B-Spline how to create control points for a curve to pass through knot values












0












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I want to create a b-spline curve that will pass through all the (knot) points I give it. How do I construct it? Do I need to find the control points for that curve? And if so - how?






spline







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  • $begingroup$
    Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
    $endgroup$
    – Oppenede
    Dec 29 '18 at 21:02
















0












$begingroup$


I want to create a b-spline curve that will pass through all the (knot) points I give it. How do I construct it? Do I need to find the control points for that curve? And if so - how?






spline







share|cite|improve this question









$endgroup$












  • $begingroup$
    Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
    $endgroup$
    – Oppenede
    Dec 29 '18 at 21:02














0












0








0





$begingroup$


I want to create a b-spline curve that will pass through all the (knot) points I give it. How do I construct it? Do I need to find the control points for that curve? And if so - how?






spline







share|cite|improve this question









$endgroup$




I want to create a b-spline curve that will pass through all the (knot) points I give it. How do I construct it? Do I need to find the control points for that curve? And if so - how?






spline




spline






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 29 '18 at 9:18









David RefaeliDavid Refaeli

1216




1216












  • $begingroup$
    Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
    $endgroup$
    – Oppenede
    Dec 29 '18 at 21:02


















  • $begingroup$
    Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
    $endgroup$
    – Oppenede
    Dec 29 '18 at 21:02
















$begingroup$
Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
$endgroup$
– Oppenede
Dec 29 '18 at 21:02




$begingroup$
Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
$endgroup$
– Oppenede
Dec 29 '18 at 21:02










2 Answers
2






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oldest

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0












$begingroup$

Suppose that your knot points (not to be confused with the B-spline knots!) are ordered in a sequence $(x_1, dots, x_m)$. Then you can compute a B-spline of order $n$ between each ordered pair $(x_i, x_{i+1})$. You can adjust $n$ in order to set the B-spline regularity.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    can you maybe show an example? say I have (x1,y1) (x2,y2) and (x3,y3). How do I construct a B-spline that passes through all these points ?
    $endgroup$
    – David Refaeli
    Dec 29 '18 at 14:50



















0












$begingroup$

Follows a MATHEMATICA script showing how to construct a special spline type. (Bézier curves) In the script $BL, BQ,BC$ are Bézier curves of first, (linear), second (quadratic) and third (cubic) order and also how them are recursively built. The points $p_1,p_2,p_2,p_4$ can be considered as control points. By properly positioning these control points, we can cause the curve to pass through determined points



$$
begin{array}{l}
{text{BL}[text{p0$_$},text{p1$_$},text{t$_$}]text{:=}(1-t)text{p0}+t text{p1}}\
{text{BQ}[text{p0$_$},text{p1$_$},text{p2$_$},text{t$_$}]text{:=}(1-t)text{BL}[text{p0},text{p1},t]+t text{BL}[text{p1},text{p2},t]}\{text{BC}[text{p0$_$},text{p1$_$},text{p2$_$},text{p3$_$},text{t$_$}]text{:=}(1-t)text{BQ}[text{p0},text{p1},text{p2},t]+t text{BQ}[text{p1},text{p2},text{p3},t]}
end{array}
$$



$$
begin{array}{l}
{p_1={0,0};}\
{p_2={1,1};}\
{p_3={-1,2};}\
{p_4={-2,0};}\
{text{gr1}=text{ParametricPlot}left[text{BC}left[p_1,p_2,p_3,p_4,tright],{t,0,1},text{PlotStyle}to {text{Blue},text{Thick}}right];}\
{text{grp}=text{Table}left[text{Graphics}left[left{text{Red},text{Disk}left[p_k,0.05right]right}right],{k,1,4}right];}\
{text{grl}=text{ListLinePlot}left[left{p_1,p_2,p_3,p_4right},text{PlotStyle}to {text{Dashed}, text{Red}}right];}\
{text{Show}[text{gr1},text{grp},text{grl},text{PlotRange}to text{All}]}
end{array}
$$



Attached a plot for this setup



enter image description here






share|cite|improve this answer









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    2 Answers
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    2 Answers
    2






    active

    oldest

    votes









    active

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    active

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    0












    $begingroup$

    Suppose that your knot points (not to be confused with the B-spline knots!) are ordered in a sequence $(x_1, dots, x_m)$. Then you can compute a B-spline of order $n$ between each ordered pair $(x_i, x_{i+1})$. You can adjust $n$ in order to set the B-spline regularity.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      can you maybe show an example? say I have (x1,y1) (x2,y2) and (x3,y3). How do I construct a B-spline that passes through all these points ?
      $endgroup$
      – David Refaeli
      Dec 29 '18 at 14:50
















    0












    $begingroup$

    Suppose that your knot points (not to be confused with the B-spline knots!) are ordered in a sequence $(x_1, dots, x_m)$. Then you can compute a B-spline of order $n$ between each ordered pair $(x_i, x_{i+1})$. You can adjust $n$ in order to set the B-spline regularity.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      can you maybe show an example? say I have (x1,y1) (x2,y2) and (x3,y3). How do I construct a B-spline that passes through all these points ?
      $endgroup$
      – David Refaeli
      Dec 29 '18 at 14:50














