Relation between a Bezier curve and B-Spline curve
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While the ideas behind Bezier curves are rather straight forward, I'm really struggling trying to understand B-Splines. I really researched quite a lot about it and still can't figure it out.
I would like to start with the objective: My aim is to chain a few cubic Bezier curves together in an efficient way. I basically have a set of points and I would like to fit a curve through all of them. There is already a nice and simple way to do so that is available online here, that uses constraints on the meeting points (equality of points, derivative and 2nd derivative) to derive the control points.
But, I heard that B-Splines are also used for that purpose, and might be better for it.
The article here shows you can derive Bezier control points out of a B-Spline (in an easy manner). It also states that:
"Whereas an open string of m Bezier curves of degree n involve nm + 1
distinct control points (shared control points counted only once),
that same string of Bezier curves can be expressed using only m + n
B-spline control points (assuming all neighboring curves are C n−1 )."
So,
1) is this the reason why I would like to use B-Splines? If not, what is the reason? Maybe I shouldn't use B-Splines at all?
2) a B-Spline is constructed using a set of control points. Which control points should I supply it, if I want it to pass through my data points?
spline bezier-curve
asked Jan 9 at 14:52
David RefaeliDavid Refaeli
1065
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add a comment |
$begingroup$
While the ideas behind Bezier curves are rather straight forward, I'm really struggling trying to understand B-Splines. I really researched quite a lot about it and still can't figure it out.
I would like to start with the objective: My aim is to chain a few cubic Bezier curves together in an efficient way. I basically have a set of points and I would like to fit a curve through all of them. There is already a nice and simple way to do so that is available online here, that uses constraints on the meeting points (equality of points, derivative and 2nd derivative) to derive the control points.
But, I heard that B-Splines are also used for that purpose, and might be better for it.
The article here shows you can derive Bezier control points out of a B-Spline (in an easy manner). It also states that:
"Whereas an open string of m Bezier curves of degree n involve nm + 1
distinct control points (shared control points counted only once),
that same string of Bezier curves can be expressed using only m + n
B-spline control points (assuming all neighboring curves are C n−1 )."
So,
1) is this the reason why I would like to use B-Splines? If not, what is the reason? Maybe I shouldn't use B-Splines at all?
2) a B-Spline is constructed using a set of control points. Which control points should I supply it, if I want it to pass through my data points?
spline bezier-curve
asked Jan 9 at 14:52
David RefaeliDavid Refaeli
1065
$endgroup$
add a comment |
$begingroup$
While the ideas behind Bezier curves are rather straight forward, I'm really struggling trying to understand B-Splines. I really researched quite a lot about it and still can't figure it out.
I would like to start with the objective: My aim is to chain a few cubic Bezier curves together in an efficient way. I basically have a set of points and I would like to fit a curve through all of them. There is already a nice and simple way to do so that is available online here, that uses constraints on the meeting points (equality of points, derivative and 2nd derivative) to derive the control points.
But, I heard that B-Splines are also used for that purpose, and might be better for it.
The article here shows you can derive Bezier control points out of a B-Spline (in an easy manner). It also states that:
"Whereas an open string of m Bezier curves of degree n involve nm + 1
distinct control points (shared control points counted only once),
that same string of Bezier curves can be expressed using only m + n
B-spline control points (assuming all neighboring curves are C n−1 )."
So,
1) is this the reason why I would like to use B-Splines? If not, what is the reason? Maybe I shouldn't use B-Splines at all?
2) a B-Spline is constructed using a set of control points. Which control points should I supply it, if I want it to pass through my data points?
spline bezier-curve
asked Jan 9 at 14:52
David RefaeliDavid Refaeli
1065
$endgroup$
While the ideas behind Bezier curves are rather straight forward, I'm really struggling trying to understand B-Splines. I really researched quite a lot about it and still can't figure it out.
I would like to start with the objective: My aim is to chain a few cubic Bezier curves together in an efficient way. I basically have a set of points and I would like to fit a curve through all of them. There is already a nice and simple way to do so that is available online here, that uses constraints on the meeting points (equality of points, derivative and 2nd derivative) to derive the control points.
But, I heard that B-Splines are also used for that purpose, and might be better for it.
The article here shows you can derive Bezier control points out of a B-Spline (in an easy manner). It also states that:
"Whereas an open string of m Bezier curves of degree n involve nm + 1
distinct control points (shared control points counted only once),
that same string of Bezier curves can be expressed using only m + n
B-spline control points (assuming all neighboring curves are C n−1 )."
So,
1) is this the reason why I would like to use B-Splines? If not, what is the reason? Maybe I shouldn't use B-Splines at all?
