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{ RCSid $Id: fovsample.cal,v 1.3 2019/04/05 23:49:59 greg Exp $ } |
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{ |
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Foveated sampling based on central direction vector, |
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magnification factor at center, and central field of view. |
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Greg Ward April 2019 |
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Preset constants: |
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Cx, Cy, Cz : central view direction |
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M : central FOV magnification factor |
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D : field of view (full angle in degrees) |
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Inputs: |
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forward = 1 if forward conversion, -1 if reverse |
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iDx, iDy, iDz = input sampling vector |
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Outputs: |
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oDx, oDy, oDz = output sample direction (normalized) |
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Other variables computed: |
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iTheta = input angle to center (radians) |
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oTheta = output angle to center (radians) |
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} |
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r : PI/360 * D; { half-angle of foveal region in radians } |
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rp : M * r; { half-angle of mapped boundary } |
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len(x,y,z) : sqrt(x*x + y*y + z*z); |
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sq(x) : x*x; |
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{ normalize central vector } |
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Clen : len(Cx,Cy,Cz); |
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nCx : Cx/Clen; |
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nCy : Cy/Clen; |
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nCz : Cz/Clen; |
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{ normalize input direction } |
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iDnf = 1/len(iDx,iDy,iDz); |
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niDx = iDx*iDnf; |
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niDy = iDy*iDnf; |
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niDz = iDz*iDnf; |
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{ check for degenerate case (center of fovea) } |
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iDot = nCx*niDx + nCy*niDy + nCz*niDz; |
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degen = iDot - cos(.05*PI/180); |
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iTheta = acos(iDot); |
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{ quadratic function coefficients mate to linear ramp } |
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qa : (PI - PI/M)/sq(PI - rp); |
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qb : 1/M - 2*PI*r*(M - 1)/sq(PI - rp); |
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qc : (PI - PI/M)/sq(PI/rp - 1); |
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oTheta = if (forward, |
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if(rp-iTheta, iTheta/M, qa*iTheta*iTheta + qb*iTheta + qc), |
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if(r-iTheta, iTheta*M, (sqrt(qb*qb - 4*qa*(qc - iTheta)) - qb)/(2*qa)) |
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); |
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{ normalized "up" vector for rotation } |
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Unf = 1/sqrt(1 - iDot*iDot); |
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nUx = (nCy*niDz - nCz*niDy)*Unf; |
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nUy = (nCz*niDx - nCx*niDz)*Unf; |
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nUz = (nCx*niDy - nCy*niDx)*Unf; |
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{ rotate central vector by oTheta to get new output } |
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rcos = cos(oTheta); |
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rsin = sqrt(1 - rcos*rcos); |
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{ foveated sample direction } |
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rDx = nCx*rcos + (nUy*nCz - nUz*nCy)*rsin; |
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rDy = nCy*rcos + (nUz*nCx - nUx*nCz)*rsin; |
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rDz = nCz*rcos + (nUx*nCy - nUy*nCx)*rsin; |
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{ substitute approximation in degenerate case } |
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oDx = if(degen, nCx+(niDx-nCx)*if(forward,1/M,M), rDx); |
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oDy = if(degen, nCy+(niDy-nCy)*if(forward,1/M,M), rDy); |
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oDz = if(degen, nCz+(niDz-nCz)*if(forward,1/M,M), rDz); |
