B-Spline how to create control points for a curve to pass through knot values
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
asked Dec 29 '18 at 9:18
David RefaeliDavid Refaeli
1065
add a comment |
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
asked Dec 29 '18 at 9:18
David RefaeliDavid Refaeli
1065
Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
– Oppenede
Dec 29 '18 at 21:02
add a comment |
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
asked Dec 29 '18 at 9:18
David RefaeliDavid Refaeli
1065
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
asked Dec 29 '18 at 9:18
David RefaeliDavid Refaeli
1065
asked Dec 29 '18 at 9:18
David RefaeliDavid Refaeli
1065
asked Dec 29 '18 at 9:18
David RefaeliDavid Refaeli
1065
asked Dec 29 '18 at 9:18
David RefaeliDavid Refaeli
1065
asked Dec 29 '18 at 9:18
David RefaeliDavid Refaeli
1065
1065
Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
– Oppenede
Dec 29 '18 at 21:02
add a comment |
Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
– Oppenede
Dec 29 '18 at 21:02
Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
– Oppenede
Dec 29 '18 at 21:02
Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
– Oppenede
Dec 29 '18 at 21:02
add a comment |
2 Answers
2
active
oldest
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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.
answered Dec 29 '18 at 10:42
mathcounterexamples.netmathcounterexamples.net
25.3k21953
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 ?
– David Refaeli
Dec 29 '18 at 14:50
add a comment |
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
answered Dec 29 '18 at 12:15
CesareoCesareo
8,4113516
add a comment |
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2 Answers
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votes
2 Answers
2
active
oldest
votes
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votes
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.
answered Dec 29 '18 at 10:42
mathcounterexamples.netmathcounterexamples.net
25.3k21953
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 ?
– David Refaeli
Dec 29 '18 at 14:50
add a comment |
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.
answered Dec 29 '18 at 10:42
mathcounterexamples.netmathcounterexamples.net
25.3k21953
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 ?
– David Refaeli
Dec 29 '18 at 14:50
add a comment |
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.
answered Dec 29 '18 at 10:42
mathcounterexamples.netmathcounterexamples.net
25.3k21953
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.
answered Dec 29 '18 at 10:42
mathcounterexamples.netmathcounterexamples.net
25.3k21953
answered Dec 29 '18 at 10:42
mathcounterexamples.netmathcounterexamples.net
25.3k21953
answered Dec 29 '18 at 10:42
mathcounterexamples.netmathcounterexamples.net
25.3k21953
answered Dec 29 '18 at 10:42
mathcounterexamples.netmathcounterexamples.net
25.3k21953
25.3k21953
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 ?
– David Refaeli
Dec 29 '18 at 14:50
add a comment |
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 ?
– David Refaeli
Dec 29 '18 at 14:50
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 ?
– David Refaeli
Dec 29 '18 at 14:50
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 ?
– David Refaeli
Dec 29 '18 at 14:50
add a comment |
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
answered Dec 29 '18 at 12:15
CesareoCesareo
8,4113516
add a comment |
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
answered Dec 29 '18 at 12:15
CesareoCesareo
8,4113516
add a comment |
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
answered Dec 29 '18 at 12:15
CesareoCesareo
8,4113516
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
answered Dec 29 '18 at 12:15
CesareoCesareo
8,4113516
answered Dec 29 '18 at 12:15
CesareoCesareo
8,4113516
answered Dec 29 '18 at 12:15
CesareoCesareo
8,4113516
answered Dec 29 '18 at 12:15
CesareoCesareo
8,4113516
8,4113516
add a comment |
add a comment |
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Is the knot vector fixed in advance, and if yes what knots are used? Also please add an example of 'knot' points
– Oppenede
Dec 29 '18 at 21:02