In article <38b7154e@news.povray.org>, "Eric Freeman" 
<eri### [at] datasynccom> wrote:

> How did you come up with this?  Luck?  Patience?  Or did you say, "hmm...
> the formula for a cactus should be blah blah blah" and just whip it up?
> Even tho I've had a couple semesters of calculus and physics (20 years 
> ago)
> I have no clue how to take an idea and make an iso-surface out of it.

No calculus or physics required. If there were, I wouldn't be able to 
accomplish anything. I am still in Algebra II.
Well, the cactus is approximately spherical, with an indentation in the 
middle and radial ridges.

I started from the equation for a sphere:
function {
   sqrt(
      sqr(x)
      + sqr(y)
      + sqr(z)
   ) - 1
}

To indent the top of the cactus and extend the bottom, I modified the y 
portion of the equation like this:
      + sqr(y-sqrt(sqr(x/2)+sqr(z/2))*1.5)
This subracts a certain amount from the y value which depends on the 
distance from the y axis. The /2 and *1.5 were just to "tune" it to the 
right proportions. Subtracting from the y value has the effect of 
raising that portion of the surface, since a higher initial value is 
required to reach the threshold value. This modification really raises 
the sides of the cactus, although a variant of it could be made to 
depress the middle.

Then to add the "ridges", I subtracted a value depending on the sine of 
a multiple of the angle around the y axis from the total density:
   - (sin(atan2(x, z)*18)*0.1)

(note that while this is slightly different from the other version, it 
is really just a different way of calling the same function.)
The angle around the y axis can be calculated by atan2(x,z), which 
returns the angle in radians. This is ok, since the sin() function takes 
radians. I then multiply by 18 to get 18 "cycles" for a full revolution.
Since the sin() function returns values between -1 and 1, I multiplied 
it's result by 0.1 to get shallower ridges. This makes the ridges extend 
from about 0.1 units "below" the surface of the original sphere-like 
shape to about 0.1 units "above" it. Because the function returns both 
positive and negative values equally, I could just have easily used 
addition to incorporate it into the equation.

I probably did a very poor job of explaining it, but that is how I came 
up with this equation. I tend to have more success visualizing the 
isosurface as a density function with a "skin" at a certain density 
level(the threshold value). I start with a basic shape and progressively 
"sculpt" the density pattern by tweaking the function, adding 
characteristics to the function, and adding other density functions into 
the mix.

-- 
Chris Huff
e-mail: chr### [at] yahoocom
Web page: http://chrishuff.dhs.org/

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