projections #

Shows orthogonal projection geometrically (a vector v, a subspace line S, a sliding candidate u∈S, then the minimizing orthogonal projection with residual r ⟂ S) and connects it to least squares via the projection operator P = A(A^T A)^{-1}A^T, highlighting that for orthogonal projections P is symmetric and idempotent (P^2 = P).

canvasclick to interact

⏮◀◀▶▶STEP0.25x1xZOOM

t=0s

practical uses #

01.Least-squares fitting / regression (Ax̂ is the projection of data b onto Col(A))
02.Computing closest points on lines/planes (graphics, robotics, constraints)
03.Signal denoising by projecting onto a chosen subspace/basis

technical notes #

Two-panel canvas: left panel renders R^2 geometry with a grid-snapped, blocky style; right panel renders P as a 2×2 matrix (using a 2×1 A so P = uu^T) and animates emphasis of P^2=P and P^T=P. Time is segmented into phases (0–1) using ease() to smoothly transition from sliding u to locking at proj_S(v) and then to operator/least-squares text.

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