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Maarten L¨
- ffler
Maarten L offler Marc van Kreveld Center for Geometry, Imaging - - PowerPoint PPT Presentation
Maarten L offler Marc van Kreveld Center for Geometry, Imaging - - PowerPoint PPT Presentation
Maarten L offler Marc van Kreveld Center for Geometry, Imaging and Virtual Environments Utrecht University ? PROPERTIES OF IMPRECISE POINTS ? PROPERTIES OF IMPRECISE POINTS connected ? PROPERTIES OF IMPRECISE POINTS connected
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?
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PROPERTIES OF IMPRECISE POINTS
?
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PROPERTIES OF IMPRECISE POINTS
- connected
?
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PROPERTIES OF IMPRECISE POINTS
- connected
- convex
?
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PROPERTIES OF IMPRECISE POINTS
- connected
- convex
- polygonal
?
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PROPERTIES OF IMPRECISE POINTS
- connected
- convex
- polygonal
- constant
?
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PROPERTIES OF IMPRECISE POINTS
- connected
- convex
- polygonal
- constant
PROPERTIES OF IMPRECISE LINES
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PROPERTIES OF IMPRECISE POINTS
- connected
- convex
- polygonal
- constant
PROPERTIES OF
- connected
IMPRECISE LINES
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PROPERTIES OF IMPRECISE POINTS
- connected
- convex
- polygonal
- constant
PROPERTIES OF
- connected
- convex?
IMPRECISE LINES
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PROPERTIES OF IMPRECISE POINTS
- connected
- convex
- polygonal
- constant
PROPERTIES OF
- connected
- convex?
- polygonal?
IMPRECISE LINES
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PROPERTIES OF IMPRECISE POINTS
- connected
- convex
- polygonal
- constant
PROPERTIES OF
- connected
- convex?
- polygonal?
- constant?
IMPRECISE LINES
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WHAT ARE CONVEX SETS OF LINES?
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
- problem: vertical lines
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
- problem: vertical lines
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
- problem: vertical lines
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
- problem: vertical lines
- different mapping?
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
- problem: vertical lines
- different mapping?
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WHAT ARE CONVEX SETS OF LINES?
- let’s use duality!
- problem: vertical lines
- different mapping?
[Rosenfeld, 1995]
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WHAT ARE CONVEX SETS OF LINES?
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WHAT ARE CONVEX SETS OF LINES?
- desirable properties of convex hull
- connectivity
- anti-exchange property
- affine transformation invariant
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WHAT ARE CONVEX SETS OF LINES?
- desirable properties of convex hull
- connectivity
- anti-exchange property
- affine transformation invariant
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WHAT ARE CONVEX SETS OF LINES?
- desirable properties of convex hull
- connectivity
- anti-exchange property
- affine transformation invariant
[Goodman, 1998]
- no such definition exists!
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WHAT ARE CONVEX SETS OF LINES?
- desirable properties of convex hull
- connectivity
- anti-exchange property
- affine transformation invariant
[Goodman, 1998]
- no such definition exists!
- drop connectivity?
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WHAT ARE CONVEX SETS OF LINES?
- desirable properties of convex hull
- connectivity
- anti-exchange property
- affine transformation invariant
[Goodman, 1998]
- no such definition exists!
- drop connectivity?
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WHAT ARE CONVEX SETS OF LINES?
- desirable properties of convex hull
- connectivity
- anti-exchange property
- affine transformation invariant
[Goodman, 1998]
- no such definition exists!
- drop connectivity?
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WHAT ARE CONVEX SETS OF LINES?
[Gates, 1993]
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WHAT ARE CONVEX SETS OF LINES?
[Gates, 1993]
- what about directed lines?
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WHAT ARE CONVEX SETS OF LINES?
[Gates, 1993]
- what about directed lines?
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WHAT ARE CONVEX SETS OF LINES?
[Gates, 1993]
- what about directed lines?
