Nevertheless, when the equations are polynomials, one can always eliminate as many variables as one would like. So our first goal is to convert the parametrization we found for the nephroid from a trigonometric parametrization to a system of polynomial equations.

Here is the idea: if we can get the trigonometric parametrization
involving *t* into one that only uses the trigonometric functions
cos(*t*) and sin(*t*), then we can substitute

c = cos(and add in the algebraic relation ct)

s = sin(t)

- How can you write cos(2
*t*) and sin(2*t*) in terms of cos(*t*), sin(*t*)? Can you similarly write cos(3*t*) and sin(3*t*) in these terms? - Rewrite your parametrization equations x=x(
*t*), y=y(*t*) in the form x=x(c,s), y=y(c,s) with c^{2}+ s^{2}= 1.

Now we are ready to eliminate variables from the parametrization equations. To do this, one can plug into the theory of Gröbner bases. This requires us to write the equations in the form

x-x(c,s)=0and then perform a lexicographic Gröbner basis computation on these polynomial equivalent system of equations. For example, one possible trigonometric parametrization for the nephroid is

y-y(c,s)=0

c^{2}+s^{2}-1=0

x=3/2cos(which we converted to the polynomial systemt)+1/2cos(3t-Pi)

y=3/2sin(t)+1/2sin(3t-Pi)

x - 3 c + 2 cby answering Question 3.^{3}= 0

y - 2 s + 2 s c^{2}= 0

c^{2}+ s^{2}- 1 = 0

One can then perform the following Gröbner basis computation in Maple, ordering the variables [c,s,x,y] so that the the output will be an equivalent polynomial system that contains some equations that do not involve c, s (if such equations exist):

> GB:=gbasis({x- 3*c + 2*c^3, y- 2* s + 2 *s* c^2, c^2+s^2-1},[c,s,x,y],plex); 2 2 2 GB := [3 c + x - 4 + y , - s x + s c + c y, 4 2 2 2 2 4 9 c x + 2 x - 7 x - 7 y - 4 + 4 y x + 2 y , 2 5 4 3 2 3 2 18 c y + 9 c - 4 x - 19 x - 4 y x + 14 x - 8 y x + 14 y x, 2 2 2 3 s - x + 1 - y , 2 4 5 3 2 2 3 18 s x - 18 s + 4 x y - 23 y + 4 y + 8 y x - 8 y x - 8 y , 4 2 2 2 4 2 9 s y - 2 - 2 y + 4 x - 4 y x - 2 x + 4 y , 4 2 2 4 6 2 2 2 4 2 4 12 x y - 4 - 15 y - 12 y + 4 y + 12 x - 24 y x + 12 y x - 12 x 6 + 4 x ]

The output Gröbner basis looks like a mess, but we get to ignore most
of it. We are only looking for the polynomials in this Gröbner basis
that *do not involve the variables c or s* which we wanted to
eliminate (see
Gröbner bases for the justification of this).
There is only one such polynomial, namely the 8th (last) one,
so this polynomial gives the desired implicit polynomial
equation in x,y for the nephroid:

>GB[8]; 4 2 2 4 6 2 2 2 4 2 4 12 x y - 4 - 15 y - 12 y + 4 y + 12 x - 24 y x + 12 y x - 12 x 6 + 4 x

We can check this equation is correct by plotting the set of
*(x,y)* values that will make this equation equal zero.
For example, this could be accomplished using
Maple's `plots[implicitplot]`

function or by using specialized software such as
Pisces. If we do this, we will
discover that the zero-set of the above equation is the same
set of points as the parametrized curve that we met in the
Introduction.

Vic Reiner <reiner@math.umn.edu> Frederick J. Wicklin <fjw@geom.umn.edu> Last modified: Sun Apr 14 17:30:01 1996