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{"sheet":"INTRODUCTORY PAGE","index":0,"n_rows":0,"n_cols":0,"rows_kept":0,"cut":false,"hidden":false,"merged":[]}

{"sheet":"PARABOLA","index":2,"n_rows":62,"n_cols":37,"rows_kept":62,"cut":false,"hidden":false,"merged":[]}
[null,null,"PARABOLIC MIRROR and Hero's law",null,null,null,null,null,"dx",1,null,"REAL FOCUS",{"$f":"1/4/kapa","v":5},"When the optical focus is set on the parabola focus"]
[null,null,"PARABOLIC MIRROR",null,null,null,null,null,"Kapa",0.05,null,"estparab",{"$f":"M1","v":5},"then the \"ray\" that passes through the focus is a reflected ray."]
[null,null,"Only when the value of focus  is equal to REAL FOCUS",null,null,null,null,null,"focus",5,"xfocus",0,"calculation of the PERPENDICULAR"]
["PARABOLA",null,"the total length is a constant",null,null,null,null,null,"xinitial",{"$f":"-25+AK4","v":-12},null,null,null,{"$f":"xinitial","v":-12},null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,13]
["x","y","and the angles with the perpendicular are equal",null,null,null,null,null,"yinitial",25,null,null,"DY/DX =",{"$f":"2*kapa*xinitial","v":-1.2000000000000002},"-1/DY/DX",{"$f":"-1/N5","v":0.83333333333333326}]
[-20,{"$f":"kapa*A6^2","v":20},null,null,null,{"$f":"xinitial","v":-12},{"$f":"yinitial","v":25},null,"yfinal",{"$f":"kapa*xinitial^2","v":7.2},null,null,null,"point of perpendicular on y axis =",null,null,{"$f":"yfinal-P5*xinitial","v":17.2}]
[{"$f":"A6+dx","v":-19},18.05,null,null,null,{"$f":"xfinal","v":-12},{"$f":"yfinal","v":7.2},null,"xfinal",{"$f":"xinitial","v":-12},null,"TO DRAW FOCUS",null,null,null,null,"Draw Perpendicular",null,null,"draw rays"]
[-18,16.2,null,null,null,{"$f":"xfocus","v":0},{"$f":"focus","v":5},null,"TOTAL LENGTH",{"$f":"(yinitial-yfinal)+SQRT((xfocus-xfinal)^2+(focus-yfinal)^2)","v":30},null,{"$f":"xfocus","v":0},{"$f":"focus","v":5},null,null,null,{"$f":"xinitial","v":-12},{"$f":"yfinal","v":7.2},null,{"$f":"xinitial","v":-12},{"$f":"yinitial","v":25}]
[-17,14.450000000000001,null,null,null,null,null,null,null,null,null,null,null,null,"change ",null,{"$f":"0","v":0},{"$f":"Q6","v":17.2},null,{"$f":"xfinal","v":-12},{"$f":"yfinal","v":7.2}]
[-16,12.8,null,null,null,null,null,null,null,null,null,null,null,"focus","xinitial",null,null,null,null,{"$f":"0","v":0},{"$f":"focus","v":5}]
[-15,11.25]
[-14,9.8000000000000007,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Calculation of angles"]
[-13,8.4500000000000011,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Metro 1","Metro2","Perpendicular"]
[-12,7.2,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"ABS(yinitial-yfinal)","v":17.8},{"$f":"SQRT((xfinal-xfocus)^2+(yfinal-focus)^2)","v":12.2},{"$f":"SQRT((Q6-yfinal)^2+xfinal^2)","v":15.620499351813308}]
[-11,6.0500000000000007,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"External products"]
[-10,5,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"(-(xfinal-xfocus)*(yinitial-yfinal))/(Q14*S14)","v":0.76822127959737596},{"$f":"-((xfocus-xfinal)*(focus-yfinal)-(Q6-yfinal)*(xfocus-xfinal))/(R14*S14)","v":0.76822127959737596}]
[-9,4.05,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"angles"]
[-8,3.2,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"DEGREES(ASIN(Q16))","v":50.194428907734817},{"$f":"DEGREES(ASIN(R16))","v":50.194428907734817}]
[-7,2.4500000000000002]
[-6,1.8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"t1 ","time froma parabola to focus"]
