Search and Intersection. O Rourke, Chapter 7 de Berg et al., Chapter 11

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1 Search and Intersection O Rourke, Chapter 7 de Berg et al., Chapter 11

2 Announcements Assignment 3 web-page has been updated: Additional extra credit Hints for managing a dynamic half-edge representation

3 Outline Review Duality Linear Programming Half-Spaces and Convex Hulls (2D)

4 Duality Definition: Given a point p = (α, β) in the plane, define the dual line to be the (non-vertical) line with equation: p = x, y y = 2αx β Given a line L = x, y y = mx + b, define the dual point to be the point with coordinates: L = m 2, b

5 Duality Properties: Given a point p and lines L, L 1, and L 2 : p = p and L = L. p L iff. L p. p L 1 L 2 iff. L 1, L 2 p. L is below/above p iff. L is above/below p. p is on the parabola y = x 2 iff. p is tangent to the parabola at p.

6 Duality Properties: Given a point p = α, β : The slope of p is the slope of the tangent to the parabola at (α, α 2 ). p passes through the point α, α 2 + α 2 β.

7 Linear Programming Goal: Given a set of linear constraints: C i = p p, n i d i and a linear energy function: E p = p, n + d we would like to find the point p that satisfies the constraints and minimizes the energy.

8 Linear Programming Approach: Since the constraints are linear, each one defines a half-space of valid solutions. The intersection of these half-spaces is convex. E Since the energy is linear, it has a constant gradient E pointing away from the minimum. The minimizer is the point in the convex region which is extreme along direction n.

9 Linear Programming Approach: Since the constraints are linear, each one defines a half-space of valid solutions. The intersection of these half-spaces is convex. Since the energy is linear, it has a constant gradient E pointing away from the minimum. The minimizer is the point in the convex region which is extreme along direction n. E How do we compute the convex hull corresponding to the intersection of half spaces?

10 Outline Review Half-Spaces and Convex Hulls (2D)

11 Half-Spaces and Convex Hulls (2D) Notation: Given a set of points, P = {p 1,, p n }: Denote the convex hull of the points as: H(P) Denote the upper hull of the points as: U(P) Denote the lower hull of the points as: L(P) P H(P) U(P) L(P)

12 Half-Spaces and Convex Hulls (2D) Goal: Given a set of half-spaces, represented by directed lines {L 1,, L n } compute the convex hull, H, corresponding to the boundary of their intersection. L k H L 3 L 1 L 2

13 Half-Spaces and Convex Hulls (2D) Approach: Consider the upper/lower hulls, U/L, independently. For the upper (resp. lower) part, assume all line segments have (0, ) (resp. (0, )) to their left. L k L 7 L 6 L 5 L 4 H L 3 L 3 L 1 L 2 U L 1 L 2

14 Half-Spaces and Convex Hulls (2D) Approach: Consider the upper/lower hulls, U/L, independently. An edge e U is the set of points on a line that are below all the other lines: e L i and Below v, L k k i, v e Dually: L i v and Below L k, v k i, v e Lines v, with v e, pass through L i and have all other L 1,, L n below. L L L L i is a vertex of: U = U L 1,, L n U L 4 L 3 L 1 L 2

15 Half-Spaces and Convex Hulls (2D) Approach: Consider the upper/lower hulls, U/L, independently. An edge e U is the set of points on a line that are below all the other lines: e L i and Below v, L k k i, v e Dually: L i v and Below L k, v k i, v e L 3 L 5 L 7 L 6 L 5 L 4 L 1 L 2 U L 4 L 6 L 7 U L 3 L 1 L 2

16 Half-Spaces and Convex Hulls (2D) Approach: Consider the upper/lower hulls, U/L, independently. A vertex v U lies on the intersection of two lines and is below all the other lines: v L i L j and Below(v, L k ) k i, j Dually: L i, L j v and Below L k, v k i, j The line segment L i L j has all other L 1,, L n L i L j is an edge of U = U L 1,, L n L 7 L 6 L 5 below. L 4 L 3 U L 1 L 2

17 Half-Spaces and Convex Hulls (2D) Approach: Consider the upper/lower hulls, U/L, independently. A vertex v U lies on the intersection of two lines and is below all the other lines: v L i L j and Below(v, L k ) k i, j Dually: L i, L j v and Below L k, v k i, j L 1 L 3 L 1 L 2 L 3 L 3 L 5 U L 4 L 5 L 5 L 6 L 6 L 6 L 7 L 7 L 7 L 6 U L 5 L 4 L 3 L 1 L 2

