{"version":9,"randomSeed":"6085bdc405a20b5cf77cb1241376e423","graph":{"viewport":{"xmin":-1.2500000000000033,"ymin":-1.2499999999999998,"xmax":1.2499999999999962,"ymax":1.2499999999999998}},"expressions":{"list":[{"type":"folder","id":"37","title":"library of useful functions","collapsed":true},{"type":"text","id":"38","folderId":"37","text":"project, R^3 -> R^2"},{"type":"expression","id":"39","folderId":"37","color":"#2d70b3","latex":"P\\left(v\\right)=\\frac{\\left(v\\left[1\\right],v\\left[2\\right]\\right)}{v\\left[3\\right]}"},{"type":"text","id":"40","folderId":"37","text":"project list, R^(3*n) -> R^3 -> R^2 -> (R^2)^n"},{"type":"expression","id":"41","folderId":"37","color":"#2d70b3","latex":"P_{L}\\left(v,o,\\phi\\right)=\\left[P\\left(R_{x}\\left(R_{y}\\left(o+\\left[v\\left[3k-2\\right],v\\left[3k-1\\right],v\\left[3k\\right]\\right],\\phi.x\\right),\\phi.y\\right)\\right)\\operatorname{for}k=\\left[1...\\frac{\\operatorname{length}\\left(v\\right)}{3}\\right]\\right]"},{"type":"text","id":"42","folderId":"37","text":"dot product, R^n -> R^n -> R"},{"type":"expression","id":"43","folderId":"37","color":"#c74440","latex":"D\\left(u,v\\right)=\\sum_{k=1}^{\\min\\left(\\operatorname{length}\\left(u\\right),\\operatorname{length}\\left(v\\right)\\right)}u\\left[k\\right]v\\left[k\\right]"},{"type":"text","id":"44","folderId":"37","text":"cross product, R^3 -> R^3"},{"type":"expression","id":"45","folderId":"37","color":"#388c46","latex":"C\\left(u,v\\right)=\\left[u\\left[2\\right]v\\left[3\\right]-u\\left[3\\right]v\\left[2\\right],u\\left[3\\right]v\\left[1\\right]-u\\left[1\\right]v\\left[3\\right],u\\left[1\\right]v\\left[2\\right]-u\\left[2\\right]v\\left[1\\right]\\right]"},{"type":"text","id":"46","folderId":"37","text":"normalise vector, R^n -> R^n"},{"type":"expression","id":"47","folderId":"37","color":"#6042a6","latex":"N\\left(v\\right)=\\frac{v}{\\sqrt{D\\left(v,v\\right)}}"},{"type":"text","id":"48","folderId":"37","text":"insert element as replacement, R^n -> N -> {R, R^2} -> R^n"},{"type":"expression","id":"49","folderId":"37","color":"#2d70b3","latex":"I\\left(L,i,x\\right)=\\left\\{i=1:\\operatorname{join}\\left(x,L\\left[2...\\right]\\right),i=\\operatorname{length}\\left(L\\right):\\operatorname{join}\\left(L\\left[1...\\operatorname{length}\\left(L\\right)-1\\right],x\\right),\\operatorname{join}\\left(L\\left[1...i-1\\right],x,L\\left[i+1...\\right]\\right)\\right\\}"},{"type":"text","id":"50","folderId":"37","text":"rotate around each axis, R^3 -> R^3"},{"type":"expression","id":"51","folderId":"37","color":"#2d70b3","latex":"R_{x}\\left(v,\\theta\\right)=\\left[v\\left[1\\right],v\\left[2\\right]\\cos\\theta-v\\left[3\\right]\\sin\\theta,v\\left[2\\right]\\sin\\theta+v\\left[3\\right]\\cos\\theta\\right]"},{"type":"expression","id":"52","folderId":"37","color":"#388c46","latex":"R_{y}\\left(v,\\theta\\right)=\\left[v\\left[1\\right]\\cos\\theta-v\\left[3\\right]\\sin\\theta,v\\left[2\\right],v\\left[1\\right]\\sin\\theta+v\\left[3\\right]\\cos\\theta\\right]"},{"type":"expression","id":"53","folderId":"37","color":"#6042a6","latex":"R_{z}\\left(v,\\theta\\right)=\\left[v\\left[1\\right]\\cos\\theta-v\\left[2\\right]\\sin\\theta,v\\left[1\\right]\\sin\\theta+v\\left[2\\right]\\cos\\theta,v\\left[3\\right]\\right]"},{"type":"folder","id":"110","title":"basic