    0












    0








    0





    $begingroup$

    Suppose that your knot points (not to be confused with the B-spline knots!) are ordered in a sequence $(x_1, dots, x_m)$. Then you can compute a B-spline of order $n$ between each ordered pair $(x_i, x_{i+1})$. You can adjust $n$ in order to set the B-spline regularity.






    share|cite|improve this answer









    $endgroup$



    Suppose that your knot points (not to be confused with the B-spline knots!) are ordered in a sequence $(x_1, dots, x_m)$. Then you can compute a B-spline of order $n$ between each ordered pair $(x_i, x_{i+1})$. You can adjust $n$ in order to set the B-spline regularity.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Dec 29 '18 at 10:42









    mathcounterexamples.netmathcounterexamples.net

    26.9k22158




    26.9k22158












    • $begingroup$
      can you maybe show an example? say I have (x1,y1) (x2,y2) and (x3,y3). How do I construct a B-spline that passes through all these points ?
      $endgroup$
      – David Refaeli
      Dec 29 '18 at 14:50


















    • $begingroup$
      can you maybe show an example? say I have (x1,y1) (x2,y2) and (x3,y3). How do I construct a B-spline that passes through all these points ?
      $endgroup$
      – David Refaeli
      Dec 29 '18 at 14:50
















    $begingroup$
    can you maybe show an example? say I have (x1,y1) (x2,y2) and (x3,y3). How do I construct a B-spline that passes through all these points ?
    $endgroup$
    – David Refaeli
    Dec 29 '18 at 14:50




    $begingroup$
    can you maybe show an example? say I have (x1,y1) (x2,y2) and (x3,y3). How do I construct a B-spline that passes through all these points ?
    $endgroup$
    – David Refaeli
    Dec 29 '18 at 14:50











    0












    $begingroup$

    Follows a MATHEMATICA script showing how to construct a special spline type. (Bézier curves) In the script $BL, BQ,BC$ are Bézier curves of first, (linear), second (quadratic) and third (cubic) order and also how them are recursively built. The points $p_1,p_2,p_2,p_4$ can be considered as control points. By properly positioning these control points, we can cause the curve to pass through determined points



    $$
    begin{array}{l}
    {text{BL}[text{p0$_$},text{p1$_$},text{t$_$}]text{:=}(1-t)text{p0}+t text{p1}}\
    {text{BQ}[text{p0$_$},text{p1$_$},text{p2$_$},text{t$_$}]text{:=}(1-t)text{BL}[text{p0},text{p1},t]+t text{BL}[text{p1},text{p2},t]}\{text{BC}[text{p0$_$},text{p1$_$},text{p2$_$},text{p3$_$},text{t$_$}]text{:=}(1-t)text{BQ}[text{p0},text{p1},text{p2},t]+t text{BQ}[text{p1},text{p2},text{p3},t]}
    end{array}
    $$



    $$
    begin{array}{l}
    {p_1={0,0};}\
    {p_2={1,1};}\
    {p_3={-1,2};}\
    {p_4={-2,0};}\
    {text{gr1}=text{ParametricPlot}left[text{BC}left[p_1,p_2,p_3,p_4,tright],{t,0,1},text{PlotStyle}to {text{Blue},text{Thick}}right];}\
    {text{grp}=text{Table}left[text{Graphics}left[left{text{Red},text{Disk}left[p_k,0.05right]right}right],{k,1,4}right];}\
    {text{grl}=text{ListLinePlot}left[left{p_1,p_2,p_3,p_4right},text{PlotStyle}to {text{Dashed}, text{Red}}right];}\
    {text{Show}[text{gr1},text{grp},text{grl},text{PlotRange}to text{All}]}
    end{array}
    $$



    Attached a plot for this setup



    enter image description here






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Follows a MATHEMATICA script showing how to construct a special spline type. (Bézier curves) In the script $BL, BQ,BC$ are Bézier curves of first, (linear), second (quadratic) and third (cubic) order and also how them are recursively built. The points $p_1,p_2,p_2,p_4$ can be considered as control points. By properly positioning these control points, we can cause the curve to pass through determined points



      $$
      begin{array}{l}
      {text{BL}[text{p0$_$},text{p1$_$},text{t$_$}]text{:=}(1-t)text{p0}+t text{p1}}\
      {text{BQ}[text{p0$_$},text{p1$_$},text{p2$_$},text{t$_$}]text{:=}(1-t)text{BL}[text{p0},text{p1},t]+t text{BL}[text{p1},text{p2},t]}\{text{BC}[text{p0$_$},text{p1$_$},text{p2$_$},text{p3$_$},text{t$_$}]text{:=}(1-t)text{BQ}[text{p0},text{p1},text{p2},t]+t text{BQ}[text{p1},text{p2},text{p3},t]}
      end{array}
      $$



