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{ RCSid $Id: disk2square.cal,v 2.5 2019/08/13 18:29:23 greg Exp $ } |
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{ |
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Convert between unit square and disk, using Shirley-Chiu mapping |
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|
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5/20/2011 G. Ward |
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|
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Inputs: |
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in_disk_x Input unit disk Cartesian coordiantes |
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in_disk_y center at (0,0) |
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or: |
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in_square_x Input unit square [0,1]^2 |
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in_square_y |
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|
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Corresponding outputs: |
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out_square_x Output unit square [0,1]^2 |
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out_square_y |
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or: |
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out_disk_x Output unit disk Cartesian coordinates |
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out_disk_y center at (0,0) |
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} |
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{ Compute disk position from square coordinates } |
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in_square_a = 2*in_square_x - 1; |
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in_square_b = 2*in_square_y - 1; |
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|
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in_square_rgn = if(in_square_a + in_square_b, |
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if(in_square_a - in_square_b, 1, 2), |
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if(in_square_b - in_square_a, 3, 4)); |
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|
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out_disk_r = select(in_square_rgn, in_square_a, in_square_b, |
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-in_square_a, -in_square_b); |
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|
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out_disk_phi = PI/4 * select(in_square_rgn, |
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in_square_b/in_square_a, |
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2 - in_square_a/in_square_b, |
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4 + in_square_b/in_square_a, |
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if(in_square_b*in_square_b, |
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6 - in_square_a/in_square_b, |
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0)); |
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|
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out_disk_x = out_disk_r*cos(out_disk_phi); |
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out_disk_y = out_disk_r*sin(out_disk_phi); |
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|
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{ Compute square position from disk coordinates } |
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norm_radians(p) : if(-p - PI/4, p + 2*PI, p); |
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in_disk_r = sqrt(in_disk_x*in_disk_x + in_disk_y*in_disk_y); |
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in_disk_phi = norm_radians(atan2(in_disk_y, in_disk_x)); |
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|
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in_disk_rgn = floor((in_disk_phi + PI/4)/(PI/2)) + 1; |
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|
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out_square_a = select(in_disk_rgn, |
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in_disk_r, |
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(PI/2 - in_disk_phi)*in_disk_r/(PI/4), |
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-in_disk_r, |
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(in_disk_phi - 3*PI/2)*in_disk_r/(PI/4), |
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in_disk_r); |
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|
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out_square_b = select(in_disk_rgn, |
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in_disk_phi*in_disk_r/(PI/4), |
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in_disk_r, |
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(PI - in_disk_phi)*in_disk_r/(PI/4), |
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-in_disk_r, -in_disk_r); |
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|
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out_square_x = (out_square_a + 1)/2; |
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out_square_y = (out_square_b + 1)/2; |
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{ |
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The following forumulas compute Shirley-Chiu bin "scbin" based on: |
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|
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RHS - right-handed system (-1 for left-handed coords) |
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Dx,Dy,Dz - Incident direction (normalized, towards surface front) |
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rNx,rNy,rNz - Surface normal (normalized, away from surface) |
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Ux,Uy,Uz - Up direction vector (does not need to be normalized) |
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|
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The SCdim variable assigns the square side dimension for bins, which are |
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ordered with the "up" direction changing fastest. |
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} |
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RHS = 1; |
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{ Compute oriented axis values } |
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inc_dz = -Dx*rNx-Dy*rNy-Dz*rNz; |
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inc_rx = -RHS*(Dx*(Uy*rNz-Uz*rNy) + Dy*(Uz*rNx-Ux*rNz) + Dz*(Ux*rNy-Uy*rNx)); |
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inc_ry = Dx*Ux+Dy*Uy+Dz*Uz + inc_dz*(rNx*Ux+rNy*Uy+rNz*Uz); |
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inc_den2 = inc_rx*inc_rx + inc_ry*inc_ry; |
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inc_radf = if(inc_den2-1e-7, sqrt((1 - inc_dz*inc_dz)/inc_den2), 0); |
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{ Pass to formulas in first section } |
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in_disk_x = inc_rx*inc_radf; |
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in_disk_y = -inc_ry*inc_radf; |
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{ Compute final bin (-1 if behind surface) } |
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scbin = if(inc_dz, |
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floor(out_square_x*SCdim)*SCdim + floor(out_square_y*SCdim), |
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-1); |
