lagrange-multipliers #

Shows a moving point constrained to g(x,y)=0 (a circle) while minimizing f(x,y)=x^2+2y^2. At the constrained optimum, the visualization highlights gradient parallelism by drawing ∇f, ∇g, and λ∇g from the same point and displaying the stationarity condition ∇f = λ∇g alongside feasibility g(x)=0.

canvasclick to interact

⏮◀◀▶▶STEP0.25x1xZOOM

t=0s

practical uses #

01.Constrained optimization in engineering design (e.g., minimize weight subject to strength constraints)
02.Economics/utility maximization under budget constraints
03.Machine learning optimization with equality constraints (e.g., normalization constraints)

technical notes #

Pure Canvas2D. The point evolves via tangential descent: v = -∇f + λ∇g with λ = (∇f·∇g)/(∇g·∇g) to make v orthogonal to ∇g, then re-projected to the circle to enforce g(x)=0. Contours are polyline approximations snapped to a pixel grid for a blocky aesthetic; animation is time-based with a 3–5s cycle and periodic restarts.

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