Transforming a Parametrization into an Implicit Algebraic Equaion

We would like to use the parametrization x=x(t), y=y(t) for the nephroid that you just computed to get an equation that only involves x and y. In other words, we wish to eliminate the variable t from these two equations. Unfortunately, elimination of variables is not always easy or even possible for arbitrary equations.

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(t)
s = sin(t)
and add in the algebraic relation c2+s2 = 1 (Why?).

Question 3

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)=0
y-y(c,s)=0
c2+s2-1=0
and then perform a lexicographic Gröbner basis computation on these polynomial equivalent system of equations. For example, one possible trigonometric parametrization for the nephroid is
x=3/2cos(t)+1/2cos(3t-Pi)
y=3/2sin(t)+1/2sin(3t-Pi)
which we converted to the polynomial system
x - 3 c + 2 c3 = 0
y - 2 s + 2 s c2 = 0
c2 + s2 - 1 = 0
by answering Question 3.

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.

Next: The Geometry of Reflecting Light Rays
Previous: Parametrizing the neprhoid
Up: Introduction
Vic Reiner <reiner@math.umn.edu>
Frederick J. Wicklin <fjw@geom.umn.edu>
Last modified: Sun Apr 14 17:30:01 1996