[-5,1.25,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"x","y parabola","dy1","dsfx"]
[-4,0.8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"-20","v":-20},{"$f":"kapa*R22^2","v":20},{"$f":"yinitial-S22","v":5},{"$f":"SQRT((R22-0)^2+(S22-1/(4*kapa))^2)","v":25},{"$f":"IF(timMMM1<=T22,R22,R22+(timMMM1-T22)*COS(ATAN((focus-S22)/(0-R22))))","v":-13.6},{"$f":"IF(timMMM1<=T22,yinitial-timMMM1,S22+(timMMM1-T22)*SIN(ATAN((focus-S22)/(0-R22))))","v":15.2},null,{"$f":"yinitial-T22","v":20}]
[-3,0.45,null,null,null,null,null,null,null,null,null,null,null,null,261,null,null,{"$f":"R22+dx","v":-19},18.05,6.9499999999999993,23.05,-14.013015184381779,14.624728850325379]
[-2,0.2,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-18,16.2,8.8000000000000007,21.2,-14.433962264150944,13.981132075471699]
[-1,0.05,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-17,14.450000000000001,10.549999999999999,19.45,-14.858611825192801,13.259640102827763]
[0,0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-16,12.8,12.2,17.8,-15.280898876404494,12.449438202247192]
[1,0.05,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-15,11.25,13.75,16.25,-15,12]
[2,0.2,null,null,null,null,null,null,null,null,null,null,null,"time",{"$f":"(P28-1)/10","v":13},131,null,-14,9.8000000000000007,15.2,14.8,-14,12]
[3,0.45,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-13,8.4500000000000011,16.549999999999997,13.45,-13,12]
[4,0.8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-12,7.2,17.8,12.2,-12,12]
[5,1.25,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-11,6.0500000000000007,18.95,11.05,-11,12]
[6,1.8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-10,5,20,10,-10,12]
[7,2.4500000000000002,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-9,4.05,20.95,9.0500000000000007,-9,12]
[8,3.2,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-8,3.2,21.8,8.1999999999999993,-8,12]
[9,4.05,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-7,2.4500000000000002,22.55,7.45,-7,12]
[10,5,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-6,1.8,23.2,6.8,-6,12]
[11,6.0500000000000007,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-5,1.25,23.75,6.25,-5,12]
[12,7.2,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-4,0.8,24.2,5.8,-4,12]
[13,8.4500000000000011,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-3,0.45,24.55,5.4499999999999993,-3,12]
[14,9.8000000000000007,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-2,0.2,24.8,5.2,-2,12]
[15,11.25,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,-1,0.05,24.95,5.05,-1,12]
[16,12.8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,0,25,5,{"$f":"IF(timMMM1<=T42,R42,R42+(timMMM1-T42)*COS(pi/2))","v":0},{"$f":"IF(timMMM1<=T42,yinitial-timMMM1,S42+(timMMM1-T42)*SIN(pi/2))","v":12}]
[17,14.450000000000001,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1,0.05,24.95,5.05,{"$f":"IF(timMMM1<=T43,R43,R43-(timMMM1-T43)*COS(ATAN((focus-S43)/(0-R43))))","v":1},{"$f":"IF(timMMM1<=T43,yinitial-timMMM1,S43-(timMMM1-T43)*SIN(ATAN((focus-S43)/(0-R43))))","v":12}]
[18,16.2,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2,0.2,24.8,5.2,2,12]
[19,18.05,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3,0.45,24.55,5.4499999999999993,3,12]
