English
The forward stereographic projection equals the explicit algebraic formula: stereoToFun v x = (2 / (1 − ⟪v,x⟫)) • P x, where P is the orthogonal projection onto the orthogonal complement of v.
Русский
Прямое стереографическое проецирование задаётся явной формулой: stereoToFun v x = (2 / (1 − ⟪v,x⟫)) • P x, где P — ортогональная проекция на ортогональное дополнение к v.
LaTeX
$$$stereoToFun\ v\ x = \left(\dfrac{2}{1 - \langle v, x\rangle}\right) \cdot P_{(\mathbb{R} \cdot v)^{\perp}}(x)$$$
Lean4
/-- A product `M × [x,y]` for `M` boundaryless has boundary `M × {x, y}`. -/
theorem boundary_product [I.Boundaryless] : (I.prod (𝓡∂ 1)).boundary (M × Icc x y) = Set.prod univ {⊥, ⊤} := by
rw [I.boundary_of_boundaryless_left, boundary_Icc]