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WHAT ARE CONVEX SETS OF LINES?
[Gates, 1993]
- what about directed lines?
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WHAT ARE CONVEX SETS OF LINES?
[Gates, 1993]
- what about directed lines?
- imprecise lines have a
“general direction”
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WHAT ARE CONVEX SETS OF LINES?
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
- there is a line d /
∈ L such that no line in L is parallel to d d
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
- there is a line d /
∈ L such that no line in L is parallel to d d
- if ℓ, ℓ′ ∈ L, all lines between
ℓ and ℓ′ are also in L
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
- there is a line d /
∈ L such that no line in L is parallel to d d
- if ℓ, ℓ′ ∈ L, all lines between
ℓ and ℓ′ are also in L
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
- there is a line d /
∈ L such that no line in L is parallel to d d
- if ℓ, ℓ′ ∈ L, all lines between
ℓ and ℓ′ are also in L
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
- there is a line d /
∈ L such that no line in L is parallel to d d
- if ℓ, ℓ′ ∈ L, all lines between
ℓ and ℓ′ are also in L
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
- there is a line d /
∈ L such that no line in L is parallel to d d
- if ℓ, ℓ′ ∈ L, all lines between
ℓ and ℓ′ are also in L
- convex hull not defined
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
- there is a line d /
∈ L such that no line in L is parallel to d d
- if ℓ, ℓ′ ∈ L, all lines between
ℓ and ℓ′ are also in L
- convex hull not defined
- given by boundary
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WHAT ARE CONVEX SETS OF LINES?
- a set of lines L is convex when:
- there is a line d /
∈ L such that no line in L is parallel to d d
- if ℓ, ℓ′ ∈ L, all lines between
ℓ and ℓ′ are also in L
- convex hull not defined
- given by boundary
- limit angle α
α
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PROPERTIES OF IMPRECISE LINES
- connected
- convex
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PROPERTIES OF IMPRECISE LINES
- connected
- convex
- polygonal
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PROPERTIES OF IMPRECISE LINES
- connected
- convex
- polygonal
- constant
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EXAMPLE: LINEAR PROGRAMMING
- important, well known problem
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EXAMPLE: LINEAR PROGRAMMING
- given set of directed lines
- important, well known problem
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EXAMPLE: LINEAR PROGRAMMING
- given set of directed lines
- determine the lowest point
to the left of all lines
- important, well known problem
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EXAMPLE: LINEAR PROGRAMMING
- given set of directed lines
- determine the lowest point
to the left of all lines
- important, well known problem
- takes O(n) time
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EXAMPLE: LINEAR PROGRAMMING
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EXAMPLE: LINEAR PROGRAMMING
- given set of imprecise directed lines
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EXAMPLE: LINEAR PROGRAMMING
- given set of imprecise directed lines
- determine all possible heights
- f the lowest point
to the left of all lines
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EXAMPLE: LINEAR PROGRAMMING
- given set of imprecise directed lines
- determine all possible heights
- f the lowest point
to the left of all lines
- lowest possible point
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EXAMPLE: LINEAR PROGRAMMING
- given set of imprecise directed lines
- determine all possible heights
- f the lowest point
to the left of all lines
- lowest possible point
- highest possible point
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HIGHEST VALUE
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HIGHEST VALUE
- only consider left borders of bundles
- find lowest point to the left of those
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HIGHEST VALUE
- only consider left borders of bundles
- apply convex programming
- takes O(n) time
- find lowest point to the left of those
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LOWEST VALUE
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LOWEST VALUE
- only consider right borders of bundles
- find lowest point to the left of those
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LOWEST VALUE
- only consider right borders of bundles
- find lowest point to the left of those
- takes Θ(n2) time
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LOWEST VALUE
- only consider right borders of bundles
- find lowest point to the left of those
- takes Θ(n2) time
- if α < 180◦ − c
it takes Θ(n log n) time
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