... back to the description

Distances are in meters, except when indicated.

And here's the (probably very ugly) Mathematica code from which the interactive view above was exported as a Computable Document Format (CDF) file. The actual code for the CDF is actually simpler than that because I'm asking to display less information for clarity.

The first part defines the default values of some variables.

    rb = 0.024;(*bulb radius*)
    rp = 0.049;(*precondenser radius*)
    fp = 0.27;(*precondenser focal length*)
    hal = 0.015;(*arc half height*)
    fc = 0.33;(*condenser lens focal length*)
    fco = 0.33;(*collector lens focal length*)
    hhc = 0.17;(*condenser lens half height*)
    margin = 1.05;(*proportion of condenser lens lit*)
    hhl = 0.17;(*LCD half height*)
    wl = 0.2125;(*LCD width*)
    lc = 0.02;(*distance condenser lens - LCD*)
    lco = 0.02;(*distance LCD-collector lens*)
    prcf = 0;(*distance projection lens - collector lens focal point*)
    fpr = 0.327;(*projection lens focal length*)
    rpr = 0.032;(*projection lens radius*)

Then sliders are created to interactively change the variables describing the different elements of the projector.

    {{"Precondenser radius=" Dynamic[rp], Slider[Dynamic[rp], {0.015, 0.2, 0.001}]},
    {"Precondenser focal length=" Dynamic[fp], Slider[Dynamic[fp], {0.03, 0.4, 0.001}]},
    {"Bulb Radius=" Dynamic[rb](*,Slider[Dynamic[rb],{0.01,0.06,0.001}]*)},
    {"Half Arc height=" Dynamic[hal](*,Slider[Dynamic[hal],{0.002,0.05,0.001}]*)},
    {"Margin factor (zone lit)=" Dynamic[margin], Slider[Dynamic[margin], {1, 1.5, 0.01}]},
    {"Condenser focal length=" Dynamic[fc], Slider[Dynamic[fc], {0.15, 0.6, 0.001}]},
    {"Condenser half height=" Dynamic[hhc = hhl](*,Slider[Dynamic[hhc],{0.05,0.25,0.001}]*)},
    {"LCDWidth=" Dynamic[wl]}}
    {{"Distance condenser-LCD=" Dynamic[lc], Slider[Dynamic[lc], {0.001, 0.05, 0.001}]},
    {"LCD half height=" Dynamic[hhl], Slider[Dynamic[hhl], {0.05, 0.25, 0.001}]},
    {"Distance LCD-coll=" Dynamic[lco], Slider[Dynamic[lco], {0.001, 0.05, 0.001}]}}
    {{"Collector focal length=" Dynamic[fco], Slider[Dynamic[fco], {0.15, 0.6, 0.001}]},
    {"Distance projector - collector focal=" Dynamic[prcf], Slider[Dynamic[prcf], {-0.05, 0.05, 0.0005}]},
    {"Projector focal length=" Dynamic[fpr], Slider[Dynamic[fpr], {0.1, 0.5, 0.001}]},
    {"Projector radius=" Dynamic[rpr], Slider[Dynamic[rpr], {0.02, 0.1, 0.001}]}}

From these variables, the values of the remaining parameters are calculated.

    (*The distance bulb-precondensor is automatically calculated for the light to cover (margin*condensor_height) by keeping the virtual bulb at the focal point of the condensor*)
    {{"Distance arc-precondenser (Mathematica equation solver)=" Dynamic[xa = -(fp (2 fp hal - fp 2*hhcm + \[Sqrt](4 fc^2 hal^2 + fp^2 (-2 hal + 2*hhcm)^2 + 4 fc fp hal (2 hal + 2*hhcm - 4 rp)) + 2 fc (hal - 2 rp)))/(4*fp hhcm + 4 fc rp)]},
    {"Virtual arc position=" Dynamic[xva = 1/(1/fp - 1/xa)]},
    {"Virtual arc half length=" Dynamic[hval = hal*fp/(fp - xa)]},
    {"Height of light around condenser=", Dynamic[hhcm = hhc*margin]},
    {"Distance precondenser-condenser=", Dynamic[pc = fc + xva]},
    {"Distance collector-virtual LCD=" Dynamic[covl = -1/(1/fco - 1/lco)]},
    {"Distance projector-virtual LCD" Dynamic[prvl = covl + fco + prcf]},
    {"Screen height=" Dynamic[hs = hhl*fco/(fco - lco)*fpr/(prvl - fpr)]},
    {"Screen width=" Dynamic[ws = hs/hhl*wl]},
    {"Throw=" Dynamic[T = prvl*fpr/(prvl - fpr)]},
    {"Collector axis offset for overlap=" Dynamic[off = hs/2*hhl/(hs + hhl)]},
    {"Virtual image height under offset axis=" Dynamic[vh1 = fco/(fco - lco)*off]},
    {"Virtual image height above offset axis=" Dynamic[vh2 = fco/(fco - lco)*(hhl - off)]},
    {"Arc image half length at collector focal=" Dynamic[ial = fco/fc*hval]}}

And finally, this big ugly piece of code takes care of all the drawing. The color calls should help figure out what does what.

