signedDist_apply — Mathlib

English

The signed distance from p to q in the direction of v is defined as the inner product of the normalized v with q−p, after projecting v onto the relevant subspace.

Русский

Знакованный километраж расстояния в направлении v равен внутреннему произведению нормализации v и вектора q−p после проекции v.

LaTeX

$$$$\\operatorname{signedDist}(v)(p,q) = \\langle \\operatorname{normalize}(v),\, q - p\\rangle.$$$$

Lean4

theorem signedDist_apply :
    signedDist v p =
      (innerSL ℝ (normalize v)).toContinuousAffineMap.comp (ContinuousAffineMap.id ℝ P -ᵥ .const ℝ P p) :=
  rfl

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