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 LeeInterpolatingNodes

Source Notebook

Generate interpolating nodes from points on a curve

Contributed by: Jan Mangaldan

ResourceFunction["LeeInterpolatingNodes"][{{x1,y1,…},{x2,y2,…},…}]

yields centripetal interpolating nodes suitable for use with Interpolation from points assumed to be lying on a space curve.

ResourceFunction["LeeInterpolatingNodes"][{{x1,y1,…},{x2,y2,…},…},a]

yields Lee's interpolating nodes with parameter a.

Details

The parameter a is usually taken to be between 0 and 1.
Typical values of a used are:
0uniform parametrization
1/2centripetal parametrization
1chord length parametrization

Examples

Basic Examples (3) 

Generate centripetal interpolating nodes from a given set of points:

In[1]:=
pts = {{1., 2.}, {0., 1.}, {2., 0.}, {2., 2.}, {3., 3.}, {5., 2.}}; tval = ResourceFunction["LeeInterpolatingNodes"][pts]
Out[1]=

Use Interpolation to get an interpolating curve:

In[2]:=
cc = Interpolation[Transpose[{tval, pts}], Method -> "Spline"]
Out[2]=

Plot the curve along with the original points:

In[3]:=
ParametricPlot[cc[t], {t, 0, 1}, Epilog -> {Directive[AbsolutePointSize[5], ColorData[97, 4]], Point[pts]}]
Out[3]=

Scope (3) 

A set of 3D points:

In[4]:=
pts = CompressedData[" 1:eJx1lHs01Hkchn/GbcY1Y2RsyShKV9V2Ia3PV1K6iFTDKkS0JdXmkkgpE5WS UJvjFlYu1ZaQUpnvb5SuYqZk5DJjzIymQSvDFKXdzp49+98+57zn/fs57zmv dfB+71AKQRDoezSJ/yCv/oPev63EXh9jLnzoV+GjVg+SQnP1SRtxxeacAx/w Ku+MT27xEtxfGBns4UgjI5aar/RKHcCfIbg3X/MKzOcfPHG/VYtslQRs1aUo sG9v6uO5yV3QYrdGYu3wF75eu3qQa96Fl5zzd2kwlcOPkxLuWsuG8Ya87Kpo /TuY2k7ElOX0gJlLANM9XIHPpLcvrZcIIEjDU+Tl8AI0gjkg/CLAqiOs0oar 3RAXPVtAu9qJLdIcy3ZO4YNd7sz0wwNd8NTA9aw9ZwQH2SgX1Kr7oSBEOGu0 owr8Z+ucZTtQyNXfYjKnwhjonS6uDIoX4rbxiBzb5VTSPL6Kb7VME1U7xe40 iZTgm4f0mtlD+mRZPs33BpuGbFSxq3Sj2/E+bW9bpbchuYgXKF4eaYykq2MC 2rKf4N2/2XAXhxuQu5MzOCGVdORsMdzlNrkYCx2XsBou0cjd1/ItfY3NUO13 /9jIm7DF11M1312b9F/iOaFm+kQ0fFh0mDnvJSTWV7cuNSfIbfdoVpaPzNBn SKu02yUC2WXCAT8awSfFC4uPH2WgbeKeMBDJoNj200b+x/c4soLYNO8bHRVs rDF/FyKF0mxuY3SmELO33YiLfm6ChkPrk6JNW+FU40W36v6LcDrZqbBxiwlS 5Edp7Om7iHVuc9sdT7TDcGHnX2flJmipA7vjZFgbNlnZV6Cu6QODUOOqzBmm yI+ZVflOLcWca2kmpRI1LNjKtPlYxkCNS7j7Z3FkmOfwsjurgkCy126ul7vM UFCnVrm8rBMXGNF9Fa+00cH3sscWkyeiCAvF8UTN57jpUH7D6DgNSUUHN1xp Y6CoCOqQj1Y5Zp9p9MEFBsi0OSLhdhAdtfF8frnvchWuvBKrrNcYokEz52vt D43QWuX85XF5z8EnsOvh+Sv6aJfq3bT1pVRkI8U6gtEOUI6n1ARd10V8fnpu XQoFKWBSdnyYBOSWUxjccA30EyOjaUj6GZyzGJb0Fy0wc9+K1cubVcAYQT7q KCWEkwLunXISHxoSZFwmhCA9eO6OsuUJiOwD5Y/SxXhlXaUq0V6Aby20T7WL fYON7y5DOnO6MV3k5x3wsxR7eCmmui9W4iLbCyPXe5owO9XapfaJDLuHI1Ha 1xF8rvJXK1YfCURqZZ3DjA5M1GonJC4mSKnjsn0jeiJgxsblLRxMhyjy1gv+ Dm1yo2tbPtKQg/miw8OMt1LY+4HanpRHI9f6vcoyWSCFb9NNDNegEehvoPOM KvVJTeZcu7sqIezMEl/2uEmgkKbglIx1emRowNrOsep6sEqol0Qc0UFapWfo zs+1SIPHne9ZOzMhPTLU9WWTHvKcOnNi08JRHMQumZLNrMHvPItc9/oaIN5k z2S/hx1YmFtxzDZNgFu0boTdF+khwqVLUf6sF4yqwL1ujxjr3Dqivu6oi+ZT D6QcuzcOxf5lD2Ql3ZiSo6w96UFB3YXrndZ90UV+IY3d+d3PcAU913ja+ChM nTZptMRvAqrxkDd2/sEHNJY9MHP7IMQxW4tc1zMQxfbpJy+WBBQxFj+cD5DA 7xCcwokyR+cN3hJe9Z0Q5VNwLbOEB2VF7BdqewtUFOjs8WYvDyrGVmyX5zTj bJY/JaiHiQa+dMSe39aEuRmcza1b5bhX16hp782JyO1S8w4Ny258dLbWSa5o CM85mpdRBqbIvE4mpdSK8Qmb1+VbB7/iHsM+w75NRqgmcvr+Sf48fIFF+B+q 1yTHsv/kBEZoI/nwjIvJCS2wIK6fOrqbShZwojRZlz5DC9/1XkqwBLJ832iH pXzfaf/T2wLtt+DU9uSAfnw7OEZLjiWBAcm6xNq4z06BF+XQhKXCanB/XF3g 9JpG7kgafzpk9Q3b8DjU9wOvcO+ErJLCYS1S7Zurtr+hQ04/pecXMU2G/+/n /wakyv8F "];