functions of complex numbers","collapsed":true},{"type":"text","id":"111","folderId":"110","text":"multiply"},{"type":"expression","id":"112","folderId":"110","color":"#c74440","latex":"M_{c}\\left(w,z\\right)=\\left(w.x\\cdot z.x-w.y\\cdot z.y,w.x\\cdot z.y+w.y\\cdot z.x\\right)"},{"type":"text","id":"113","folderId":"110","text":"reciprocate"},{"type":"expression","id":"114","folderId":"110","color":"#388c46","latex":"R_{c}\\left(z\\right)=\\frac{\\left(z.x,-z.y\\right)}{z.x^{2}+z.y^{2}}"},{"type":"text","id":"115","folderId":"110","text":"square"},{"type":"expression","id":"116","folderId":"110","color":"#2d70b3","latex":"Q_{c}\\left(z\\right)=M_{c}\\left(z,z\\right)"},{"type":"text","id":"117","folderId":"110","text":"cube"},{"type":"expression","id":"118","folderId":"110","color":"#000000","latex":"C_{c}\\left(z\\right)=M_{c}\\left(Q_{c}\\left(z\\right),z\\right)"},{"type":"text","id":"119","folderId":"110","text":"naturally exponentiate"},{"type":"expression","id":"120","folderId":"110","color":"#c74440","latex":"E_{c}\\left(z\\right)=\\left(e^{z.x}\\cos\\left(z.y\\right),e^{z.x}\\sin\\left(z.y\\right)\\right)"},{"type":"text","id":"121","folderId":"110","text":"naturally logarithmise"},{"type":"expression","id":"122","folderId":"110","color":"#2d70b3","latex":"L_{c}\\left(z\\right)=\\left(\\ln\\left(\\operatorname{distance}\\left(\\left(0,0\\right),z\\right)\\right),\\arctan\\left(z.y,z.x\\right)\\right)"},{"type":"expression","id":"54","color":"#c74440","latex":"f\\left(z\\right)=E_{c}\\left(z\\right)","hidden":true},{"type":"text","id":"58","text":"screen-space from disc indices"},{"type":"expression","id":"65","color":"#000000","latex":"r=30","hidden":true,"slider":{"hardMin":true,"hardMax":true,"min":"10","max":"50","step":"1"}},{"type":"expression","id":"108","color":"#6042a6","latex":"C_{is}\\left(x\\right)=\\left(\\cos x,\\sin x\\right)"},{"type":"expression","id":"56","color":"#388c46","latex":"s\\left(a,b,c\\right)=\\frac{a}{r}\\cdot C_{is}\\left(\\frac{2\\pi}{2a+1-c}b\\right)","labelSize":"medium"},{"type":"text","id":"61","text":"sphere-space from disc indices"},{"type":"expression","id":"63","color":"#388c46","latex":"\\alpha=0","slider":{"hardMin":true,"hardMax":true,"loopMode":"LOOP_FORWARD","min":"-2\\pi","max":"2\\pi"}},{"type":"expression","id":"64","color":"#6042a6","latex":"\\beta=-1.5708","slider":{"hardMin":true,"hardMax":true,"loopMode":"LOOP_FORWARD","min":"-\\pi","max":"\\pi"}},{"type":"expression","id":"62","color":"#2d70b3","latex":"g\\left(a,b\\right)=R_{x}\\left(R_{y}\\left(\\left[s\\left(a,b,0\\right).x,s\\left(a,b,0\\right).y,\\sqrt{1-\\left(\\frac{a}{r}\\right)^{2}}\\right],\\alpha\\right),\\beta\\right)"},{"type":"text","id":"67","text":"plane-space