Function Repository Resource:

ApproximatedCurve

Source Notebook

Get an approximation to a parametric curve

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

ResourceFunction["ApproximatedCurve"][c,{t,t₀,t_f,n}]

computes a line of n points approximating the parametric curve c with t from t₀ to t_f.

ResourceFunction["ApproximatedCurve"][c,{t,t₀,t_f,n},"type"]

gives a result based on "type".

Details and Options

Possible values of "type" are: "Coordinates", "Point", "Line", "PointLine", "Arrow", "Polygon", "BezierCurve", "BSplineCurve", "Tube", "BSplineCurveTube", "Region" or "DiscretizeRegion".

Examples

Basic Examples (5)

Approximate cycles around a figure-eight curve:

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Approximate the curve with lines:

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Show the lines:

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Follow a path with arrows:

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$Graphics[ ResourceFunction[ "ApproximatedCurve"][{Sin[t], Sin[t] Cos[t]}, {t, 0, 2 \[Pi], 30}, "Arrow"]]$

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Define a curve in three dimensions:

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Polygonal approximation to a curve:

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Different approximation to a curve:

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Convert points in the following closed curve into a tube. It requires less file space than a full parametrization plot:

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x[t_] := Cos[3 t]
y[t_] := Cos[4 t + Sqrt[2]]
z[t_] := Cos[5 t + Sqrt[3]]

In[9]:=

Graphics3D[
Tube[ResourceFunction[
"ApproximatedCurve"][{x[t], y[t], z[t]}, {t, 0, 10, 15}, BSplineCurve[#, SplineClosed -> True] &], .025]]

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Applications (6)

Get a curve similar to a handwritten curve:

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Using AnglePath gives a different curve:

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Define an epicycloid:

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Modify it using Accumulate:

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$GraphicsRow[{ParametricPlot[Evaluate[ec], {t, 0, 7 \[Pi]}], Graphics[ Line[Accumulate[ N[ResourceFunction["ApproximatedCurve"][ec, {t, 0, 8 \[Pi], 200}, "Coordinates"]]]]]}]$

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Make a spline from the points of a curve:

In[14]:=

Graphics[
FilledCurve@
ResourceFunction[
"ApproximatedCurve"][{Sin[t], Sin[t] Cos[t]}, {t, 0, 20, 10}, BSplineCurve[#, SplineClosed -> True] &]]

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Make a design from an epicycloid:

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$Show[Graphics[{ResourceFunction["ApproximatedCurve"][ ec, {t, 0, 4 Pi, 20}, "Polygon"]}], ParametricPlot[Evaluate[ec], {t, 0, 4 \[Pi]}]]$

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Define an epicycloid:

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Get a region:

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$ResourceFunction[ "ApproximatedCurve"][ec, {t, 0, 4 \[Pi], 20}, "Region"]$

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Test if it is a region:

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$RegionQ[ResourceFunction["ApproximatedCurve"][ec, {t, 0, 4 \[Pi], 20}, "Region"]]$

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Discretize the region:

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$ResourceFunction[ "ApproximatedCurve"][ec, {t, 0, 4 \[Pi], 20}, "DiscretizeRegion"]$

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Compute some properties:

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Create and visualize a nephroid:

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$Graphics[ Table[Scale[{ColorData["DarkBands"][.025 n], ResourceFunction["ApproximatedCurve"][ Entity["PlaneCurve", "Nephroid"]["ParametricEquations"][1][ t], {t, 0, 2 \[Pi], 12}, "FilledCurve"]}, Log[n]^(-1/2)], {n, 2, 32}]]$

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Properties and Relations (5)

A crude approximation to arc length:

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$Plus @@ ResourceFunction[ "ApproximatedCurve"][{Sin[t], Sin[t] Cos[t]}, {t, 0, 2 \[Pi], 60}, EuclideanDistance @@@ Partition[N@#, 2, 1] &] // Timing$

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In[24]:=

$ArcLength[{Sin[t], Sin[t] Cos[t]}, {t, 0., 2. \[Pi]}] // Timing$

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Successive approximation to arc length versus the number of steps:

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$ListPlot[ Table[Plus @@ ResourceFunction[ "ApproximatedCurve"][{Sin[t], Sin[t] Cos[t]}, {t, 0, 2 \[Pi], n}, EuclideanDistance @@@ Partition[N@#, 2, 1] &], {n, 2, 20}]]$

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Forced undersampling with MaxRecursion and PlotPoints:

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$ParametricPlot[{Sin[t], Sin[t] Cos[t]}, {t, 0, 2 \[Pi]}, MaxRecursion -> 0, PlotPoints -> 15]$

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Create a similar result using ApproximatedCurve:

In[27]:=

$Graphics@ ResourceFunction[ "ApproximatedCurve"][{Sin[t], Sin[t] Cos[t]}, {t, 0, 2 \[Pi], 15}, "Line"]$

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A polygon gives a similar result to FilledCurve:

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$Graphics[ ResourceFunction[ "ApproximatedCurve"][{Sin[4 t], Sin[5 t]}, {t, 0, 2 \[Pi], 150}, "Polygon"]]$

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Generalize CirclePoints:

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$ellipse[a_, b_, t_] := {a Cos[t], b Sin[t]}$

In[30]:=

$Show[Graphics[ ResourceFunction["ApproximatedCurve"][ ellipse[2, 1, t], {t, 0, 2 \[Pi], 10}, "Line"]], ParametricPlot[Evaluate[ellipse[2, 1, t]], {t, 0, 2 \[Pi]}]]$

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Distances between successive points are different:

In[31]:=

$EuclideanDistance @@@ Partition[ ResourceFunction["ApproximatedCurve"][ ellipse[2, 1, t], {t, 0, 2 \[Pi], 10}, "Coordinates"] // N, 2, 1]$

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Show the curve using points:

In[32]:=

$Graphics[ Table[{PointSize[.05 n], Hue[1/n], ResourceFunction["ApproximatedCurve"][ ellipse[2, 1, t], {t, 0, 2 \[Pi], 20}, "Point"]}, {n, 5, 1, -1}]]$