Generate an interpolating curve with uniform parametrization:

In[5]:=
c1 = Interpolation[ Transpose[{ResourceFunction["LeeInterpolatingNodes"][pts, 0], pts}], Method -> "Spline"]; ParametricPlot3D[c1[t], {t, 0, 1}]
Out[5]=

Generate an interpolating curve with chord length parametrization:

In[6]:=
c1 = Interpolation[ Transpose[{ResourceFunction["LeeInterpolatingNodes"][pts, 1], pts}], Method -> "Spline"]; ParametricPlot3D[c1[t], {t, 0, 1}]
Out[6]=

Applications (3) 

A set of 3D points:

In[7]:=
cpts = CompressedData[" 1:eJx1lPs70wscgGfjLHUeCdXpPBVKUul0kUvE59sZorvSRTno4lJiiuQpRbqI qFglp9ZFcktMW+Ekn68kUxK2sZWzuaZGmjEMrdPT7+d9nvd5/4PXdC9ziz+V QqEQP6T9MO8nSnz4s1SS8hMSx3x8DHOeynHipEUrS8toZMaExTMKE9/ikkov vaATbdikX5q4K5hGLl9blh/m0YR6oixvp6BGjBG4FWubU8kln7IGiwta0FMS /uTo1GyMOFjXtNibQjYNXhxJ75Yh37Uq5dpxPugIDdJffBxHB6k8rWOzFG1t xWXNSgkc1ESf/WykRp7Y+x7PQ4zbOC/amUQ7CCy6mfoMFX5On1uqCa3DX9qS VlcZdYHTVo/dtof70ULnhsuNP4tQWOGv+ZrZCVcv321Ybd+DBolzjg/YlMHg g2rgNbeCjm12OW9dB85Ms6QwlzcAf+cye30rIbhyOt5kbBahK5P6yXuBGDan sGusV90ERejFfmujW5haAQsHxiVgwh2uCe8Qopn2ps8pX4XAOPltyZRaIbzu W16ol/IRZ1+o52od6gZvi6fRp5JfQOtfCbPmP1bgBDrbIbdGCZwNYyGF27hY 4B88YarvIK7RHGPNgVHgB4y5uLMb8N01japjpQprCsxSzL5SCH7agUi/BRJM Nsysjq1RIpEtYoWwtYlpXE2va6AE8xvsNhSm9KCyYsh+iKQTflaXtK60CnCu AUzaursFmd2sUDe5LmGpXQW3p1Zh/Z2DQ70BN0HgF1H5NXciMaHHKV43+Sxc F0mtba7LQK76Z7pVmS6huq1+pMOshqJdzisWdPZCVPvVk26H6cSXWzeNvpQI wbdDkeu8YgBGHId+zbLTJn4bWpduyJVAXs5Y0rINKnhvLPi34jSFyDzMUO/l iGHEft/dSKNBiHNZ1Tbkr4aW4Eam1KcegiuCrBrmK+DKHasrBSP98Em+9WjU cC6MvlJ3H3LogiZ8RP/ysQuSYj+N8BKqcCWTHmVj1Qj73WNN5ts1AnV00GtH hQj1C5qlqWOFaC5XpLB3cdB23r5C7oz3OHNynfUvoyK8knAmKcCrGdtZIY28 9GYUHyiZ7XmoDfMf353hYtOJazHV4MYf9bgzMcTURtqJ4ZqNzjHTe3EKIyK0 0rYYjXaFOpV/7kQ3n4/PhNJ+ZLh7a9XueAqaNLpM/2Ubiq2i6x9xVJjz0Hj7 W696MI7YbrE/QYzHdA26zsepUbDTvy5+kQS0n+vY2i2sQn1ZttrX8xuuF6wz Jcuk4JOQ6swOfAxRSvrY36kUsvJOxj3ZCxnILn6QGi4XQpHh/cC5W6lkm3sL 58O0Fkh47dNqHNkOhg8sFubH0Uj3B5VO+adEcCrSi/VQ0ANLL2cuxCIaybVx Da2RvQHzRc31sZsHIMi4dJ8qjErmTwPj9JjncOZsdeesQDUMPHH19zD6jkUv 4wrrFuQAGluuZDC/g+Zb7SxR0xBS9pbMPzbsjpoAvU1UFyphdL/Oyz5Wjg+H +ZldFrlozRH32/fSiL4dGH7e5g36i/vCF7uW41PVNV7Eem3i/bAZg93XBD1h XIWJfS2aGDTOPL+RRtRmDfgej+uBCPnkKu8xIf7utUcnREkhHMP4UcuODIK8 rk8LL0uQfi2+ZUvYKJjrOhjXXFfDOYnjnSOOEkw/4B6dtE0BE3c47B8eHYeD yarJJk0N+Iz/jr9H2ABixunG0CQNrMho5R2I4aHjS9az6ksyzC9v7aGu08B7 zeyjLZoKOKI28w0LGMBz1V4crs042KXVTmbdEkLezCy/d2/H8f5z+XWaxwiw I0sti9USWCSKCdFFLXLT5YbxkmIlJNvRspU5Yjj66nsepZRGbl+xTdeP0Q1s TeATuaUA3Nobg956apMlG2IEs6xfw5k84l21Mx/0gif2rGHRyJsv6bZasWI8 UTvvvunxElDPeRLpcVeLZAUITpZTFTgaXyJOTL8Hpy7dGpLWfsM9CoG83lON e6JDMrjLLmC3lGW6NG8Yq3PDLPkxFPKZufvs3XYc/L/P/wdogDkU "];

