smoothLimitInside

Once continuously differentiable limitation of x to [l + deltaX, u - deltaX], strictly inside [l, u] (for arguments of functions singular at the bounds, as acos at -1 and 1)

FastSmoothBuildings.Utilities.smoothLimitInside, function (source)

Purpose

Limitation of x strictly inside [l, u], once continuously differentiable: for the argument of a function singular at the bounds, as acos and asin at −1 and 1.

Equations

With δ = deltaX,

y = smoothMin(smoothMax(x, l + δ, δ), u − δ, δ),

with smoothMax and smoothMin. For u − l ≥ 4 δ: y = l + δ for x < l, y = x between l + 2 δ and u − 2 δ, y = u − δ for x > u. In the blends the result passes at most 0.087 δ beyond l + δ and u − δ, so it stays at least 0.91 δ away from the bounds. C1.

Use

The weather and the windows bound the cosines they give to acos and asin with it, whose derivatives are infinite at −1 and 1.

Revisions

• October 2026: moved from trano's FastModel.Utilities.