      $$
      begin{array}{l}
      {p_1={0,0};}\
      {p_2={1,1};}\
      {p_3={-1,2};}\
      {p_4={-2,0};}\
      {text{gr1}=text{ParametricPlot}left[text{BC}left[p_1,p_2,p_3,p_4,tright],{t,0,1},text{PlotStyle}to {text{Blue},text{Thick}}right];}\
      {text{grp}=text{Table}left[text{Graphics}left[left{text{Red},text{Disk}left[p_k,0.05right]right}right],{k,1,4}right];}\
      {text{grl}=text{ListLinePlot}left[left{p_1,p_2,p_3,p_4right},text{PlotStyle}to {text{Dashed}, text{Red}}right];}\
      {text{Show}[text{gr1},text{grp},text{grl},text{PlotRange}to text{All}]}
      end{array}
      $$



      Attached a plot for this setup



      enter image description here






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Follows a MATHEMATICA script showing how to construct a special spline type. (Bézier curves) In the script $BL, BQ,BC$ are Bézier curves of first, (linear), second (quadratic) and third (cubic) order and also how them are recursively built. The points $p_1,p_2,p_2,p_4$ can be considered as control points. By properly positioning these control points, we can cause the curve to pass through determined points



        $$
        begin{array}{l}
        {text{BL}[text{p0$_$},text{p1$_$},text{t$_$}]text{:=}(1-t)text{p0}+t text{p1}}\
        {text{BQ}[text{p0$_$},text{p1$_$},text{p2$_$},text{t$_$}]text{:=}(1-t)text{BL}[text{p0},text{p1},t]+t text{BL}[text{p1},text{p2},t]}\{text{BC}[text{p0$_$},text{p1$_$},text{p2$_$},text{p3$_$},text{t$_$}]text{:=}(1-t)text{BQ}[text{p0},text{p1},text{p2},t]+t text{BQ}[text{p1},text{p2},text{p3},t]}
        end{array}
        $$



        $$
        begin{array}{l}
        {p_1={0,0};}\
        {p_2={1,1};}\
        {p_3={-1,2};}\
        {p_4={-2,0};}\
        {text{gr1}=text{ParametricPlot}left[text{BC}left[p_1,p_2,p_3,p_4,tright],{t,0,1},text{PlotStyle}to {text{Blue},text{Thick}}right];}\
        {text{grp}=text{Table}left[text{Graphics}left[left{text{Red},text{Disk}left[p_k,0.05right]right}right],{k,1,4}right];}\
        {text{grl}=text{ListLinePlot}left[left{p_1,p_2,p_3,p_4right},text{PlotStyle}to {text{Dashed}, text{Red}}right];}\
        {text{Show}[text{gr1},text{grp},text{grl},text{PlotRange}to text{All}]}
        end{array}
        $$



        Attached a plot for this setup



        enter image description here






        share|cite|improve this answer









        $endgroup$



        Follows a MATHEMATICA script showing how to construct a special spline type. (Bézier curves) In the script $BL, BQ,BC$ are Bézier curves of first, (linear), second (quadratic) and third (cubic) order and also how them are recursively built. The points $p_1,p_2,p_2,p_4$ can be considered as control points. By properly positioning these control points, we can cause the curve to pass through determined points



        $$
        begin{array}{l}
        {text{BL}[text{p0$_$},text{p1$_$},text{t$_$}]text{:=}(1-t)text{p0}+t text{p1}}\
        {text{BQ}[text{p0$_$},text{p1$_$},text{p2$_$},text{t$_$}]text{:=}(1-t)text{BL}[text{p0},text{p1},t]+t text{BL}[text{p1},text{p2},t]}\{text{BC}[text{p0$_$},text{p1$_$},text{p2$_$},text{p3$_$},text{t$_$}]text{:=}(1-t)text{BQ}[text{p0},text{p1},text{p2},t]+t text{BQ}[text{p1},text{p2},text{p3},t]}
        end{array}
        $$



        $$
        begin{array}{l}
        {p_1={0,0};}\
        {p_2={1,1};}\
        {p_3={-1,2};}\
        {p_4={-2,0};}\
        {text{gr1}=text{ParametricPlot}left[text{BC}left[p_1,p_2,p_3,p_4,tright],{t,0,1},text{PlotStyle}to {text{Blue},text{Thick}}right];}\
        {text{grp}=text{Table}left[text{Graphics}left[left{text{Red},text{Disk}left[p_k,0.05right]right}right],{k,1,4}right];}\
        {text{grl}=text{ListLinePlot}left[left{p_1,p_2,p_3,p_4right},text{PlotStyle}to {text{Dashed}, text{Red}}right];}\
        {text{Show}[text{gr1},text{grp},text{grl},text{PlotRange}to text{All}]}
        end{array}
        $$



        Attached a plot for this setup



        enter image description here







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 29 '18 at 12:15









        CesareoCesareo

        10k3518




        10k3518






























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