Generate the corresponding knots and control points for a B-spline curve:

In[8]:=
With[{deg = 3, n = Length[cpts]}, tvals = ResourceFunction["LeeInterpolatingNodes"][cpts]; kp = Join[ConstantArray[0, deg + 1], MovingAverage[ArrayPad[tvals, -1], deg], ConstantArray[1, deg + 1]]; cp = LinearSolve[ Table[BSplineBasis[{deg, kp}, j - 1, tp], {tp, tvals}, {j, n}], cpts]] // Short
Out[8]=

Visualize the B-spline curve and the points:

In[9]:=
Graphics3D[{{Directive[ColorData[97, 1], AbsoluteThickness[2]], BSplineCurve[cp, SplineDegree -> 3, SplineKnots -> kp]}, {Directive[ColorData[97, 4], AbsolutePointSize[6]], Point[cpts]}}, PlotRange -> All]
Out[9]=

Neat Examples (1) 

Demonstrate the effect of varying the parameter for the Lee interpolating nodes:

In[10]:=
Manipulate[ With[{iF = Interpolation[ Transpose[{ResourceFunction["LeeInterpolatingNodes"][pex, a], pex}], Method -> "Spline"]}, ParametricPlot[iF[t], {t, 0, 1}, Epilog -> {Directive[AbsolutePointSize[5], ColorData[97, 4]], Point[pex]}, Sequence[ Frame -> True, PlotRange -> {{0, 31}, {0, 31}}, PlotRangePadding -> Scaled[0.05]]]], {{a, 1/2}, 0, 1}, Initialization :> (pex = {{0., 0.}, {1.34, 5.}, {5., 8.66}, {10., 10.}, {10.6, 10.4}, {10.7, 12.}, {10.7, 28.6}, {10.8, 30.2}, { 11.4, 30.6}, {19.6, 30.6}, {20.2, 30.2}, {20.3, 28.6}, {20.3, 12.}, {20.4, 10.4}, {21., 10.}, {26., 8.66}, {29.66, 5.}, {31., 0.}};), SaveDefinitions -> True]
Out[10]=

Version History

  • 1.0.0 – 05 April 2021

Source Metadata

  • Citation:
    • Lee E.T.Y., "Choosing nodes in parametric curve interpolation." Computer-Aided Design, vol. 21, no. 6, 363–370, 1989. DOI: 10.1016/0010-4485(89)90003-1

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