18 Half-Spaces and Convex Hulls (2D) Implementation: UpperHull( {L 1,, L n } ) {v 1,, v m } UpperHull( {L 1,, L n } ) return v m,, v 1 This gives the lines in the (CCW) order in which they appear on the intersection of half-spaces. L 1 L 3 L 1 L 2 L 3 L 3 L 5 U L 4 L 5 L 5 L 6 L 6 L 6 L 7 L 7 L 7 L 6 U L 5 L 4 L 3 L 1 L 2

19 Half-Spaces and Convex Hulls (2D) Implementation: UpperHull( {L 1,, L n } ) {v 1,, v m } UpperHull( {L 1,, L n } ) return v m,, v 1 LowerHull( {L 1,, L n } ) {v 1,, v m } LowerHull( {L 1,, L n } ) return v 1,, v m L 1 L 1 L 2 L L 2 L 3 L 2 L 4 L 4 L 1 L 2 L L 3 L 4

20 Half-Spaces and Convex Hulls (2D) Implementation: UpperHull( {L 1,, L n } ) {v 1,, v m } UpperHull( {L 1,, L n } ) return v m,, v 1 LowerHull( {L 1,, L n } ) {v 1,, v m } LowerHull( {L 1,, L n } ) return v 1,, v m Taking the intersection (in linear time), we get the convex hull. U L

21 Half-Spaces and Convex Hulls (2D) Implementation: UpperHull( We have {Lto 1, separately, L n } ) compute the upper and lower {v 1,, hulls v m } because UpperHull( the dual {L 1, map, L n } is ) undefined return v (discontinuous) m,, v 1 as lines approach vertical. LowerHull( Is there an {L alternate 1,, L n definition } ) of duality that would allow {v 1,, us v m to } compute LowerHull( the {L hulls 1, simultaneously?, L n } ) return v No! The 1, convex, v m hull of the dual cannot be Taking empty, the but intersection the intersection of half-spaces can be. L (in linear time), we get the convex hull. U

22 Half-Spaces and Convex Hulls Writing p R n as: p = α 1,, α n 1, β = α, β we can define duality in n-dimensional space. Given a point p = ( α, β), define the dual hyperplane to be the (non-vertical) plane: p = x, y y = 2 x, α β Given a plane H = x, y y = m, x + b, define the dual point to be the point with coordinates: H = m 2, b

23 Half-Spaces and Convex Hulls Writing p R n as: p = α 1,, α n 1, β = α, β we The can same define properties duality hold in in n-dimensional n-dimensions: space. p = p and H = H. Given p a H point iff. H p = p ( α,. β), define the dual hyperplane p to H be the (non-vertical) plane: 1 H p 2 iff. H 1, H 2 p. H is below/above = x, p y iff. y H = is 2 above/below x, α β p. p = ( x, y) is on the parabola y = x 2 iff. p is tangent to the parabola at p. Given a plane H = x, y y = m, x + b, define the dual point to be the point with coordinates: H = m 2, b

24 Half-Spaces and Convex Hulls In 3D: Let {H 1,, H m } be oriented hyper-planes in R 3 with (0, 0, ) to the left and let U be the intersection of the associate half-spaces. H i, H j, H k is a triangle of U H 1,, H m H i H j H k is a vertex of U.

25 Half-Spaces and Convex Hulls In 3D: Let {H 1,, H m } be oriented hyper-planes in R 3 with (0, 0, ) to the left and let U be the intersection of the associate half-spaces. H i, H j is an edge of U H 1,, H m H i H j contains one edge of U.

26 Half-Spaces and Convex Hulls In 3D: Let {H 1,, H m } be oriented hyper-planes in R 3 with (0, 0, ) to the left and let U be the intersection of the associate half-spaces. H i is a vertex of U H 1,, H m H i contains one face of U.

27 Half-Spaces and Convex Hulls More Generally: Let {H 1,, H m } be oriented hyper-planes in R n with (0,, 0, ) to the left and let U be the intersection of the associate half-spaces. H i1,, H ik is a k-simplex of U H 1,, H m k j=1 H ij contains one (n d)-dimensional face of U.