Vertex Kaleidoscope shader

Question

I'm attempting to translate a fragment shader into a vertex shader (for mobile optimisation)

As you can see in the image below, the vertices of the center and right edge are buggy. (This is a plane with 11 x 11 vertices)

The UVs are currently mapped from the right, and wrapped around the center (with radial rotation). I'm guessing multiple vertices in the middle half the same value which is creating a hole? And then the right side is cross-fading the first UV values to the final values which creates a stretched effect.

The question is how to override or fix those. (They'll probably be 2 separate problems?)

uniform vec2 _Offset;
uniform float _Multiply;
varying vec4 position;
varying vec4 vert;
varying vec2 tex;
varying vec2 uv;
varying float ar;
#ifdef VERTEX  
void main()
{          
    gl_Position = gl_ModelViewProjectionMatrix * gl_Vertex;
    vec2 p = -1.0 + 2.0 * tex.xy;
    float a = atan(p.y,p.x) ;
    float r = sqrt(dot(p,p));

    uv.x = _Multiply*a/3.1416 * 2.0 + 0.1;

    float sx = sin(_Multiply*r+_Offset.x);
    float sy = _Multiply*.1*cos(_Offset.y+_Multiply*a);
    uv.y = -_Offset.x + sx + sy;

}
#endif  

#ifdef FRAGMENT

uniform sampler2D _MainTex;
void main(void)
{

    gl_FragColor = texture2D(_MainTex,uv*.5);
}
#endif                          
ENDGLSL    

Answer 1

Okay, essentially rewriting the shader I was able to get expected results. And for advice to newcomers: the higher the number of vertices in your plane, the less pixel distortion will occur.

GLSLPROGRAM                          
#ifdef VERTEX  
varying vec2 position;
uniform float _Multiply;
uniform vec2 _Offset;
void main()
{          
    gl_Position = gl_ModelViewProjectionMatrix * gl_Vertex;
    vec2 tex = vec2(gl_MultiTexCoord0);

    vec2 p = tex.xy - 0.5;
    float r = length(p);
    float a = atan(p.y, p.x);
    float sides = _Multiply;
    float tau = 2.0 * 3.1416;
    a = mod(a, tau/sides);
    a = abs(a - tau/sides/2.0);
    position = r * vec2(cos(a), sin(a)) + 0.5  + _Offset;
}
#endif  
#ifdef FRAGMENT
varying vec2 position;
uniform sampler2D _MainTex;
void main(void)
{
    gl_FragColor = texture2D(_MainTex, position);
}
#endif                          
ENDGLSL