CS133 Computational Geometry Convex Hull 1 Convex Hull Given a - - PowerPoint PPT Presentation

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CS133 Computational Geometry Convex Hull 1 Convex Hull Given a - - PowerPoint PPT Presentation

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CS133 Computational Geometry Convex Hull 1 Convex Hull Given a set of n points, find the minimal convex polygon that contains all the points 2 Convex Hull Properties 3 Convex Hull Representation The convex hull is represented by all

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CS133

Computational Geometry

Convex Hull

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Convex Hull

Given a set of n points, find the minimal convex polygon that contains all the points

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Convex Hull Properties

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ΞΈ

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Convex Hull Representation

The convex hull is represented by all its points sorted in CW/CCW order Special case: Three collinear points

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NaΓ―ve Convex Hull Algorithm

Iterate over all possible line segments A line segment is part of the convex hull if all

  • ther points are to its left

Emit all segments in a CCW order Running time 𝑃 π‘œ3

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NaΓ―ve Convex Hull Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Graham Scan Algorithm

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Example

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Graham Scan Pseudo Code

Select the point with minimum 𝑧 Sort all points in CCW order π‘ž0, π‘ž1, … , π‘žπ‘œ 𝑇 = π‘ž0, π‘ž1 For 𝑗 = 2 to π‘œ

While |𝑇| > 2 && π‘žπ‘— is to the right of π‘‡βˆ’2, π‘‡βˆ’1

𝑇.pop

𝑇.push(π‘žπ‘—)

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Monotone Chain Algorithm

Has some similarities with Graham scan algorithm Instead of sorting in CCW order, it sorts by

  • ne coordinate (e.g., x-coordinates)

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Example

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Pseudo Code

Sort 𝑇 by 𝑦 𝑉 = {𝑇0} For 𝑗 = 1 to π‘œ

while |𝑉| > 1 && 𝑇𝑗 is to the left of π‘‰βˆ’2π‘‰βˆ’1

𝑉.pop

𝑉.push(𝑇𝑗)

𝑀 = {𝑇0}

While |𝑀| > 1 && 𝑇𝑗 is to the right of π‘€βˆ’2π‘€βˆ’1

𝑀.pop

𝑀.push(𝑇𝑗)

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Gift Wrapping Algorithm

Start with a point on the convex hull Find more points on the hull one at a time Terminate when the first point is reached back Also knows as Jarvi’s March Algorithm

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Gift Wrapping Example

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Gift Wrapping Example

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Gift Wrapping Example

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Gift Wrapping Example

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Gift Wrapping Example

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Gift Wrapping Example

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Gift Wrapping Example

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Gift Wrapping Example

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Gift Wrapping Example

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Gift Wrapping Pseudo Code

Gift Wrapping(S)

CH= {} CH << Left most point do

Start point = CH.last End point = CH[0] For each point s ∈ S

If Start point = End Point OR s is to the left of 𝑇𝑒𝑏𝑠𝑒 π‘žπ‘π‘—π‘œπ‘’, πΉπ‘œπ‘’ π‘žπ‘π‘—π‘œπ‘’ End point = s

CH << End point

Running time 𝑃(π‘œ β‹… 𝑙)

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Divide & Conquer Convex Hull

ConvexHull(S)

Splits S into two subsets S1 and S2 Ch1 = ConvexHull(S1) Ch2 = ConvexHull(S2) Return Merge(Ch1, Ch2)

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Divide & Conquer Convex Hull

ConvexHull(S)

Splits S into two subsets S1 and S2 Ch1 = ConvexHull(S1) Ch2 = ConvexHull(S2) Return Merge(Ch1, Ch2)

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 

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Merge: Upper Tangent

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Merge: Upper Tangent

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Merge: Upper Tangent

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Merge: Lower Tangent

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Merge: Lower Tangent

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Merge: Lower Tangent

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Merge: Lower Tangent

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Merge: Lower Tangent

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Merge: Lower Tangent

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Merge Step

Upper Tangent(𝑀, 𝑆)

𝑄𝑗 = Right most point in 𝑀 π‘„π‘˜ = Left most point in 𝑆 Do

Done = true While 𝑄

𝑗+1 is to the right of π‘žπ‘˜π‘žπ‘—

𝑗 + +; done = false

While 𝑄

π‘˜βˆ’1 is to the left of π‘žπ‘—π‘žπ‘˜

π‘˜ βˆ’ βˆ’; done = false

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Analysis

Sort step: 𝑃 π‘œ β‹… log π‘œ Merge step: 𝑃 π‘œ Recursive part

π‘ˆ π‘œ = 2π‘ˆ

π‘œ 2 + 𝑑 β‹… π‘œ

π‘ˆ π‘œ = 𝑃 π‘œ β‹… log π‘œ

Overall running time 𝑃 π‘œ β‹… log π‘œ

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Incremental Convex Hull

Start with an initial convex hull Add one additional point to the convex hull Given a convex hull CH and a point p, how to compute the convex hull of {CH, p}? Think: Insert an element into a sorted list

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Case 1: p inside CH

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Case 2: p on CH

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Case 3: p outside CH

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Case 3: p outside CH

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Analysis of the Insert Function

Test whether the point is inside, outside, or

  • n the polygon O(n)

Find the two tangents O(n) A more efficient algorithm can have an amortized running time of O(log n)

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Quick Hull

If we can have a divide-and-conquer algorithm similar to merge sort … why not having an algorithm similar to quick sort? Sketch

Find a pivot Split the points along the pivot Recursively process each side

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Quick Hull

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Quick Hull

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How to split the points across the line segment?

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Quick Hull

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How to select the farthest point?

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Quick Hull

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Quick Hull

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Quick Hull

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How to split the points into three subsets?

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Quick Hull

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Quick Hull

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Quick Hull

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Quick Hull

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Quick Hull

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Quick Hull

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Quick Hull

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Example

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Running Time Analysis

π‘ˆ π‘œ = π‘ˆ π‘œ1 + π‘ˆ π‘œ2 + 𝑃 π‘œ Worst case π‘œ1 = π‘œ βˆ’ 𝑙 or π‘œ2 = π‘œ βˆ’ 𝑙, where 𝑙 is a small constant (e.g., k=1)

π‘ˆ π‘œ = 𝑃 π‘œ2

Best case π‘œ1 = 𝑙 and π‘œ2 = 𝑙, where 𝑙 is a small constant

In this case, most of the points are pruned π‘ˆ π‘œ = 𝑃 π‘œ

Average case, π‘œ1 = π›½π‘œ and π‘œ2 = π›Ύπ‘œ, where 𝛽 < 1 and 𝛾 < 1

π‘ˆ π‘œ = 𝑃 π‘œ log π‘œ

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