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:

In[1]:=
circ = Entity["PlaneCurve", "Circle"]["ParametricEquations"][a][t]
Out[1]=
In[2]:=
ResourceFunction["CesaroEquation"][circ, t, {\[Kappa], s}]
Out[2]=

Cesàro equation for the logarithmic spiral:

In[3]:=
ResourceFunction["CesaroEquation"][
 Entity["PlaneCurve", "LogarithmicSpiral"]["ParametricEquations"][a, b][t], t, {\[Kappa], s}]
Out[3]=

Equations can be simplified using assumptions:

In[4]:=
FullSimplify[%, {s, \[Kappa], a, b} \[Element] PositiveReals]
Out[4]=

Cesàro equations for several curves:

In[5]:=
{#, 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
Out[5]=

Properties and Relations (2) 

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

In[6]:=
Entity["PlaneCurve", "TschirnhausenCubic"][
    EntityProperty["PlaneCurve", "CesaroEquation"]][a][
  s, \[Rho]] // Expand
Out[6]=

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

In[7]:=
tc = Entity["PlaneCurve", "TschirnhausenCubic"][
     "ParametricEquations"][a][t];
In[8]:=
ResourceFunction["CesaroEquation"][tc, t, {1/\[Rho], s}] // Expand
Out[8]=

Cesàro equation for the Cornu spiral:

In[9]:=
cs = Entity["PlaneCurve", "CornuSpiral"]["ParametricEquations"][a][t]
Out[9]=
In[10]:=
ResourceFunction["CesaroEquation"][cs, t, {\[Kappa], s}]
Out[10]=

Curvature is linear:

In[11]:=
Simplify[ResourceFunction["Curvature"][cs, t], a > 0]
Out[11]=

Alfred Gray’s generalization of the Cornu spiral:

In[12]:=
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]

The clothoid for n, a both set to unity gives the Cornu spiral:

In[13]:=
clothoid[1, 1][t]
Out[13]=

Plot the clothoid:

In[14]:=
ParametricPlot[Evaluate[clothoid[1, 1][t]], {t, -10, 10}, Ticks -> None]
Out[14]=

Different values give other types of special functions:

In[15]:=
clothoid[3, 1][t]
Out[15]=

The clothoid for general values n, a:

In[16]:=
cl = clothoid[n, a][t]
Out[16]=

Find the Cesàro equation for the clothoid:

In[17]:=
FullSimplify[
 ResourceFunction["CesaroEquation"][cl, t, {\[Kappa], s}], {n, a, t} \[Element] PositiveReals]
Out[17]=

The curvature takes the simpler form:

In[18]:=
FullSimplify[
 ResourceFunction["Curvature"][clothoid[n, a][t], t], {n, a, t} \[Element] PositiveReals]
Out[18]=

A plot of quadratic intrinsic curvature:

In[19]:=
ResourceFunction["CurvaturePlot"][ t^2, {t, -10, 10}]
Out[19]=

Publisher

Enrique Zeleny

Version History

  • 1.0.0 – 01 June 2020

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