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R^3"},{"type":"expression","id":"10","folderId":"2","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":"11","folderId":"2","text":"normalise vector, R^n -> R^n"},{"type":"expression","id":"12","folderId":"2","color":"#6042a6","latex":"N\\left(v\\right)=\\frac{v}{\\sqrt{D\\left(v,v\\right)}}"},{"type":"text","id":"13","folderId":"2","text":"insert element as replacement, R^n -> N -> {R, R^2} -> R^n"},{"type":"expression","id":"14","folderId":"2","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":"15","folderId":"2","text":"rotate around each axis, R^3 -> R^3"},{"type":"expression","id":"16","folderId":"2","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":"17","folderId":"2","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":"18","folderId":"2","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":"expression","id":"19","color":"#c74440","latex":"\\alpha=-1.256","slider":{"hardMin":true,"hardMax":true,"animationPeriod":8000,"loopMode":"LOOP_FORWARD","isPlaying":true,"min":"-\\pi","max":"\\pi"}},{"type":"expression","id":"20","color":"#2d70b3","latex":"\\beta=-1.015","slider":{"hardMin":true,"hardMax":true,"animationPeriod":11428.57142857143,"loopMode":"LOOP_FORWARD","isPlaying":true,"min":"-\\pi","max":"\\pi"}},{"type":"expression","id":"21","color":"#388c46","latex":"f\\left(p\\right)=\\left[p.x,p.x^{2}-p.y^{2},p.y\\right]"},{"type":"expression","id":"22","color":"#fa7e19","latex":"\\left[P\\left(\\left[0,0,3\\right]+R_{x}\\left(R_{y}\\left(f\\left(\\left(t,y_{1}\\right)\\right),\\alpha\\right),\\beta\\right)\\right)\\operatorname{for}y_{1}=\\left[-1,-0.75,...1\\right]\\right]","hidden":true,"parametricDomain":{"min":"-1","max":"1"},"domain":{"min":"-1","max":"1"}},{"type":"expression","id":"24","color":"#fa7e19","latex":"\\left[P\\left(\\left[0,0,3\\right]+R_{x}\\left(R_{y}\\left(f\\left(\\left(x_{1},t\\right)\\right),\\alpha\\right),\\beta\\right)\\right)\\operatorname{for}x_{1}=\\left[-1,-0.75,...1\\right]\\right]","hidden":true,"parametricDomain":{"min":"-1","max":"1"},"domain":{"min":"-1","max":"1"}},{"type":"expression","id":"28","color":"#000000","latex":"\\left[P\\left(\\left[0,0,3\\right]+R_{x}\\left(R_{y}\\left(f\\left(r_{1}\\cdot\\left(\\cos 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