Adam Coffman wrote:

> 
> 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.

  Thank you for taking the time to comment on this and I appreciate
you expanding on the brief description to give it meaning. I wish I
could follow more of what you said but some of it did sink in. The Pov
representation of the tear drop shape you provided is a real treasure
among the mathematically challenged, like myself, and is likewise very
much appreciated.

-- 
Ken Tyler

mailto://tylereng@pacbell.net

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