Conditions for two B-Splines to represent the same curve
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What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space?
Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($t_0,t_1,t_2,dots,t_{m_1}$) and ($u_0,u_1,u_2,dots,u_{m_2}$) respectively, and the control points are $P_0,P_1,dots,P_{n_1}$ and $Q_0,Q_1,dots,Q_{n_2}$ respectively.
differential-geometry parametric constraints spline bezier-curve
asked Nov 21 at 17:55
Suraj L
82
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What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space?
Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($t_0,t_1,t_2,dots,t_{m_1}$) and ($u_0,u_1,u_2,dots,u_{m_2}$) respectively, and the control points are $P_0,P_1,dots,P_{n_1}$ and $Q_0,Q_1,dots,Q_{n_2}$ respectively.
differential-geometry parametric constraints spline bezier-curve
asked Nov 21 at 17:55
Suraj L
82
If we exclude curves containing at least one straight line segment, then there exists a unique representation for all orders $k$ which equal or exceed the order of the highest order polynomial piece. If there is a straight there will be control points which are on a straight line. The non-endpoints can be moved along the line with no geometric consequence and the same is true for those knots which only affect basis functions whose corresponding control point is on the line. Edit: I forgot you can of course rescale the knot sequence however you wish without changing the geometry.
– Oppenede
2 days ago
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down vote
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up vote
0
down vote
favorite
What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space?
Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($t_0,t_1,t_2,dots,t_{m_1}$) and ($u_0,u_1,u_2,dots,u_{m_2}$) respectively, and the control points are $P_0,P_1,dots,P_{n_1}$ and $Q_0,Q_1,dots,Q_{n_2}$ respectively.
differential-geometry parametric constraints spline bezier-curve
asked Nov 21 at 17:55
Suraj L
82
What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space?
Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($t_0,t_1,t_2,dots,t_{m_1}$) and ($u_0,u_1,u_2,dots,u_{m_2}$) respectively, and the control points are $P_0,P_1,dots,P_{n_1}$ and $Q_0,Q_1,dots,Q_{n_2}$ respectively.
differential-geometry parametric constraints spline bezier-curve
differential-geometry parametric constraints spline bezier-curve
asked Nov 21 at 17:55
Suraj L
82
asked Nov 21 at 17:55
Suraj L
82
asked Nov 21 at 17:55
Suraj L
82
asked Nov 21 at 17:55
Suraj L
82
asked Nov 21 at 17:55
Suraj L
82
82
If we exclude curves containing at least one straight line segment, then there exists a unique representation for all orders $k$ which equal or exceed the order of the highest order polynomial piece. If there is a straight there will be control points which are on a straight line. The non-endpoints can be moved along the line with no geometric consequence and the same is true for those knots which only affect basis functions whose corresponding control point is on the line. Edit: I forgot you can of course rescale the knot sequence however you wish without changing the geometry.
– Oppenede
2 days ago
add a comment |
If we exclude curves containing at least one straight line segment, then there exists a unique representation for all orders $k$ which equal or exceed the order of the highest order polynomial piece. If there is a straight there will be control points which are on a straight line. The non-endpoints can be moved along the line with no geometric consequence and the same is true for those knots which only affect basis functions whose corresponding control point is on the line. Edit: I forgot you can of course rescale the knot sequence however you wish without changing the geometry.
– Oppenede
2 days ago
If we exclude curves containing at least one straight line segment, then there exists a unique representation for all orders $k$ which equal or exceed the order of the highest order polynomial piece. If there is a straight there will be control points which are on a straight line. The non-endpoints can be moved along the line with no geometric consequence and the same is true for those knots which only affect basis functions whose corresponding control point is on the line. Edit: I forgot you can of course rescale the knot sequence however you wish without changing the geometry.
– Oppenede
2 days ago
If we exclude curves containing at least one straight line segment, then there exists a unique representation for all orders $k$ which equal or exceed the order of the highest order polynomial piece. If there is a straight there will be control points which are on a straight line. The non-endpoints can be moved along the line with no geometric consequence and the same is true for those knots which only affect basis functions whose corresponding control point is on the line. Edit: I forgot you can of course rescale the knot sequence however you wish without changing the geometry.
– Oppenede
2 days ago
add a comment |
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If we exclude curves containing at least one straight line segment, then there exists a unique representation for all orders $k$ which equal or exceed the order of the highest order polynomial piece. If there is a straight there will be control points which are on a straight line. The non-endpoints can be moved along the line with no geometric consequence and the same is true for those knots which only affect basis functions whose corresponding control point is on the line. Edit: I forgot you can of course rescale the knot sequence however you wish without changing the geometry.
– Oppenede
2 days ago