Computational Geometry Part 2: Line Sweep, Convex Hulls Lucca - PowerPoint PPT Presentation

Computational Geometry Part 2: Line Sweep, Convex Hulls Lucca Siaudzionis and Jack Spalding-Jamieson 2020/02/27 University of British Columbia

Announcements • A3 is due Sundayyyyy! • Project topics are due Tuesday! • Presentation dates will be on March 31st and April 2nd . • If you can’t present in one of those two days, let us know by March 4th so we can schedule you on the other (with a decent reason). • Otherwise, dates will be randomly assigned. • Extra office hours tomorrow from 2-4PM in ICCSX237. 1

Discussion Problem: Remember How to do Line Sweep? Jack is constantly busy with meetings that take up all his time. Many of his meetings even overlap! Given a list of 1 ≤ n ≤ 10 6 meetings along with their start and end times, find out how much time Jack spends overwhelmed with meetings. 2

Discussion Problem: Remember How to do Line Sweep? – Insight We can process the “events” (start and end times) in increasing order (the “ x ”-coordinate in Line Sweep), keeping track of how many intervals are currently under the sweep line. 3

Line sweep: General Plan The general method for solving sweep problems: • Identify important “events” that occur as we move from left to right. • Keep track of some auxiliary data structure that lets us • find the answer, and • update efficiently at each event. 4

Line Sweep: Pachinko Given N ≤ 10 5 non-intersecting, non-horizontal line segment obstacles, where does the pinball end up? 5

Line Sweep: Pachinko We only need to know for each segment, which “next segment” the pinball will fall onto. We’ll find these by sweeping from left to right. 6

Line Sweep: Pachinko Data structure: A set of lines, sorted by their order on the y -axis. Events: The line segment endpoints. Actions: • Left endpoint: Insert line into the set • Right endpoint: Remove line from the set • Lower endpoint of segment: Query for the next line below the point 9

Convex Sets A set S is convex ⇔ ∀ x , y ∈ S , 0 ≤ λ ≤ 1 , λ x + (1 − λ ) y ∈ S ⇔ the line segment x → y is inside the set Figure 1: In a convex set, the line segment between any 2 points lies in a set 10

Convex Hulls: Definition A convex hull of a set is the smallest convex set containing the set. Figure 2: Some sets and their convex hulls Why do we care about convex hulls? They give us nice convex polygons, which we can optimize over easier. 11

Convex Hulls: Finite Point Sets If we have a finite set of points, we define the following about a point p and the convex hull H : • p may be in the interior, i.e. p is not found on H at all. • p may be a vertex of the convex hull, i.e. removing p would make the convex hull smaller. • p could lie on an edge in the convex hull. When finding a convex hull, we need to know which of these sets we’re looking for (just the vertices? or all the points on the convex hull). 12

Convex Hulls: Applications Convex Hulls can be used for many real-world applications, such as: • In game programming, they can be used to speed up collision-detection, similarly to bounding-boxes. • Can be used to find the shortest paths around physical objects (remember the circle problem from last class?). • And much more... 13

Convex Hulls: Gift Wrapping How do we find the convex hull of a finite point-set with n points? 14

Convex Hulls: Gift Wrapping How do we find the convex hull of a finite point-set with n points? We can use the gift wrapping algorithm : • Start at some point that is guaranteed to be a vertex in the convex hull, e.g. the leftmost point (break ties by height). • Look at all of the other points, and choose the one that maximizes the angle with the line segment pointing downwards from our original point. • The point chosen is the next point in the convex hull. • Do this iteratively until we hit a point already added, but with the most recent line segment added to the convex hull. Overall runtime: O ( n 2 ) 14

Convex Hulls: Gift Wrapping Example (1) 15

Convex Hulls (Aside): Gift Wrapping – Output Sensitivity Although the runtime of gift wrapping is in fact a tight Θ( n 2 ), we can actually give a better bound! This is using something called output sensitivity : If we know that there will be at most h < n points on the eventual convex hull, we can actually bound the algorithm with complexity O ( nh ). 25

Convex Hulls: Faster Algorithms Are there faster algorithms for solving this? 26

Convex Hulls: Faster Algorithms Are there faster algorithms for solving this? Yes. Lots. No less than eight: • Graham Scan ( O ( n log n )) • Monotone Chain/Andrew’s Algorithm ( O ( n log n )) • Quickhull ( O ( n log n ) average case) • Divide & Conquer via axis split ( O ( n log n )), extendable to 3D! • Divide & Conquer with arbitrary merge and rotating calipers ( O ( n log n )) • Kallay’s incremental algorithm ( O ( n log n )) • Kirkpatrick-Seidel Algorithm ( O ( n log h )), aka the ”ultimate planar convex hull algorithm” • Chan’s Algorithm ( O ( n log h )) We’re going to talk about Monotone Chain/Andrew’s Algorithm, although the algorithm is very similar to Graham’s algorithm. 26

Monotone Chain Algorithm Step 1: scan from left to right, “wrap” around the top Step 2: scan from right to left, “wrap” around the bottom. Figure 3: Upper and lower hulls built by monotone chain algorithm 27

Monotone Chain Algorithm 28

Monotone Chain Algorithm Algorithm summary • Sort all points by x -coordinate • Start from leftmost point (i.e. smallest x -coordinate) • If there are ties, you can take the uppermost one (largest y -coordinate) • Iteratively add points into the hull • If adding a point causes CCW turn, pop points from hull until this is no longer the case before adding! • Repeat the above steps from right to left. 29

Monotone Chain Algorithm Edge cases • Convex hull of 1, 2, 3 points • Collinear points 30

Monotone Chain Algorithm Time complexity analysis • Sort all points by x -coordinate: O ( n log n ) • Each point is added and removed at most once: O ( n ) ⇒ Time complexity: O ( n log n ) Without looking at output sensitivity, can we do better? 31

Monotone Chain Algorithm Time complexity analysis • Sort all points by x -coordinate: O ( n log n ) • Each point is added and removed at most once: O ( n ) ⇒ Time complexity: O ( n log n ) Without looking at output sensitivity, can we do better? With angle-based comparison, we can use the sorting lower bound: Arrange n points on a circle ⇒ equivalent to sorting in some sense, Ω( n log n ). 31

Discussion Problem: Visibility Input : A point-set D of 1 ≤ n ≤ 10 5 key-points representing some convex geometric object, guaranteed to include all boundary points. Additionally, there are 1 ≤ q ≤ 10 5 queries. For each query, a point p is provided, guaranteed to be outside the boundary of the object. Output : For each query, output the angles over which a camera positioned at point p would not be able to see past the object. P 32

Discussion Problem: Visibility – Insight (1) Observation 1 : The solution for each query lies on the convex hull. Observation 2 : Pick an arbitrary point on the hull, draw a line through, then one extremum is on each side of the line. P 33

Discussion Problem: Visibility – Insight (2) Binary search to find start/end vertices on each half of the hull, and then binary search on each half of the hull for when P → p i → p i +1 changes from CCW to CW (or vice versa). Time complexity: O ( N log N + Q log N ) This problem has many applications, including game programming, graphics, and more broad computational geometry problems (such as euclidean shortest path problems). 34

Computational Geometry Part 2: Line Sweep, Convex Hulls Lucca - PowerPoint PPT Presentation

Computational Geometry Part 2: Line Sweep, Convex Hulls Lucca Siaudzionis and Jack Spalding-Jamieson 2020/02/27 University of British Columbia Announcements A3 is due Sundayyyyy! Project topics are due Tuesday! Presentation dates

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