[20,20,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,4,0.8,24.2,5.8,4,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,5,1.25,23.75,6.25,5,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,6,1.8,23.2,6.8,6,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,7,2.4500000000000002,22.55,7.45,7,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,8,3.2,21.8,8.1999999999999993,8,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,9,4.05,20.95,9.0500000000000007,9,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,10,5,20,10,10,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11,6.0500000000000007,18.95,11.05,11,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,12,7.2,17.8,12.2,12,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,13,8.4500000000000011,16.549999999999997,13.45,13,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,14,9.8000000000000007,15.2,14.8,14,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,15,11.25,13.75,16.25,15,12]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,16,12.8,12.2,17.8,15.280898876404494,12.449438202247192]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,17,14.450000000000001,10.549999999999999,19.45,14.858611825192801,13.259640102827763]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,18,16.2,8.8000000000000007,21.2,14.433962264150944,13.981132075471699]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,19,18.05,6.9499999999999993,23.05,14.013015184381779,14.624728850325379]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,20,20,5,25,13.6,15.2]

{"sheet":"ELLIPSE","index":4,"n_rows":106,"n_cols":34,"rows_kept":106,"cut":false,"hidden":false,"merged":[]}
[null,null,null,null,null,null,null,"ESTIA1",{"$f":"-CE","v":-25}]
[null,null,null,null,null,null,null,"ESTIA2",{"$f":"CE","v":25},null,null,null,"-",{"$f":"ALPHA/50","v":0.57999999999999996}]
[null,null,null,null,null,null,null,"ALPHA",29]
[null,null,null,null,null,null,null,"CE",25,null,null,null,38]
["X","Y","y2",null,null,null,null,"BETA",{"$f":"SQRT(ALPHA^2-CE^2)","v":14.696938456699069},null,null,null,"to right focus"]
[{"$f":"-ALPHA","v":-29},{"$f":"BETA*SQRT(1-A6^2/ALPHA^2)","v":0},{"$f":"-B6","v":0},null,null,null,null,"Point on ellipse",null,null,null,null,null,42,null,null,"The following are for the calculation of the angles"]
[{"$f":"A6+DEX","v":-28.42},2.9246538256689405,{"$f":"-B7","v":-2.9246538256689405},null,null,null,null,"xarxiko",{"$f":"-ALPHA+N6","v":13},null,"dy/dx",{"$f":"-(xarxiko/yarxiko)*(BETA/ALPHA)^2","v":-0.25414852495448043},null,null,null,null,"metro1","metro2"]
[-27.840000000000003,4.1151427678757315,-4.1151427678757315,null,null,null,null,"yarxiko",{"$f":"BETA*SQRT(1-(xarxiko/ALPHA)^2)","v":13.13752375149314},null,"perpendicular",{"$f":"-1/L7","v":3.934707077993493},"determine point on ellipse",null,null,null,{"$f":"SQRT((xarxiko-ESTIA1)^2+yarxiko^2)","v":40.206896551724142},{"$f":"SQRT((xarxiko-ESTIA2)^2+yarxiko^2)","v":17.793103448275861}]
[-27.260000000000005,5.0142197797862771,-5.0142197797862771,"total Length",null,{"$f":"SQRT((xarxiko-ESTIA1)^2+yarxiko^2)+SQRT((xarxiko-ESTIA2)^2+yarxiko^2)","v":58},null,null,{"$f":"ESTIA1","v":-25},0,"perp-x axis",{"$f":"IF(L7<>0,xarxiko+yarxiko*L7,0)","v":9.6611177170035667},"(xarxiko)",null,null,null,null,"metro perp"]
[-26.680000000000007,5.7599999999999927,-5.7599999999999927,null,null,null,null,null,{"$f":"ESTIA2","v":25},0,"tangent",{"$f":"IF(L7<>0,xarxiko-yarxiko/L7,0)","v":64.692307692307679},null,null,null,null,null,{"$f":"SQRT((xarxiko-perp)^2+yarxiko^2)","v":13.555171161617762}]
