Computational Geometry

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Computational Geometry
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What is Computational Geometry?

• Deals with problems involving geometric
  primitives
• points
• lines
• polygons

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Geometric Primitives

• Points
• a geometric element determined by an ordered set
  of coordinates
• 2D (x,y)
• 3D (x,y,z)
• 4D (x,y,z,w)

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Geometric Primitives

• Line
• a set of points satisfying equation
• P t A (1-t) B Parametric equation
• in 2D
• y m x c Analytic equation
• Line segment
• closed subset of a line contained between two
  points a and b which are called end points
• P t A (1-t) B where 0 ? t ? 1 Parametric
  equation

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Geometric Primitives

• Polygons
• circular sequence (cycle) of points (vertices)
  and segments (edges)

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Some Geometric Problems
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Segment Intersection

• Test whether segments (a,b) and (c,d) intersect.
  How do we do it?

a
d
c
b

• We could start by writing down the equations of
  the lines through the segments, then test whether
  the lines intersect, then ...

• An alternative (and simpler) approach is based in
  the notion of orientation of an ordered triplet
  of points in the plane

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Orientation

• Clockwise

if slope2 gt slope1 then counterclockwise else
if slope2 lt slope1 clockwise else
colinear
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Intersection and Orientation
(p1,q1,p2), (p1,q1,q2), (p2,q2,p1), (p2,q2,q1)
(p1,q1,p2), (p1,q1,q2), (p2,q2,p1), (p2,q2,q1)
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Intersection and Orientation
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Intersection and Orientation

• Two segments (p1,q1) and (p2,q2) intersect iff
• general case (p1,q1,p2) and (p1,q1,q2) have
  different orientations and (p2,q2,p1) and
  (p2,q2,q1) have different orientations
• special case (p1,q1,p2), (p1,q1,q2), (p2,q2,p1),
  and (p2,q2,q1) are all collinear and the
  x-projections of (p1,q1) and (p2,q2) intersect,
  the y-projections of (p1,q1) and (p2,q2) intersect

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Intersection and Orientation(special cases)
q
2
p
2
q
q
2
1
p
2
(p1,q1,p2), (p1,q1,q2), (p2,q2,p1), (p2,q2,q1)
p
(p1,q1,p2), (p1,q1,q2), (p2,q2,p1), (p2,q2,q1)
1
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How to Compute the Orientation?

• slope of segment (p1,p2) ? (y2-y1) / (x2-x1)
• slope of segment (p2,p3) ? (y3-y2) / (x3-x2)

p3
p2

• Orientation test
• counterclockwise (left turn) ? lt ?
• clockwise (right turn) ? gt ?
• collinear ? ?

p1

• The orientation depends on whether the expression
  (y3-y2) (x2-x1) - (y2-y1) (x3-x2) is positive,
  negative, or zero.

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Point Inclusion

• given a polygon and a point
• is the point inside ?
• or
• outside the polygon?

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Point Inclusion

• Draw a horizontal line to the right of each point
  and extend it to infinity
• Count the number of times a line intersects the
  polygon.
• -even number ? point is outside
• -odd number ? point is inside

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Point Inclusion

• Draw a horizontal line to the right of each point
  and extend it to infinity
• Count the number of times a line intersects the
  polygon.
• -even number ? point is outside
• -odd number ? point is inside

a
b
c
d
e
f
g
What about points d and g ?? Special Case !
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Segment-Segment Intersection
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Segment-Segment Intersection

• Problem Definition
• Let n of input segments in a plane
• Find all intersection points among segments.

How many intersections k, possible?
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Brute Force Intersect Algorithm

• Algorithm Brute Force Intersect(S)
• Test all pairs (si, sj), i ? j.
• If si intersects sj, report si intersects sj.
• Complexity
• -Independent of the number of intersections.

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Goal

• Spend time depending on the of intersection ?

