpandBrackets(Coef(Coef(Coef(POLY, x, %d), y, %d), z, %d))",
combin[1L], combin[2L], combin[3L]
))
if(coef == "0") return(NULL)
coef <- gsub("([ab])\\^(\\d+)", "pow(\\1,\\2)", x = coef)
paste0(xyz, coef, ",")
}
# for example:
f(c(2L, 6L, 0L))
## [1] "xyz(2, 6, 0): 2*pow(b,2)+2*b*a,"

Then we get the POV-Ray code as follows:

povray <- apply(combins, 2L, f)
cat(povray, sep = "\n", file = "SolidMobiusStrip.pov")

The file SolidMobiusStrip.pov:

xyz(4, 0, 0): pow(a,2)*pow(b,2)-2*pow(a,2)*b+pow(a,2),
xyz(6, 0, 0): (-2)*pow(a,2)*b-2*pow(a,2),
xyz(8, 0, 0): pow(a,2),
xyz(2, 2, 0): 2*pow(b,2)*pow(a,2)-2*pow(b,2)*a+(-2)*b*pow(a,2)+2*b*a,
xyz(4, 2, 0): (-2)*pow(b,2)*a+(-4)*b*pow(a,2)-4*b*a-2*pow(a,2),
xyz(6, 2, 0): 2*pow(a,2)+2*a*b,
xyz(0, 4, 0): pow(b,2)*pow(a,2)-2*pow(b,2)*a+pow(b,2),
xyz(2, 4, 0): (-4)*pow(b,2)*a-2*pow(b,2)+(-2)*b*pow(a,2)-4*b*a,
xyz(4, 4, 0): pow(b,2)+4*b*a+pow(a,2),
xyz(0, 6, 0): (-2)*pow(b,2)*a-2*pow(b,2),
xyz(2, 6, 0): 2*pow(b,2)+2*b*a,
xyz(0, 8, 0): pow(b,2),
xyz(3, 1, 1): 4*pow(a,2)*b-4*pow(a,2)+(-4)*a*pow(b,2)+4*a*b,
xyz(5, 1, 1): 4*pow(a,2)-4*a*b,
xyz(1, 3, 1): 4*pow(a,2)*b+(-4)*a*pow(b,2)-4*a*b+4*pow(b,2),
xyz(3, 3, 1): 4*pow(a,2)-4*pow(b,2),
xyz(1, 5, 1): 4*a*b-4*pow(b,2),
xyz(4, 0, 2): (-2)*a*pow(b,2)+2*a*b,
xyz(6, 0, 2): 2*b*a,
xyz(2, 2, 2): (-2)*pow(b,2)*a+6*pow(b,2)+(-2)*b*pow(a,2)-8*b*a+6*pow(a,2),
xyz(4, 2, 2): (-2)*pow(a,2)+10*a*b-2*pow(b,2),
xyz(0, 4, 2): (-2)*b*pow(a,2)+2*b*a,
xyz(2, 4, 2): (-2)*pow(a,2)+10*a*b-2*pow(b,2),
xyz(0, 6, 2): 2*a*b,
xyz(3, 1, 3): 4*pow(b,2)-4*b*a,
xyz(1, 3, 3): (-4)*pow(a,2)+4*a*b,
xyz(4, 0, 4): pow(b,2),
xyz(2, 2, 4): 2*a*b,
xyz(0, 4, 4): pow(a,2)

Now we will restrict the $$405$$ combinations. There are only $$29$$ combinations of exponents in the polynomial expansion. How to get them? With spray. We don’t care if there are rational numbers in the polynomial because we will take the exponents only.

library(spray)
x <- lone(1L, 5L)
y <- lone(2L, 5L)
z <- lone(3L, 5L)
a <- lone(4L, 5L)
b <- lone(5L, 5L)
P <- ((x*x+y*y+1)*(a*x*x+b*y*y) + z*z*(b*x*x+a*y*y) -
2*(a-b)*x*y*z - a*b*(x*x+y*y))^2 -
4*(x*x+y*y)*(a*x*x+b*y*y-x*y*z*(a-b))^2
exponents <- index(P)
exponents_xyz <- unique(exponents[, c(1L, 2L, 3L)])
dim(exponents_xyz)
## [1] 29  3

Indeed, there are $$29$$ combinations. Now we can proceed as before and get the POV-Ray within a couple of seconds:

povray <- apply(exponents_xyz, 1L, f)
cat(povray, sep = "\n", file = "SolidMobiusStrip.pov")
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