Computational Geometry β Voronoi Diagram COMPUTATIONAL GEOMETRY -- VORONOI DIAGRAM Sophie Che.
Computational Geometry
Transcript of Computational Geometry
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Computational Geometry
Point-Line DualityMichael Goodrich
with slides from Carola Wenk
Point-Line DualityLet π = π!, β¦ , π" β β# be a set of π points. Now define a set πβ = π!β, β¦ , π"βof π lines as follows:
Primal planePoint: π = (π% , π&)Line: π: π¦ = ππ₯ + π
Dual planeLine: πβ: π¦ = π%π₯ β π&Point: πβ = (π,βπ)
p1
p2
p3
p4
p*1
p*2
p*4
p*3
l2
l*2
l*1
l1
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Properties
p1
p2
p3
p4
p*1
p*2
p*4
p*3
l2
l*2
l*1
l1
β’ πβ β = πβ’ π β π β πβ β πβ incidence-preservingβ’ π lies above π β πβ lies above πβ
Primal Dual
β’ p1 Γl2Γ l*2 Γp*1β’ p3 is above l1Γ l*1 is above p*3
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Primal planePoint: π = (π% , π&)Line: π: π¦ = ππ₯ + π
Dual planeLine: πβ: π¦ = π%π₯ β π&Point: πβ = (π,βπ)
PropertiesPrimal Dual
p1
p2
l1
p*1
p*2
l*1
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β’ Points π! and π# lie on line π!
β’ Lines π!β and π#β contain point π!β
Primal planePoint: π = (π% , π&)Line: π: π¦ = ππ₯ + π
Dual planeLine: πβ: π¦ = π%π₯ β π&Point: πβ = (π,βπ)
Point-Line Duality Puzzle
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l1 l2p1
p2p3
p4
p*1
p*2
p*4
l*2
l*1
p*3
π! lies above π!π# lies on π!π# lies on π#π' lies below π!π( lies above π#
π!βlies aboveπ!β
π#βlies onlies onπ#β
lies belowlies above
π!βπ#β
π'βπ!βπ(βπ#β
βββββ
LCH @ UEPrimal Dual
p1
p2
p3p4
p5
p*1p*2
p*3
p*4
p*5
β’ LCH = p1, p2, p3, p4, p5β’ UE = p*1, p*2, p*3, p*4, p*5
LCH lower convex hull UE upper envelope (= pointwise maximum)= halfplane intersection (of upper halfplanes)
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Primal planePoint: π = (π% , π&)Line: π: π¦ = ππ₯ + π
Dual planeLine: πβ: π¦ = π%π₯ β π&Point: πβ = (π,βπ)