    Dynamic[
    Graphics[{
    Lighter[Orange],
    Polygon[{{xva-(fc+lc+xva),-hval},{xva-(fc+lc+xva),hval},{0-(fc+lc+xva),rp},{0-(fc+lc+xva),-rp}}],
    Lighter[Yellow],
    Rectangle[{-lc,-hhc},{0,hhc}],
    Polygon[{{-xa-(fc+lc+xva),-hal},{-xa-(fc+lc+xva),hal},{-xa-(fc+lc+xva),hal},{0-(fc+lc+xva),rp},{fc+xva-(fc+lc+xva),rp-(rp-hval)/xva(fc+xva)},{fc+xva-(fc+lc+xva),-rp+(rp-hval)/xva(fc+xva)},{0-(fc+lc+xva),-rp}}],
    Lighter[Lighter[Red]],
    Rectangle[{0,0},{lco,hhl}],
    Polygon[{{lco,0},{lco+fco,off-fcohval/fc},{lco+fco,off+fcohval/fc},{lco,hhl}}],
    Polygon[{{lco+fco+prcf,off-fcohval/fc-prcf(hhl-off+fcohval/fc)/fco},{lco+fco,off-fcohval/fc},{lco+fco,off+fcohval/fc},{lco+fco+prcf,off+fcohval/fc+prcf(off+fcohval/fc)/fco}}],
    Lighter[Lighter[Blue]],
    Rectangle[{0,0},{lco,-hhl}],
    Polygon[{{lco,0},{lco+fco,-off+fcohval/fc},{lco+fco,-off-fcohval/fc},{lco,-hhl}}],
    Polygon[{{lco+fco+prcf,-off+fcohval/fc+prcf(hhl-off+fcohval/fc)/fco},{lco+fco,-off+fcohval/fc},{lco+fco,-off-fcohval/fc},{lco+fco+prcf,-off-fcohval/fc-prcf(off+fcohval/fc)/fco}}],
    Thick,
    Gray,
    Arrowheads[{-rp/2,rp/2}],
    Arrow[{{0-(fc+lc+xva),-rp},{0-(fc+lc+xva),rp}}],
    Arrowheads[{-hhc/5,hhc/5}],
    Arrow[{{fc+xva-(fc+lc+xva),-hhc},{fc+xva-(fc+lc+xva),hhc}}],
    Arrowheads[{-hhl/5,hhl/5}],
    Arrow[{{lco,0},{lco,hhl}}],
    Arrowheads[{-hhl/5,hhl/5}],
    Arrow[{{lco,-hhl},{lco,0}}],Line[{{lco-0.005,off},{lco+0.005,off}}],
    Line[{{lco-0.005,-off},{lco+0.005,-off}}],
    Arrowheads[{-rpr/2,rpr/2}],
    Arrow[{{lco+fco+prcf,off-rpr},{lco+fco+prcf,off+rpr}}],
    Arrowheads[{-rpr/2,rpr/2}],Arrow[{{lco+fco+prcf,-off-rpr},{lco+fco+prcf,-off+rpr}}],
    Darker[Cyan],
    Circle[{-xa-(fc+lc+xva),(hal+rb/2+0.8rp)},rb,{0,Pi}],
    Circle[{-xa-(fc+lc+xva),-(hal+rb/2+0.8rp)},rb,{-Pi,0}],
    Line[{{-xa-rb-(fc+lc+xva),-(hal+rb/2+0.8rp)},{-xa-rb-(fc+lc+xva),(hal+rb/2+0.8rp)}}],
    Line[{{-xa+rb-(fc+lc+xva),-(hal+rb/2+0.8rp)},{-xa+rb-(fc+lc+xva),(hal+rb/2+0.8rp)}}],
    Magenta,Line[{{0,-hhl},{0,hhl}}],
    Black,
    Line[{{0,0},{lco,0}}],
    Line[{{-lc,hhc},{-lc,marginhhc}}],
    Line[{{-lc,-hhc},{-lc,-marginhhc}}],
    Thin,
    Dashed,
    Line[{{lco,0},{Max[lco+fco+prcf+0.01,lco+fco],0}}],
    Line[{{lco,off},{Max[lco+fco+prcf+0.01,lco+fco],off}}],
    Line[{{lco,-off},{Max[lco+fco+prcf+0.01,lco+fco],-off}}],
    Arrowheads[{-0.02,0.02}],
    Arrow[{{-xa-(fc+lc+xva),0},{0-(fc+lc+xva),0}}],
    Text[Round[1000xa] mm,{-xa/2-(fc+lc+xva),0.01}],
    Arrowheads[{-0.02,0.02}],
    Arrow[{{0-(fc+lc+xva),0},{pc-(fc+lc+xva),0}}],
    Text[Round[1000pc] mm,{pc/2-(fc+lc+xva),0.01}],
    Arrowheads[{-0.02,0.02}],
    Arrow[{{Max[lco+fco+prcf+0.01,lco+fco],0},{Max[lco+fco+prcf+0.01,lco+fco],off}}],
    Text[Round[off1000] mm,{Max[lco+fco+prcf+0.01,lco+fco]+0.01,off/2},
    Automatic,{0,-1}],
    Arrowheads[{-0.02,0.02}],
    Arrow[{{-rb-fc-lc-xva-xa,Max[hhc,hhl]margin+0.01},{Max[lco+fco+prcf,lco+fco],Max[hhc,hhl]margin+0.01}}],
    Text[Round[1000(lc+fc+xva+xa+rb+lco+fco+prcf)] mm,{0,Max[hhc,hhl]*margin+0.02}]},
    ImageSize->1200]]