6
$\begingroup$

I am trying to find de parametric equations of this curve (the curve in that sector of circle): enter image description here

What I've tried:

data = {{2.75, 2.7}, {3.1, 2.84}, {3.45, 2.97}, {3.76, 2.82}, {3.9, 2.45}, {3.72, 2.21}, {3.33, 2.13}, {3.36, 1.87}, {3.65, 1.63}, {3.93, 1.39}, {4.22, 1.15}, {4.55, 0.97}, {4.89, 0.8}, {5.25, 0.7}, {5.41, 1}, {5.4, 1.37}, {5.35, 1.74}, {5.11, 1.96}, {4.8, 1.74}, {4.51, 1.5}, {4.28, 1.21}, {4.06, 0.91}, {3.89, 0.57}, {4.11, 0.27}, {4.45, 0.11}, {2.75, 2.7}, {2.93, 2.79}, {3.18, 2.87}, {3.39, 2.95}, {3.63, 2.96}, {3.78, 2.78}, {3.88, 2.53}, {3.89, 2.27}, {3.63, 2.2}, {3.42, 2.15}, {3.23, 2.08}, {3.34, 1.9}, {3.52, 1.72}, {3.73, 1.58}, {3.91, 1.41}, {4.09, 1.29}, {4.25, 1.12}, {4.49, 1}, {4.74, 0.86}, {4.95, 0.78}, {5.18, 0.69}, {5.34, 0.73}, {5.39, 0.93}, {5.43, 1.11}, {5.41, 1.32}, {5.39, 1.55}, {5.34, 1.77}, {5.24, 1.97}, {5.07, 1.94}, {4.88, 1.81}, {4.71, 1.66}, {4.53, 1.52}, {4.38, 1.37}, {4.24, 1.15}, {4.07, 0.93}, {3.95, 0.75}, {3.89, 0.55}, {4, 0.36}, {4.2, 0.22}, {4.45, 0.11}}

And this is what I've got: enter image description here

Now I've tried to do some Bezier interpolation:

f = SplineFit[data, Bezier]
ParametricPlot[f[x], {x, 0, 62}]

And this is what I've got: enter image description here

This seems to be quite ok. But I don't know how to extract parametric equations from here. Any idea? Thanks!

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ Welcome to the community. I'm not sure what version you're using but executing your code on version 14, gives a different output. Regardless of that quoting documentation "A Bézier curve can be constructed from the sum of the Bernstein polynomials:" 7th example under the "Properties & Relations" section. $\endgroup$
    – Ben Izd
    Commented Mar 3 at 18:18
  • $\begingroup$ @BenIzd i'm executing this code on version 14 too. But from that documentation I don't understand how I can view the parametric equations. $\endgroup$
    – MathLearner
    Commented Mar 3 at 18:28
  • 1
    $\begingroup$ Maybe your data is different. One could use PiecewiseExpand@ Module[{n = Length[data] - 1}, Sum[data[[i + 1]] BernsteinBasis[n, i, x], {i, 0, n}] ] to get the equations. $\endgroup$
    – Ben Izd
    Commented Mar 3 at 18:43

1 Answer 1

Reset to default
6
$\begingroup$
  • Your data have some duplicate points. We can use Mathematica to import the image and right click to Get Coordinates then use Ctrl + c to get the points.
img = Import["https://i.sstatic.net/MKUMm.png"]

enter image description here

bezierFunction[pts_][t_] := 
  Module[{n = Length@pts}, 
   Sum[Binomial[n - 1, i - 1] t^(i - 1) (1 - t)^(n - 1 - (i - 1))*
     pts[[i]], {i, 1, n}]];

Clear["Global`*"];
data = {{313.2857142857143, 
    163.29310825892853`}, {307.92857142857144`, 
    166.50739397321425`}, {302.92857142857144`, 
    170.4359654017857}, {299.7142857142857, 
    176.50739397321425`}, {303.2857142857143, 
    184.00739397321425`}, {307.2142857142857, 
    186.8645368303571}, {310.42857142857144`, 
    191.50739397321425`}, {316.14285714285717`, 
    196.50739397321425`}, {320.7857142857143, 
    201.50739397321425`}, {325.07142857142856`, 
    204.3645368303571}, {329.3571428571429, 
    207.9359654017857}, {332.57142857142856`, 
    205.07882254464286`}, {334., 
    201.50739397321425`}, {334.7142857142857, 
    195.4359654017857}, {334.7142857142857, 
    189.3645368303571}, {334.7142857142857, 
    184.00739397321425`}, {331.5, 
    178.29310825892853`}, {325.07142857142856`, 
    178.29310825892853`}, {320.42857142857144`, 
    181.15025111607142`}, {315.07142857142856`, 
    184.3645368303571}, {310.42857142857144`, 
    186.50739397321425`}, {305.7857142857142, 
    190.79310825892856`}, {301.14285714285705`, 
    195.07882254464283`}, {296.8571428571428, 
    198.29310825892858`}, {291.14285714285717`, 
    202.9359654017857}, {287.2142857142857, 
    206.50739397321425`}, {285.4285714285714, 
    209.36453683035717`}, {290.07142857142856`, 
    212.9359654017857}, {292.9285714285714, 
    212.57882254464283`}, {297.57142857142856`, 
    214.00739397321425`}, {300.7857142857143, 
    216.8645368303571}, {298.6428571428571, 
    223.29310825892856`}, {295.7857142857143, 
    229.0073939732143}, {291.14285714285717`, 
    230.07882254464283`}, {284.71428571428567`, 
    230.4359654017857}, {280.4285714285714, 
    227.9359654017857}, {276.1428571428571, 
    227.22167968750003`}, {273.2857142857143, 226.15025111607142`}};
n = Length@data;
center = {210.78571428571428`, 163.29310825892853`};
Needs["Splines`"];
f = SplineFit[data, Bezier];
plot1 = ParametricPlot[f[x], {x, 0, Length@data}];
(*bezierFunction[pts_][t_]:=Module[{n=Length@pts},Sum[BernsteinBasis[\
n-1,i-1,t]*pts[[i]],{i,1,n}]];*)
bezierFunction[pts_][t_] := 
  Module[{n = Length@pts}, 
   Sum[Binomial[n - 1, i - 1]   t^(i - 1)   (1 - t)^(n - 1 - (i - 1))*
     pts[[i]], {i, 1, n}]];
expr = bezierFunction[data][t];
plot2 := 
  ParametricPlot[expr, {t, 0, 1}, 
   PlotStyle -> 
    Directive@{RandomColor[], Opacity[.5], AbsoluteThickness[2]}];
{plot1, plot2}

expr

enter image description here

  • For the full graphics.
p1 = data[[1]];
p2 = data[[-1]]; 
pts = 
 SequenceFoldList[
  ReflectionTransform[Cross[#2 - center], center]@#1 &, {p1, p2}, 
  Range[5], 2];
Graphics[{Line[{center, #}] & /@ pts, First@plot2, 
  GeometricTransformation[First@plot2, 
   FoldList[RightComposition, 
    Table[ReflectionTransform[Cross[pts[[i]] - center], center], {i, 
      Range[Length@pts - 1]}]]]}, Axes -> True]

enter image description here

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thank you! This is perfect! $\endgroup$
    – MathLearner
    Commented Mar 4 at 21:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.