Function Repository Resource:

CesaroEquation

Source Notebook

Compute the Cesàro equation for a planar curve

Contributed by: Wolfram Staff (original content by Alfred Gray)

ResourceFunction["CesaroEquation"][c,t,{κ,s}]

computes the Cesàro equation for curvature κ and arc length s of a planar curve c parametrized by t.

Examples

Basic Examples (3)

Cesàro equation for the circle:

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$ResourceFunction["CesaroEquation"][circ, t, {\[Kappa], s}]$

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Cesàro equation for the logarithmic spiral:

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$ResourceFunction["CesaroEquation"][ Entity["PlaneCurve", "LogarithmicSpiral"]["ParametricEquations"][a, b][t], t, {\[Kappa], s}]$

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Equations can be simplified using assumptions:

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$FullSimplify[%, {s, \[Kappa], a, b} \[Element] PositiveReals]$

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Cesàro equations for several curves:

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${#, ResourceFunction["CesaroEquation"][#["ParametricEquations"][a][t], t, {\[Kappa], s}]} & /@ {Entity["PlaneCurve", "Catenary"], Entity["PlaneCurve", "CornuSpiral"], Entity["PlaneCurve", "CircleInvolute"], Entity["PlaneCurve", "Tractrix"], Entity["PlaneCurve", "Astroid"]} // FullSimplify // Grid$

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Properties and Relations (2)

For some curves, entities can be used to get the Cesàro equation:

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$Entity["PlaneCurve", "TschirnhausenCubic"][ EntityProperty["PlaneCurve", "CesaroEquation"]][a][ s, \[Rho]] // Expand$

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This can instead be computed by CesaroEquation with κ=1/ρ:

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$ResourceFunction["CesaroEquation"][tc, t, {1/\[Rho], s}] // Expand$

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Cesàro equation for the Cornu spiral:

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$ResourceFunction["CesaroEquation"][cs, t, {\[Kappa], s}]$

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Curvature is linear:

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Alfred Gray’s generalization of the Cornu spiral:

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$clothoidprime[n_, a_][t_] := {a Sin[t^(n + 1)/(n + 1)], a Cos[t^(n + 1)/(n + 1)]} clothoid[n_, a_][s_] := Integrate[clothoidprime[n, a][ss], {ss, 0, s}, Assumptions -> t > 0 && n > 0]$