Here you find more info about tweaking spherical projections of radial functions using my POV-Ray package »Sphericals.zip (simply unpack it to a subdirectory of your choice)

I want to show some possible tweaks for these parametric objects by discussing my model script SphericalHarmonics_Twisted.pov

Theory: A radial function gets projected onto a sphere:

R(φ,θ)= sinb(aθ)+cosd(cθ) +  sinf(eφ)+cosh(gφ)

x = R(φ,θ)cos(φ)×sin(θ) 
y = R(φ,θ)sin(φ)×sin(θ)            0 ≤ φ < 2π , 0 ≤ θ ≤π 
z = R(φ,θ)      ×cos(θ)

So the script runs as follows:

// POV-Ray source for modified spherical harmonics 
// Wolfgang.Urban@schule.at
// lots of examples: http://www.lifesmith.com/spharmin.html

   
global_settings {assumed_gamma 1.0}     

#include "math.inc"
#include "colors.inc"
#include "textures.inc"
#include "stones.inc"
#include "woods.inc"

// Values for        a,b,c,d,e,f,g,h  from formula  
#declare M=array[8] {7,1,3,2,4,5,6,4}

// ==========================================================================  
// set radial function and size for automatic scaling
// ==========================================================================  

#declare R=function(phi,theta){
           pow(sin(M[0]*theta),M[1]) + pow(cos(M[2]*theta),M[3])
          +pow(sin(M[4]*phi),  M[5]) + pow(cos(M[6]*phi)  ,M[7]) 
} 
                                 
#declare SH_SIZE  = 18;  // if SH_SIZE != 0: rescaled into [-SH_SIZE,+SH_SIZE]^3

// ==========================================================================  
// set tweaking values if necessary
// ==========================================================================  
 
#declare U_Steps = 300;   // 0..2pi
#declare V_Steps = 150;   // 0..pi    
    
#declare SH_Mode   = 1;   // 0=plain spheric, 1=twisted spheric, 2=cylindric, 
                          // 3=rectangle (no warps)

#declare SH_CZ     = 1;   // relative scaling in height, <0 flips vertical
#declare SH_R0     = 0;   // push out / pull back
#declare SH_Twist  = 0;   // twist along vertical
#declare SH_Spiral = 0;   // horizontal spiraling
#declare SH_RaiseR = 0;   // vertical shift depending on radius 
#declare SH_RaiseRA= 0;   // like SH_RaiseR but using abs(R) 
#declare SH_RaiseH = 0;   // vertical shift depending on theta
 
#include "spherical.inc" 

// ==========================================================================  
// texture
// ========================================================================== 
 
#include "sphericalTextures.inc" 

// UVTex_Phi          red lines along phi
// UVTex_Theta        blue lines along theta
// UVTex_Phi_Trans    transparency + red lines 
// UVTex_Theta_Trans  transparency + blue lines 
// UVTex_Grid         lines for phi and theta   
// UVTex_Grad         gradient along theta   
// UVTex_Grad_Agate   gradient along theta, big agate normal                         
// UVTex_Grad_AgateV  gradient along theta, big agate normal along theta
// UVTex_Grad_Marble  gradient along theta, stripes of marble normal         
// UVTex_Grad_Glow    gradient theta, nice glowing green (try without normal) 
// UVTex_RainbowV     rainbow along theta, ripple normal           
// UVTex_RainbowU     cyclic rainbow along phi 
// UVTex_Line         transparency with diagonal line   
// UVTex_SignOfR      R<0 : red, R>0 : green  

// UVTex_1            blue, red, and a white line  
// UVTex_Checker      chess board
     
// ==========================================================================  
// the scene
// ==========================================================================  
                                                                
camera {
        right    x*image_width/image_height 
        location <0,0,-80>
        look_at  <0,0,0> 
        angle    40      
} 
                     
sky_sphere { pigment { color rgb 0 } }
                     
light_source { < 500,100,-500> rgb <1.0,0.9,0.8> }
light_source { <-100,100,-500> rgb <0.3,0.3,0.5> }
 
                   
object { Surface  
    texture { 
       UVTex_RainbowV   
    }
    rotate  0*y
    rotate 40*x 
}

Render that script to see one of the sherical harmonics as described by Paul Bourke.

Let's step through the script. For easy demonstration I simplify the radial function to
R = cos²(4θ) × sin(2φ)

Its flat mapping looks like

and the spherical projection becomes

Now let's see what those parameters can accomplish:

#declare SH_CZ     = 1;   // relative scaling in height, <0 flips vertical

SH_CZ scales the vertical axis. Here come CZ=0.5 and CZ=2. Not that automatic scaling changes a low height to a greater width.

#declare SH_R0     = 0;   // push out / pull back

A constant R0 is added to R. So a positive value expands the figure and makes the positive values (green) dominant. A negative value pulls the points back below zero(red).
Here come R=0.4 and R=-0.2

#declare SH_Twist  = 0;   // twist along vertical

That twist rotates points around the vertical axis by adding a portion of theta to phi. Therefore the blobs get stretched horizontally as in this example with SH_Twist = 3;

#declare SH_Spiral = 0;   // horizontal spiraling

Spiraling rotates points according to their R-Value. I use the opposite turning direction for negative Rs, as it looks better by giving nice intersections. An example with SH_Spiral = 5

#declare SH_RaiseR = 0;   // vertical shift depending on radius

Why not modify the theta-value? This can be done according to R. First pic has 2, second -2. I zoomed in to reveal some details.

#declare SH_RaiseRA= 0;   // like SH_RaiseR but using abs(R)

Same method, but modifying proportional to abs(R). Value is 2.

#declare SH_RaiseH = 0;   // vertical shift depending on theta

Or simply by rescaling theta itself. See the effect of 2 and -2. The first stretches, the second shrinks.

#declare U_Steps = 300;   // 0..2pi
#declare V_Steps = 150;    // 0..pi

A higher resolution like this gives smooth objects. The second image has both numbers set to 30 and shows some nice edges.

Doing so can turn our nice little function into a wild bastard!

Now it's up to you - roll your own pics. And try to use these parameters in some morphing sequences to do some wild animations!