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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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