why a Bezier curve is guaranteed to lie within the convex hull of its control points?
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why if Bernstein basis polynomials are non-negative ($ B_{k,n}(x) geq 0 $) and also due to the Partition of Unity/sum up to one ($ sum_{k=0}^n B_{k,n}(x) = 1, for all x in [0,1] $) implies Bezier curve is guaranteed to lie within the convex hull of its control points $ CH=({p_{0},p_{1},p_{2},...,p_{n}})$
differential-geometry bezier-curve
asked Nov 17 at 2:13
GreenQuestioner
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why if Bernstein basis polynomials are non-negative ($ B_{k,n}(x) geq 0 $) and also due to the Partition of Unity/sum up to one ($ sum_{k=0}^n B_{k,n}(x) = 1, for all x in [0,1] $) implies Bezier curve is guaranteed to lie within the convex hull of its control points $ CH=({p_{0},p_{1},p_{2},...,p_{n}})$
differential-geometry bezier-curve
asked Nov 17 at 2:13
GreenQuestioner
777
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
why if Bernstein basis polynomials are non-negative ($ B_{k,n}(x) geq 0 $) and also due to the Partition of Unity/sum up to one ($ sum_{k=0}^n B_{k,n}(x) = 1, for all x in [0,1] $) implies Bezier curve is guaranteed to lie within the convex hull of its control points $ CH=({p_{0},p_{1},p_{2},...,p_{n}})$
differential-geometry bezier-curve
asked Nov 17 at 2:13
GreenQuestioner
777
why if Bernstein basis polynomials are non-negative ($ B_{k,n}(x) geq 0 $) and also due to the Partition of Unity/sum up to one ($ sum_{k=0}^n B_{k,n}(x) = 1, for all x in [0,1] $) implies Bezier curve is guaranteed to lie within the convex hull of its control points $ CH=({p_{0},p_{1},p_{2},...,p_{n}})$
differential-geometry bezier-curve
differential-geometry bezier-curve
asked Nov 17 at 2:13
GreenQuestioner
777
asked Nov 17 at 2:13
GreenQuestioner
777
edited Nov 17 at 11:34
asked Nov 17 at 2:13
GreenQuestioner
777
asked Nov 17 at 2:13
GreenQuestioner
777
asked Nov 17 at 2:13
GreenQuestioner
777
777
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1 Answer
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Here are the facts:
A convex combination of the points $p_{0},p_{1},p_{2},...,p_{n}$ is a point of the form
$
sum_{k=0}^n lambda_k p_k
$.
where $lambda_k ge 0$ and $sum_{k=0}^n lambda_k = 1$.The convex hull of the points $p_{0},p_{1},p_{2},...,p_{n}$ is the set of all convex combination of $p_{0},p_{1},p_{2},...,p_{n}$.
A point on a Bézier curve is a convex combination of its control points and so is in their convex hull.
answered Nov 17 at 11:40
lhf
161k9165384
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Here are the facts:
A convex combination of the points $p_{0},p_{1},p_{2},...,p_{n}$ is a point of the form
$
sum_{k=0}^n lambda_k p_k
$.
where $lambda_k ge 0$ and $sum_{k=0}^n lambda_k = 1$.The convex hull of the points $p_{0},p_{1},p_{2},...,p_{n}$ is the set of all convex combination of $p_{0},p_{1},p_{2},...,p_{n}$.
A point on a Bézier curve is a convex combination of its control points and so is in their convex hull.
answered Nov 17 at 11:40
lhf
161k9165384
add a comment |
up vote
0
down vote
Here are the facts:
A convex combination of the points $p_{0},p_{1},p_{2},...,p_{n}$ is a point of the form
$
sum_{k=0}^n lambda_k p_k
$.
where $lambda_k ge 0$ and $sum_{k=0}^n lambda_k = 1$.The convex hull of the points $p_{0},p_{1},p_{2},...,p_{n}$ is the set of all convex combination of $p_{0},p_{1},p_{2},...,p_{n}$.
A point on a Bézier curve is a convex combination of its control points and so is in their convex hull.
answered Nov 17 at 11:40
lhf
161k9165384
add a comment |
up vote
0
down vote
up vote
0
down vote
Here are the facts:
A convex combination of the points $p_{0},p_{1},p_{2},...,p_{n}$ is a point of the form
$
sum_{k=0}^n lambda_k p_k
$.
where $lambda_k ge 0$ and $sum_{k=0}^n lambda_k = 1$.The convex hull of the points $p_{0},p_{1},p_{2},...,p_{n}$ is the set of all convex combination of $p_{0},p_{1},p_{2},...,p_{n}$.
A point on a Bézier curve is a convex combination of its control points and so is in their convex hull.
answered Nov 17 at 11:40
lhf
161k9165384
Here are the facts:
A convex combination of the points $p_{0},p_{1},p_{2},...,p_{n}$ is a point of the form
$
sum_{k=0}^n lambda_k p_k
$.
where $lambda_k ge 0$ and $sum_{k=0}^n lambda_k = 1$.The convex hull of the points $p_{0},p_{1},p_{2},...,p_{n}$ is the set of all convex combination of $p_{0},p_{1},p_{2},...,p_{n}$.
A point on a Bézier curve is a convex combination of its control points and so is in their convex hull.
answered Nov 17 at 11:40
lhf
161k9165384
answered Nov 17 at 11:40
lhf
161k9165384
answered Nov 17 at 11:40
lhf
161k9165384
answered Nov 17 at 11:40
lhf
161k9165384
161k9165384
add a comment |
add a comment |
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