[-26.100000000000009,6.406246951218777,-6.406246951218777,null,null,null,null,"line from one focus to the other",null,null,"perpendicular",null,null,null,null,null,"exoteriko1","exoteriko2"]
[-25.52000000000001,6.9806589946795041,-6.9806589946795041,null,"Angles with perpendicular",null,null,"",{"$f":"ESTIA1","v":-25},0,{"$f":"xarxiko","v":13},{"$f":"yarxiko","v":13.13752375149314},null,null,null,null,{"$f":"((xarxiko-ESTIA1)*yarxiko-(xarxiko-perp)*yarxiko)/Q8/R10","v":0.83550782021672565},{"$f":"((xarxiko-ESTIA2)*yarxiko-(xarxiko-perp)*yarxiko)/R8/R10","v":-0.83550782021672587}]
[-24.940000000000012,7.4997599961598667,-7.4997599961598667,null,{"$f":"DEGREES(ASIN(Q12))","v":56.668752986985417},{"$f":"-DEGREES(ASIN(R12))","v":56.668752986985439},null,null,{"$f":"xarxiko","v":13},{"$f":"yarxiko","v":13.13752375149314},{"$f":"L9","v":9.6611177170035667},0]
[-24.360000000000014,7.9743589084013422,-7.9743589084013422,null,"angle 1","angle 2",null,null,{"$f":"ESTIA2","v":25},0,"tangent"]
[-23.780000000000015,8.4119914407944929,-8.4119914407944929,null,"Left ray","Right Ray",null,null,null,null,{"$f":"xarxiko","v":13},{"$f":"yarxiko","v":13.13752375149314}]
[-23.200000000000017,8.818163074019429,-8.818163074019429,null,null,null,null,null,null,null,{"$f":"L10","v":64.692307692307679},0]
[-22.620000000000019,9.197043003052654,-9.197043003052654]
[-22.04000000000002,9.5518793962235389,-9.5518793962235389]
[-21.460000000000022,9.885261756777096,-9.885261756777096]
[-20.880000000000024,10.199294093220361,-10.199294093220361]
[-20.300000000000026,10.495713410721528,-10.495713410721528]
[-19.720000000000027,10.775973273908939,-10.775973273908939]
[-19.140000000000029,11.041304270782494,-11.041304270782494,null,null,null,null,null,null,null,null,null,"t",{"$f":"(O23-1)/10","v":6.6},67]
[-18.560000000000031,11.292758741777837,-11.292758741777837,null,null,null,null,null,null,null,null,null,null,null,null,null,"dtheta",10]
[-17.980000000000032,11.531244512193803,-11.531244512193803,null,null,null,null,null,null,null,null,null,null,null,null,null,"points of a wave",null,"points of intersection"]
[-17.400000000000034,11.757550765359243,-11.757550765359243,null,null,null,null,null,null,null,null,null,null,null,null,null,"angle theta",null,"x point of cut","x - ","y+","y-","Ds+","Ds-","Ds+2","Ds-2","Klisi +2","klisi -2","x+","y+"]
[-16.820000000000036,11.972368186787429,-11.972368186787429,null,null,null,null,null,null,null,null,null,null,null,null,null,0,{"$f":"RADIANS(Q27)","v":0},{"$f":"ALPHA*(-CE*ALPHA*SIN($R27)^2+BETA*BETA*COS($R27))/(ALPHA^2*SIN($R27)^2+BETA^2*COS($R27)^2)","v":29},{"$f":"ALPHA*(CE*ALPHA*SIN($R27)^2-BETA*BETA*COS($R27))/(ALPHA^2*SIN($R27)^2+BETA^2*COS($R27)^2)","v":-29},{"$f":" (BETA*SQRT(1-S27^2/ALPHA^2))","v":0},{"$f":"BETA*SQRT(1-T27^2/ALPHA^2)","v":0},{"$f":"SQRT((S27+CE)^2+U27^2)","v":54},{"$f":"SQRT((T27+CE)^2+V27^2)","v":4},{"$f":"SQRT((S27-CE)^2+U27^2)","v":4},{"$f":"SQRT((T27-CE)^2+V27^2)","v":54},{"$f":"ATAN((U27-0)/(CE-S27))","v":0},{"$f":"ATAN((V27-0)/(CE-T27))","v":0},{"$f":"IF(t<W27,-CE+COS(R27)*t,S27-COS(AA27)*(t-W27))","v":-18.399999999999999},{"$f":"IF(t<W27,SIN(R27)*t,U27+SIN(AA27)*(t-W27))","v":0},{"$f":"IF(t<X27,-CE-COS(R27)*t,T27+COS(AB27)*(t-X27))","v":-26.4},{"$f":"IF(t<X27,SIN(R27)*t,V27-SIGN(AB27)*SIN(AB27)*(t-X27))","v":0}]
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