Difficulties for GPU rendering of a cone using parametric equation

I'm trying to render a cone at the gpu for some effects. I'm using the parametric equation provided here. I ended up with the shader below intended to work with an unorderedaccessview target (DX11). I'm blocked in the rotating effect I want to have, the cone seems to rotate but it remains flat on the screen shown in the above picture.

In the shader the rotation is applied to the direction D and to the two vectors U and V defining the plane of the circle. Maybe these U and V vectors are not correct or it is my calculs for the rotation. The rotation is in the matrix world (rotation in the y axis only). What does totally not work is when I apply the full transformation to the calculated cone point in the "a" loop. The calcul I use is set as comment.

[numthreads(32,32,1)]
void CS_PostDeferred( uint3 nGid : SV_GroupID, uint3 nDTid : SV_DispatchThreadID, uint3 nGTid : SV_GroupThreadID )
{
    float2 Tex = float2(nDTid.x/SM_SCREENX, nDTid.y/SM_SCREENY);
    //this is a brut force shader for test. The cone is displayed only at point screen(0.25,0.25) 
    if ((Tex.x>0.249)&&(Tex.x<0.25)) 
    if ((Tex.y>0.2475)&&(Tex.y<0.25)) 
    {
           matrix M = World*ViewProjection;//can't work with yet. see below
           float3 O = float3(0,0,0);//top of cone
           float R = 200;//radius of cone at the base
           float H = 255;//height of cone
           float3 D = float3(0,0,1);//direction of the cone
           float3 U = float3(0,1,0);//vector in the plane of the cone circle if cone oriented in z axis
           float3 V = float3(1,0,0);//vector in the plane of the cone circle if cone oriented in z axis
           //some rotation at the start whitout rotating the pixel position 
           //gives some effective rotation but the cone is not correctly rotating
           D =  normalize(mul( D, (float3x3)World ));
           U =  normalize(mul( U, (float3x3)World ));
           V =  normalize(mul( V, (float3x3)World ));
           float3 P;
           uint h;//current position along the total height H
           for ( h = 0; h< H; h++)
           {
               float hH = h/H;
               float RhH = R * hH;
               uint a;//angle used to draw the circle at height h
               for ( a = 0; a < 72; a++ )//make a circle at height h
               {
                   float T = 2*3.1415f*a/72.0f;
                   float S = sin(T);  
                   float C = cos(T);  
                   P.x = O.x*hH*D.x + RhH * (C * U.x + S * V.x);
                   P.y = O.y*hH*D.y + RhH * (C * U.y + S * V.y);
                   P.z = O.x*hH*D.z + RhH * (C * U.z + S * V.z);
                   //this should rotate the cone point but the cone disappears
                   //P.xy = mul( float4(P,1), M ).xy*2-1;
                   //P.xy*=float2(SM_SCREENX,SM_SCREENY);
                   int2 XY = int2(P.xy);
                   //colorize the pixel based on P.xy value (should have z as blue also)
                   UAVDiffuse0[nDTid.xy+XY]= XY.x | (XY.y<<8)|0x000000FF;
               }
           }
       }
   }