Isobars© Mike Williams 2003,2004 

Isobar AnalogyIn order to get a feel for what isosurfaces are, it might be useful to think about something that you might be more familiar with, the isobars that are drawn on weather maps. Weather stations gather information about the air pressure at different points on the ground, and then the meteorologists draw lines through the points where the numbers are the same.
As well as isobars, which connect points of equal pressure, there are
isotherms for temperature, isoclines for inclination, isogonics for
magnetic declination etc. They are all curves that join points where
some numeric value is the same.
The first image shows a ground level with numerical values
at various points. In second image there are lines joining points where the numerical values are 20, 30, 40, 50 and 60.
In the third image, we remove the limitation that the numerical values
are only on the ground. Suppose there are numerical values (not shown)
at every point in space. We can now join points that have the same numerical value
with 3d surfaces. In this case the surface shown connects all the points
where the numerical value is 30. Rather than using external numbers that represent something real, like air pressure, the numerical values that are used to generate POVRay isosurfaces come from mathematical functions. A function is simply a formula that generates a numerical value for points in 3d space. For example if we take function { x*x + y*y + z*z  4 }, then POVRay can calculate the value at a point like <1,2,0> by plugging those x,y,z values into the formula 1*1 + 2*2 + 0*0  4 giving a value of 1.
By carefully choosing the mathematical formula, we can make the associated isosurface
take up a vast range of shapes that would be difficult to achieve otherwise.

