Function Repository Resource:

TangentialAngle

Compute the tangential angle of a curve

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

ResourceFunction["TangentialAngle"][c,t]

computes the tangential angle of the curve c parameterized by variable t.

Details and Options

The tangential angle φ of a plane curve is defined as ρ dφ=ds, where ρ is the radius of curvature and s the arc length. Integration gives

, where κ=1/ρ.

Gray (1997) calls it the turning angle.

Examples

Basic Examples (4)

Tangential angle for a curve:

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Tangential angle for several curves:

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$Grid[FullSimplify[({#1, ResourceFunction[ "TangentialAngle"][#1["ParametricEquations"][a][t], t]} &) /@ {Entity["PlaneCurve", "Circle"], Entity["PlaneCurve", "CornuSpiral"], Entity["PlaneCurve", "Cardioid"], Entity["PlaneCurve", "Nephroid"]}, \[Pi]/2 > t > 0]]$

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A more complicated result:

In[3]:=

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The ellipse pedal curve:

In[4]:=

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Plot the curve:

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The tangential angle:

In[6]:=

ResourceFunction["TangentialAngle"][
Entity["PlaneCurve", "EllipsePedalCurve"]["ParametricEquations"][a, b, c, d][t], t] // FullSimplify

Out[6]=

Properties and Relations (2)

For some named curves, it is possible to get the tangential angle using entities, but the output can be written in a different way:

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Evaluating at one point:

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Possible Issues (1)

Some cases may need some time to compute:

In[10]:=

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