{"version":11,"randomSeed":"6cae3094795e7c7ceee28b9cf322ba02","graph":{"viewport":{"xmin":-10,"ymin":-10,"xmax":10,"ymax":10}},"expressions":{"list":[{"type":"text","id":"18","text":"play the animation"},{"type":"expression","id":"7","color":"#c74440","latex":"T=2.37","slider":{"hardMin":true,"hardMax":true,"min":"0","max":"N+1"}},{"type":"text","id":"12","text":"sides on the polygon"},{"type":"expression","id":"1","color":"#c74440","latex":"N=5","slider":{"hardMin":true,"hardMax":true,"min":"1","step":"1"}},{"type":"text","id":"14","text":"\"radius\" of the polygon"},{"type":"expression","id":"6","color":"#000000","latex":"R=6","slider":{"hardMin":true,"hardMax":true,"min":"0"}},{"type":"expression","id":"44","color":"#c74440","latex":"r=1","hidden":true,"slider":{"hardMin":true,"hardMax":true,"min":"0","max":"R"}},{"type":"expression","id":"51","color":"#388c46","latex":"C\\left(x\\right)=\\left(\\cos x,\\sin x\\right)"},{"type":"text","id":"16","text":"point number k around the polygon"},{"type":"expression","id":"8","color":"#2d70b3","latex":"P\\left(k\\right)=RC\\left(\\frac{k\\tau}{N}\\right)"},{"type":"expression","id":"4","color":"#6042a6","latex":"\\operatorname{polygon}\\left(\\left[P\\left(i\\right)\\operatorname{for}i=\\left[1...N\\right]\\right]\\right)","fill":false},{"type":"text","id":"20","text":"walk around the polygon with linear interpolation and annoying angle arithmetic"},{"type":"expression","id":"53","color":"#000000","latex":"K\\left(x\\right)=3u^{2}-2u^{3}\\operatorname{with}u=\\max\\left(\\min\\left(\\frac{3}{2}\\operatorname{mod}\\left(x,1\\right)-\\frac{1}{4},1\\right),0\\right)","hidden":true},{"type":"expression","id":"55","color":"#2d70b3","latex":"A\\left(t\\right)=\\frac{\\tau}{2}+\\frac{\\tau}{2}\\left(1-\\frac{2}{N}\\right)\\left(-t-\\frac{1}{2}\\right)","hidden":true},{"type":"image","id":"5","image_url":"data:image/png;base64,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","name":"blazer_small.png","center":"P\\left(\\operatorname{floor}\\left(T\\right)\\right)+K\\left(T\\right)\\left(P\\left(\\operatorname{ceil}\\left(T\\right)\\right)-P\\left(\\operatorname{floor}\\left(T\\right)\\right)\\right)","angle":"A\\left(\\operatorname{floor}\\left(T-\\frac{1}{2}\\right)+K\\left(T-\\frac{1}{2}\\right)\\right)","width":"-1","height":"1.5"},{"type":"text","id":"22","text":"angle visuals (to display when crossed)"},{"type":"expression","id":"49","color":"#c74440","latex":"\\left(P\\left(\\left[1...N\\right]\\right)\\right)+rC\\left(t+\\left(\\left[1...N\\right]+\\frac{1}{2}+\\frac{1}{4}N\\right)\\frac{\\tau}{N}\\right)\\left\\{\\left[1...N\\right]\\le\\operatorname{floor}\\left(T\\right)\\right\\}","parametricDomain":{"min":"","max":"\\frac{\\tau}{2}\\left(1-\\frac{2}{\\left(N\\right)}\\right)"},"domain":{"min":"0","max":"\\frac{\\tau}{2}\\left(1-\\frac{2}{\\left(N\\right)}\\right)"}}]}}