Idea Avoid testing all pairs. Do not Intersect
if they are far apart.
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Sweep-line
Sweep line
ab
Status
Event Points
a
b a
a b
a
a1 b1 b2 a2
ab b2 a2
b2 a2
a2
b1 b2 a2
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Plane Sweeping Algorithm

• Sweep the L over the plane stopping at events.
• Three types of events
• The left end point is reached
• The right end point is reached
• An intersection point is reached

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Plane Sweeping Algorithm

• Sweep the L over the plane stopping at events.
• Three types of events
• The left end point is reached Insert segment
  
• The right end point is reached Remove segment
• An intersection point is reached Switch adjacent
  segments

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Plane Sweeping Algorithm
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Plane Sweep Algorithm

• Analysis
• Total of events 2 n k
• Event is inserted once and deleted once from
  event queue Q
• Cost of maintaining Q O((nk) log(nk))
• k is at most n2.
• So cost of maintaining Q O((nk) log n)
• Cost of maintaining segment list is O(n log n)

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Plane Sweep Algorithm

• Analysis (contd.)
• Total cost of maintaining Q O((nk) log n)
• Cost of maintaining segment list
• n segments inserted and deleted at O(log n) cost
  each.
• At every event there are at most 2 new segment
  adjacencies.
• Total intersection calls O(nk).
• Thus total cost of maintaining segment list O(n
  log n)
• Total overall cost O((nk) log n)

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Simple Closed Path
ProblemGiven a set of points ...
Connect the dots without crossings
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Simple Closed Path
1. Pick the bottom most point as the anchor
point
2. For each point p, compute the angle(p) of
the segment (a,p) with respect to the x-axis
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Simple Closed Path

• Traverse the points by increasing angle

a
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Simple Closed Path
How do we compute angles?
a

• Option 1 use trigonometry (e.g., tan-1).
• the computation is inefficient since
  trigonometric functions are not in the normal
  instruction set of a computer and need a call to
  a math-library routine.

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Simple Closed Path
How do we compute angles?
a

• Observation
• we dont care about the actual values of the
  angles. We just want to sort by angle.
• Option 2 use orientation to compare angles
  without actually computing them!!

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Simple Closed Path

• Orientation to sort by angles ?

• angle(p) lt angle(q) ? orientation of (a,p,q) is
  counterclockwise
• We obtain an O(n log n)-time algorithm for the
  simple closed path problem on n points

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Convex Hull
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Convex Polygon

• A convex polygon is a nonintersecting polygon
  whose internal angles are all convex (i.e., less
  than 180º)
• In a convex polygon, a segment joining two
  vertices of the polygon lies entirely inside the
  polygon

convex
nonconvex
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Convex Hull

• Definition The convex hull of a set of points is
  the smallest convex polygon containing the points
• Think of a rubber band snapping around the points

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Special Cases

• The convex hull is a segment
• All the points are collinear
• The convex hull is a point
• All the points are coincident

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Applications

• Motion planning
• Find an optimal route that avoids obstacles for a
  robot
• Geometric algorithms
• Convex hull is like a two-dimensional sorting

obstacle
start
end
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The Package Wrapping Algorithm
Idea Think of wrapping a package. Put the paper
in contact with the package and continue to wrap
around from one surface to the next until you get
all the way around.
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Package Wrap

• Question Given the current point, how do we
  compute the next point to be wrapped?
• Set up an orientation tournament using the
  current point as the anchor-point.
• The next point is selected as the point that
  beats all other points at CCW orientation, i.e.,
  for any other point, we have orientation(c, p,
  q) CCW

c
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Time Complexity of Package Wrap

• For every point on the hull we examine all the
  other points to determine the next point
• Notation
• N number of points
• M number of hull points (M ? N)
• Time complexity ?(M N)
• Worst case ?(N2)
• (All the points are on the hull, and thus M
  N)
• Average case ?(N log N ) ?(N4/3)
• for points randomly distributed inside a square,
  M ?(log N) on average
• for points randomly distributed inside a circle,
  M ?(N1/3) on average

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Computing the Convex Hull

• The following method computes the convex hull of
  a set of points
• Phase 1 Find the lowest point (anchor point)
• Phase 2 Form a nonintersecting polygon by
  sorting the points counterclockwise around the
  anchor point
• Phase 3 While the polygon has a nonconvex
  vertex, remove it

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Removing Nonconvex Vertices

• Testing whether a vertex is convex can be done
  using the orientation function
• Let p, q and r be three consecutive vertices of a
  polygon, in counterclockwise order
• q convex ? orientation(p, q, r) CCW
• q nonconvex ? orientation(p, q, r) CW or COLL

r
r
q
q
p
p
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Graham Scan

• The Graham scan is a systematic procedure for
  removing nonconvex vertices from a polygon
• The polygon is traversed counterclockwise and a
  sequence H of vertices is maintained

• for each vertex r of the polygon
• Let q and p be the last and second last vertex of
  H
• while orientation(p, q, r) CW or COLLremove q
  from Hq ? pp ? vertex preceding p in H
• Add r to the end of H

