Ken wrote:

>     This describes the parametric equations which approximatly model a
>    drop of water, for example, a tear drop.

...

>    The equations as functions of longitute phi and lattitude theta are:
>
>    x = 0.5 *(1-cos(8)) sin(8) cos(Note:1)
>    y = 0.5 *(1-cos(8)) sin(8) sin(Note:1)
>    z = cos(8)

An implicit equation for the aforementioned tear drop is:

1 - 4x^2 - 4y^2 - 2z + 2z^3 - z^4 = 0,

or, it simplifies a bit as 4(x^2+y^2)=(1+z)(1-z)^3, which is a surface
of revolution bounded by -1 <= z <= 1.  The POV command is:

quartic{ <
  0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0,
  -1, 2, 0, -2, 1> }


Sometimes, a parametric map involving trig functions can be converted
to an algebraic implicit equation by using new variables:

u=cos(t),
v=sin(t),

and introducing a new implicit relation:

u^2+v^2=1.


The idea is to make the parametric equations into polynomials, at the
cost of increasing the number of variables and equations.  Then u and
v can (sometimes) be eliminated from the system of polynomial
equations to get an implicit equation in x,y,z.

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