{"version":8,"graph":{"viewport":{"xmin":-2.7967162196891127,"ymin":-8.280815209010285,"xmax":24.2849068577037,"ymax":10.484680327538932}},"randomSeed":"54a3317fc97092a22ee0fc9ea433434d","expressions":{"list":[{"type":"text","id":"69","text":"Dot product (3D)"},{"type":"expression","id":"65","color":"#388c46","latex":"p_{3}\\left(m,n\\right)=m\\left[1\\right]n\\left[1\\right]+m\\left[2\\right]n\\left[2\\right]+m\\left[3\\right]n\\left[3\\right]"},{"type":"text","id":"80","text":"Dot product \"squaring\""},{"type":"expression","id":"78","color":"#2d70b3","latex":"q_{3}\\left(m\\right)=p_{3}\\left(m,m\\right)"},{"type":"text","id":"71","text":"Ray angle (normalised vector)"},{"type":"expression","id":"58","color":"#c74440","latex":"a=\\left[0.3676,-0.2941,0.8823\\right]"},{"type":"text","id":"73","text":"Sphere centre"},{"type":"expression","id":"74","color":"#6042a6","latex":"s=\\left[1,-2,3\\right]"},{"type":"text","id":"82","text":"Sphere radius"},{"type":"expression","id":"83","color":"#000000","latex":"r=1.3","hidden":true},{"type":"text","id":"97","text":"SDF (modified, but zeroes are preserved) based on point"},{"type":"expression","id":"98","color":"#388c46","latex":"d_{p}\\left(p\\right)=q_{3}\\left(p-s\\right)-r^{2}","hidden":true},{"type":"text","id":"76","text":"SDF (modified similarly) based on ray angle"},{"type":"expression","id":"77","color":"#388c46","latex":"d\\left(x\\right)=d_{p}\\left(ax\\right)"},{"type":"text","id":"85","text":"Solving the SDF"},{"type":"expression","id":"86","color":"#2d70b3","latex":"0=d\\left(x\\right)"},{"type":"expression","id":"87","color":"#388c46","latex":"q_{3}\\left(a\\right)x^{2}-2p_{3}\\left(a,s\\right)x+q_{3}\\left(s\\right)-r^{2}"},{"type":"expression","id":"88","color":"#6042a6","latex":"x=\\frac{2p_{3}\\left(a,s\\right)+\\sqrt{4p_{3}\\left(a,s\\right)^{2}-4\\left(q_{3}\\left(s\\right)-r^{2}\\right)}}{2}","hidden":true},{"type":"expression","id":"90","color":"#c74440","latex":"x=p_{3}\\left(a,s\\right)+\\sqrt{p_{3}\\left(a,s\\right)^{2}-q_{3}\\left(s\\right)+r^{2}}"},{"type":"expression","id":"93","color":"#c74440","latex":"x=p_{3}\\left(a,s\\right)-\\sqrt{p_{3}\\left(a,s\\right)^{2}-q_{3}\\left(s\\right)+r^{2}}","lines":true},{"type":"text","id":"92","text":"The raycasting formula's a-dot-s / plus or minus the square root / of a-dot-s, squared, minus s-dot-s / plus radius squared"},{"type":"text","id":"95","text":"https://www.youtube.com/watch?v=2lbABbfU6Zc if you don't get the reference (warning, will get stuck in your head)"}]}}