2) a B-Spline is constructed using a set of control points. Which control points should I supply it, if I want it to pass through my data points?
spline bezier-curve
spline bezier-curve
asked Jan 9 at 14:52
David RefaeliDavid Refaeli
1065
asked Jan 9 at 14:52
David RefaeliDavid Refaeli
1065
asked Jan 9 at 14:52
David RefaeliDavid Refaeli
1065
asked Jan 9 at 14:52
David RefaeliDavid Refaeli
1065
asked Jan 9 at 14:52
David RefaeliDavid Refaeli
1065
1065
add a comment |
add a comment |
1 Answer
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The reason to use B-spline curve instead of multiple Bezier curves are
1) B-spline curve has a more compact representation, i.e., it uses far fewer control points to represent the same shape.
2) B-spline curve representation has built-in continuity, which will not be destroyed when moving control points around. For example, a cubic B-spline curve with two Bezier segments have 5 control points. You are free to move these 5 control points and the $C^2$ continuity between the two segments is always maintained. If the same curve is represented by two Bezier curves with 7 control points ($P_0,P_1,...,P_6$ with $P_3$ the shared control point), moving any of $P_2$, $P_3$ or $P_4$ will destroy $C^1$ continuity unless you ensure that $P_3=(P_2+P_4)/2$.
If you would like to create a B-spline curve to pass through a given set of points, you can search against "B-spline interpolation" on internet.
answered Jan 10 at 2:38
fangfang
2,462166
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1 Answer
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1 Answer
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active
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$begingroup$
The reason to use B-spline curve instead of multiple Bezier curves are
1) B-spline curve has a more compact representation, i.e., it uses far fewer control points to represent the same shape.
2) B-spline curve representation has built-in continuity, which will not be destroyed when moving control points around. For example, a cubic B-spline curve with two Bezier segments have 5 control points. You are free to move these 5 control points and the $C^2$ continuity between the two segments is always maintained. If the same curve is represented by two Bezier curves with 7 control points ($P_0,P_1,...,P_6$ with $P_3$ the shared control point), moving any of $P_2$, $P_3$ or $P_4$ will destroy $C^1$ continuity unless you ensure that $P_3=(P_2+P_4)/2$.
If you would like to create a B-spline curve to pass through a given set of points, you can search against "B-spline interpolation" on internet.
answered Jan 10 at 2:38
fangfang
2,462166
$endgroup$
add a comment |
$begingroup$
The reason to use B-spline curve instead of multiple Bezier curves are
1) B-spline curve has a more compact representation, i.e., it uses far fewer control points to represent the same shape.
2) B-spline curve representation has built-in continuity, which will not be destroyed when moving control points around. For example, a cubic B-spline curve with two Bezier segments have 5 control points. You are free to move these 5 control points and the $C^2$ continuity between the two segments is always maintained. If the same curve is represented by two Bezier curves with 7 control points ($P_0,P_1,...,P_6$ with $P_3$ the shared control point), moving any of $P_2$, $P_3$ or $P_4$ will destroy $C^1$ continuity unless you ensure that $P_3=(P_2+P_4)/2$.
If you would like to create a B-spline curve to pass through a given set of points, you can search against "B-spline interpolation" on internet.
answered Jan 10 at 2:38
fangfang
2,462166
$endgroup$
add a comment |
$begingroup$
The reason to use B-spline curve instead of multiple Bezier curves are
1) B-spline curve has a more compact representation, i.e., it uses far fewer control points to represent the same shape.
2) B-spline curve representation has built-in continuity, which will not be destroyed when moving control points around. For example, a cubic B-spline curve with two Bezier segments have 5 control points. You are free to move these 5 control points and the $C^2$ continuity between the two segments is always maintained. If the same curve is represented by two Bezier curves with 7 control points ($P_0,P_1,...,P_6$ with $P_3$ the shared control point), moving any of $P_2$, $P_3$ or $P_4$ will destroy $C^1$ continuity unless you ensure that $P_3=(P_2+P_4)/2$.
If you would like to create a B-spline curve to pass through a given set of points, you can search against "B-spline interpolation" on internet.
answered Jan 10 at 2:38
fangfang
2,462166
$endgroup$
The reason to use B-spline curve instead of multiple Bezier curves are
1) B-spline curve has a more compact representation, i.e., it uses far fewer control points to represent the same shape.
2) B-spline curve representation has built-in continuity, which will not be destroyed when moving control points around. For example, a cubic B-spline curve with two Bezier segments have 5 control points. You are free to move these 5 control points and the $C^2$ continuity between the two segments is always maintained. If the same curve is represented by two Bezier curves with 7 control points ($P_0,P_1,...,P_6$ with $P_3$ the shared control point), moving any of $P_2$, $P_3$ or $P_4$ will destroy $C^1$ continuity unless you ensure that $P_3=(P_2+P_4)/2$.
If you would like to create a B-spline curve to pass through a given set of points, you can search against "B-spline interpolation" on internet.
answered Jan 10 at 2:38
fangfang
2,462166
answered Jan 10 at 2:38
fangfang
2,462166
answered Jan 10 at 2:38
fangfang
2,462166
answered Jan 10 at 2:38
fangfang
2,462166
2,462166
add a comment |
add a comment |
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