Nekar Xenos wrote:

> Ok, so what's the isosurface formula for each of these?  =)

Parametric:

   This short note describes the parametric equations which give
   rise to an approximate model of 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(circle with verticle line through it)
  y = 0.5 *(1-cos(8)) sin(8) sin(circle with verticle line through it)
  z = cos(8)
  
    where 0 <= circ w/line tru/it  <= 2pi
    and 0   <= 8 <= pi


   When theta is 0 there is a discontinuity at the apex where 
   x = 0  y = 0  z = 1

  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.

-- 
Ken Tyler

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