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solving laplace's equation ∇⋅(∇f)=Δf=0 for a height field f given three known points

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Freya Holmér

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49,306 görüntüleme • 4 ay önce •via X (Twitter)

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Still excited about these 3Blue1Brown-style videos generated by AI. Here’s a beautiful illustration of a classic analysis problem: Let f be convex, nonnegative on [0,∞), with f(0)>0 and f(x)→0 as x→∞. Place a light source at (0,b) with 0<b<f(0). Rays reflect off the graph of f and the x-axis following the usual laws of optics. Question: can any ray escape to +∞, or do all trajectories eventually come back? Prompt to video
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Still excited about these 3Blue1Brown-style videos generated by AI. Here’s a beautiful illustration of a classic analysis problem: Let f be convex, nonnegative on [0,∞), with f(0)>0 and f(x)→0 as x→∞. Place a light source at (0,b) with 0<b<f(0). Rays reflect off the graph of f and the x-axis following the usual laws of optics. Question: can any ray escape to +∞, or do all trajectories eventually come back? Prompt to video

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for(float i,z,d,f;i++&lt;1e2;o+=vec4(4,6,8.+z,0)/f-min(dFdx(z)*r.y+z,0.)/exp(d*d/.1)){vec3 p=z*(FC.rgb*2.-r.xyy)/r.y,c=p;p.z+=8.;c.z*=3.;for(f=1.;f++&lt;9.;c+=sin(c.yzx*f+z+t*.5)/f);z+=min(f=.1+abs(.2*c.y+abs(p.y+.8)),d=max(length(p)-3.,.9-length(p-vec3(-1,1,3))))/7.;}o=tanh(o/2e3);
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for(float i,z,d,f;i++&lt;1e2;o+=vec4(4,6,8.+z,0)/f-min(dFdx(z)*r.y+z,0.)/exp(d*d/.1)){vec3 p=z*(FC.rgb*2.-r.xyy)/r.y,c=p;p.z+=8.;c.z*=3.;for(f=1.;f++&lt;9.;c+=sin(c.yzx*f+z+t*.5)/f);z+=min(f=.1+abs(.2*c.y+abs(p.y+.8)),d=max(length(p)-3.,.9-length(p-vec3(-1,1,3))))/7.;}o=tanh(o/2e3);

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