from sphere-space"},{"type":"expression","id":"68","color":"#2d70b3","latex":"p\\left(c\\right)=\\left(\\frac{c\\left[1\\right]}{1-c\\left[2\\right]},\\frac{c\\left[3\\right]}{1-c\\left[2\\right]}\\right)"},{"type":"text","id":"73","text":"plane-centre of disc-indexed square"},{"type":"expression","id":"74","color":"#c74440","latex":"p_{rc}\\left(a,b\\right)=p\\left(g\\left(a,b\\right)\\right)","labelSize":"medium"},{"type":"text","id":"70","text":"colouring (abysmal)"},{"type":"expression","id":"81","color":"#388c46","latex":"L\\left(a\\right)=\\frac{2}{\\pi}\\arctan\\left(a\\right)","hidden":true},{"type":"expression","id":"104","color":"#000000","latex":"\\gamma=1","slider":{"hardMin":true,"min":"0"}},{"type":"expression","id":"89","color":"#c74440","latex":"H_{g}\\left(a,l\\right)=\\operatorname{hsv}\\left(180\\operatorname{mod}\\left(\\frac{a}{\\pi}+0.8,2\\right),1-l^{2},\\sqrt{l}\\right)"},{"type":"expression","id":"75","color":"#2d70b3","latex":"H\\left(z\\right)=H_{g}\\left(\\arctan\\left(z.y,z.x\\right),\\frac{2}{\\pi}\\arctan\\left(\\operatorname{distance}\\left(\\left(0,0\\right),z\\right)\\right)\\right)"},{"type":"expression","id":"105","color":"#c74440","latex":"J\\left(u\\right)=\\operatorname{floor}\\left(\\sqrt{u}\\right)","hidden":true},{"type":"expression","id":"106","color":"#2d70b3","latex":"Q\\left(u\\right)=\\left(u-\\operatorname{floor}\\left(\\sqrt{u}\\right)\\right)","hidden":true},{"type":"expression","id":"71","color":"#6042a6","latex":"L_{C}=\\left[H\\left(f\\left(p_{rc}\\left(J\\left(k\\right),Q\\left(k\\right)\\right)\\right)\\right)\\operatorname{for}k=\\left[0...\\left(r^{2}-1\\right)\\right]\\right]"},{"type":"expression","id":"96","color":"#000000","latex":"\\operatorname{polygon}\\left(\\left(-99,-99\\right),\\left(-99,99\\right),\\left(99,99\\right),\\left(99,-99\\right)\\right)","fillOpacity":"1"},{"type":"expression","id":"55","color":"#fa7e19","latex":"\\left[\\operatorname{polygon}\\left(s\\left(J\\left(k\\right),Q\\left(k\\right),0\\right),s\\left(J\\left(k\\right),Q\\left(k\\right)+\\frac{1}{3},0\\right),s\\left(J\\left(k\\right),Q\\left(k\\right)+\\frac{2}{3},0\\right),s\\left(J\\left(k\\right),Q\\left(k\\right)+1,0\\right),s\\left(J\\left(k\\right)+1,Q\\left(k\\right)+1,2\\right),s\\left(J\\left(k\\right)+1,Q\\left(k\\right)+\\frac{2}{3},2\\right),s\\left(J\\left(k\\right)+1,Q\\left(k\\right)+\\frac{1}{3},2\\right),s\\left(J\\left(k\\right)+1,Q\\left(k\\right),2\\right)\\right)\\operatorname{for}k=\\left[0...r^{2}-1\\right]\\right]","lines":false,"fill":true,"labelSize":"medium","colorLatex":"L_{C}","fillOpacity":"1"},{"type":"expression","id":"86","color":"#388c46","latex":"\\left(\\frac{\\alpha}{4},\\frac{\\beta}{4}\\right)","colorLatex":"W"},{"type":"expression","id":"123","color":"#c74440","latex":"W=\\operatorname{rgb}\\left(255,255,255\\right)"}]}}