{"version":8,"graph":{"viewport":{"xmin":-3.6507512618755737,"ymin":-4.561325581649294,"xmax":9.432853753340744,"ymax":10.08254254932987}},"randomSeed":"75953d8c41577c45511118c14bdf50a6","expressions":{"list":[{"type":"text","id":"6","text":"from https://iquilezles.org/www/articles/minispline/minispline.htm"},{"type":"expression","id":"16","color":"#6042a6","latex":"D\\left(C,V\\right)=C\\left[1\\right]V\\left[1\\right]+C\\left[2\\right]V\\left[2\\right]+C\\left[3\\right]V\\left[3\\right]+C\\left[4\\right]V\\left[4\\right]"},{"type":"expression","id":"1","color":"#c74440","latex":"p_{0}=\\left(7.77,3.85\\right)"},{"type":"expression","id":"2","color":"#fa7e19","latex":"p_{1}=\\left(5.13,-0.44\\right)"},{"type":"expression","id":"7","color":"#000000","latex":"p_{2}=\\left(0.82,1.93\\right)"},{"type":"expression","id":"8","color":"#2d70b3","latex":"p_{3}=\\left(-1.76,1.41\\right)"},{"type":"expression","id":"17","color":"#000000","latex":"p_{v}=\\left[p_{0},p_{1},p_{2},p_{3}\\right]","hidden":true},{"type":"text","id":"19","text":"P(0) = p1 => a = p1"},{"type":"expression","id":"13","color":"#c74440","latex":"a=D\\left(\\left[0,1,0,0\\right],p_{v}\\right)","hidden":true},{"type":"text","id":"21","text":"P'(0) = (p2 - p0) / 2 => b = (p2 - p0) / 2"},{"type":"expression","id":"12","color":"#000000","latex":"b=D\\left(\\left[-\\frac{1}{2},0,\\frac{1}{2},0\\right],p_{v}\\right)","hidden":true},{"type":"text","id":"23","text":"P(1) = p2 => a + b + c + d = p2 => c + d = 1/2 p0 - p1 + 1/2 p2"},{"type":"text","id":"25","text":"P'(1) = (p3 - p1) / 2 => b + 2c + 3d = (p3 - p1) / 2 => 2c + 3d = 1/2 p0 -1/2 p1 - 1/2 p2 + 1/2 p3"},{"type":"text","id":"31","text":"[1, 1 | 2, 3] ^ -1 = [3, -1 | -2, 1]"},{"type":"text","id":"33","text":"c = 3(1/2 p0 - p1 + 1/2 p2) - (1/2 p0 - 1/2 p1 - 1/2 p2 + 1/2 p3) = p0 - 5/2 p1 + 2p2 - 1/2 p3"},{"type":"text","id":"35","text":"d = -2(1/2 p0 - p1 + 1/2 p2) + (1/2 p0 - 1/2 p1 - 1/2 p2 + 1/2 p3) = -1/2 p0 + 3/2 p1 - 3/2 p2 + 1/2 p3"},{"type":"expression","id":"11","color":"#6042a6","latex":"c=\\frac{1}{2}D\\left(\\left[2,-5,4,-1\\right],p_{v}\\right)","hidden":true},{"type":"expression","id":"9","color":"#2d70b3","latex":"d=\\frac{1}{2}D\\left(\\left[-1,3,-3,1\\right],p_{v}\\right)","hidden":true},{"type":"expression","id":"14","color":"#2d70b3","latex":"P\\left(t\\right)=a+bt+ct^{2}+dt^{3}"},{"type":"expression","id":"15","color":"#388c46","latex":"P\\left(t\\right)","parametricDomain":{"min":"-10","max":"10"},"domain":{"min":"-10","max":"10"}}]}}