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Analysis

• Computing the convex hull of a set of points
  takes O(n log n) time
• Finding the anchor point takes O(n) time
• Sorting the points counterclockwise around the
  anchor point takes O(n log n) time
• Use the orientation comparator and any sorting
  algorithm that runs in O(n log n) time (e.g.,
  heap-sort or merge-sort)
• The Graham scan takes O(n) time
• Each point is inserted once in sequence H
• Each vertex is removed at most once from sequence
  H

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Quick Hull Algorithm
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Quick Hull Example
J
O
K
P
A
L
N
A B C D E F G H I J K L M N O P
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Quick Hull Example
J
O
K
P
A
L
N
A B C D E F G H I J K L M N O P
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Quick Hull Example
J
O
K
P
A
L
N
A B D F G H J K M O P
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Quick Hull Example - Maximum
J
O
K
P
A
L
N
A B D F G H J K M O P
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Quick Hull Example - Filter
J
O
K
P
A
L
N
A B F J J O P
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Quick Hull Example - Maximum
J
O
B
K
P
A
L
N
A B F J J O P
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Quick Hull Example - Base Case
J
O
B
K
P
A
L
N
A B B J J O P
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Quick Hull Example - Done
J
O
B
K
P
A
L
N
A B B J J O O P
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Divide and Conquer Algorithm
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Divide and Conquer Algorithm
B
A
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Divide and Conquer Algorithm
B
A
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Divide and Conquer Algorithm
Upper Tangent
B
A
Lower Tangent
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Divide and Conquer Algorithm
Upper Tangent
B
A
Lower Tangent
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Divide and Conquer Algorithm-Find Lower Tangent-
b
B
A
a
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Divide and Conquer Algorithm -Find Lower Tangent-
b
a1
B
A
a
a
a-1
a-1
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Divide and Conquer Algorithm -Find Lower Tangent-
b
b-1
b1
b
B
A
a
a
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Divide and Conquer Algorithm -Find Lower Tangent-
b
B
A
a
a-1
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Divide and Conquer Algorithm -Find Lower Tangent-
B
b
A
a
a-1
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Divide and Conquer Algorithm -Find Lower Tangent-
B
b
A
a
a-1
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Divide and Conquer Algorithm -Find Lower Tangent-
B
A
b
a
a-1
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Divide and Conquer Algorithm -Find Lower Tangent-
B
A
b
b1
a
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Divide and Conquer Algorithm -Find Lower Tangent-
B
A
b
a
ab is a lower tangent if (a-1) and (a1) are
above/isLeftOn ab and (b-1) and (b1) are
above/isLeftOn ab
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Dynamic Convex Hull

• The basic convex hull algorithms were fairly
  interesting, but you may have noticed that you
  cant draw the hull until after all of the points
  have been specified.
• Is there an interactive way to add points to the
  hull and redraw it while maintaining an optimal
  time complexity?

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Two Halves One Hull

• For this algorithm, we consider a convex hull as
  being two parts
• An upper hull...

and a lower hull...
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Adding points Case 1

• Case 1 the new point is within the horizontal
  span of the hull
• Upper Hull 1a
• If the new point is above the upper hull,
  then it should be added to the upper hull and
  some upper-hull points may be removed.
• .

Upper Hull 1b If the new point is below
the upper hull, no changes need to be made
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Case 1 (cont.)

• The cases for the lower hull are similar.
• Lower Hull 1a
• If the new point is below the lower hull,
  then it is added to the lower hull and some
  lower-hull points may be removed.

Lower Hull 1b If the added point is above
the existing point, it is inside the existing
lower hull, and no changes need be made.
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Adding Points Case 2

• Case 2 the new point is outside the horizontal
  span of the hull
• We must modify both the upper and lower hulls
  accordingly.

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Hull Modification

• In Case 1, we determine the vertices l and r of
  the upper or lower hulls immediately
  preceding/following the new point p in the
  x-order.

If p has been added to the upper hull, examine
the upper hull rightward starting at r. If p
makes a CCW turn with r and its right neighbor,
remove r. Continue until there are no more CCW
turns. Repeat for point l examining the upper
hull leftward. The computation for the bottom
hull is similar
Bad
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Analysis

• Let n be the current size of the convex hull
• Operation locate takes O(log n) time
• Operation insert takes O((1 k)log n) time,
  where k is the number of vertices removed
• Operation hull takes O(n) time
• The amortized running time of operation insert is
  O(log n)