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Keep the direction that carries the variation.

Move a three-dimensional point cloud onto its first principal component and inspect the lost directions.

Edit this visualization

Preparing the 3D view…

Find the long direction

The centered synthetic cloud is built from independent, symmetric offsets along three orthogonal axes. Its largest variance is along u = (1,1,0)/√2, shown in green.

Project onto one component

Move the control to one to replace each point with u(uᵀx). Faint segments connect the original points to their projections. Intermediate values animate this geometric transition; they are not alternate fitted PCA models.

Understand what is discarded

PCA preserves variance, not class labels or task relevance. The principal axis is exact for this authored dataset. This example does not estimate components from an uploaded dataset or demonstrate a downstream quality improvement.

The equation behind the motion

u = (1,1,0)/√2; x̂ = u(uᵀx)

Synthetic inputs with explicit geometric assumptions. Rotate the view and inspect the model above.

Read the primary source