Computational Geometry

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Computational Geometry

• Some useful algorithms/formulas for solving
  geometric problems

2
Overview

• Representation
• Basic geometric problems
• Other issues

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Representation

• Points
• (x, y) Cartesian coordinates
• (r, T) Polar coordinates
• Segments
• Position of the end points
• Lines
• 2-tuple (m, c) using y mx c
• 3-tuple (a, b, c) using ax by c
• Two points P0, P1 on the line using P(t)
  (1-t)P0 t P1
• Polygon
• Ordered list of points

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Basic geometric problems

• Area of polygon
• CCW
• Line segment intersection
• Distance between a point and a line
• Closest point
• Angle between two lines
• Point in a polygon
• Convex hull

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Area of polygon

• Given a simple polygon, find its area.
• Area 1/2 (x1.y2 x2.y3 xn.y1)
  - (y1.x2 y2.x3 yn, x1)

(x2, y2)

(x1, y1)
(xn, yn)
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CCW
3

• Basic geometric primitive
• Returns 1 if the points are ccw
• Returns -1 if the points are cw
• Returns 0 if the points are collinear
• Area given in the previous slide is positive when
  the points are listed in counter-clockwise
  direction and negative otherwise

2
1
ccw(p1, p2, p3) returns 1
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Line segment intersection (I)

• Given two line segments l1 and l2, determine if
  they intersect ?
• isIntersect ccw(l1.p1, l1.p2, l2.p1)
• ccw(l1.p1, l1.p2, l2.p2) lt 0
  ccw(l2.p1, l2.p2, l1.p1)
  ccw(l2.p1, l2.p2, l1.p2) lt 0

l1.p2
l2.p1
l2.p2
l1.p1
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Distance between a point and a line

• Given a point and a line, determine the shortest
  distance between them.
• Select two points A, B on the line
• ½ AB h area of triangle ABP

P
h
B
A
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Closest point (I)

• Given a line L and a point P, find the point on L
  closest to P
• Recall that if u is a unit vector and v is any
  vector. The projection of v on u is given by
  (u.v)u
• u.v is the number k which minimized v ku

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Closest point (II)

• The theorem in the previous slide can only be
  applied to lines which pass through the origin
• This is sufficient to compute the general case as
  we can apply a translation if the line does not
  pass through the origin

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Angle between two lines

• Represent lines as vectors u (a, b) and v (c, d)
• Definition of dot product
• u.v uvcos T ac bd
• u sqrt(u.u)
• T cos-1(u.v / uv)

u
T
v
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Point in a polygon (I)

• Given a point and a closed polygon, determine
  whether the point lies inside the polygon.
• Most algorithms extends a ray from the given
  point and considers the interaction of the
  polygon with the ray
• Such algorithms needs to handle the following
  boundary cases

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Point in a polygon (II)

• A standard way to resolve this issue is to adopt
  the following set of rules
• Edge crossing rules
• an upward edge includes its starting endpoint and
  excludes its endpoint
• a downward edge excludes its starting endpoint
  and includes its endpoint
• horizontal edges are excluded
• the edge-ray intersection point must be strictly
  right of the point P

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Point in a polygon (III)

• One of the simplest algorithm is to compute the
  winding number of the point

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Point in a polygon (IV)

• Winding number
• number of times the polygon winds around the
  point
• Point is outside iff WN 0
• Upward edges gt WN
• Downward edges gt WN--

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Point in a polygon (V)

• //      Input   P a point,//              
  V vertex points of a polygon with
  VnV0//      Return wn the winding number
  (0 only if P is outside V)int
  wn_PointInPoly(Point P, Point V, int n)   
  int wn 0    // the winding number
  counter    // loop through all edges of the
  polygon    for (int i0 iltn i)    // edge
  from Vi to Vi1        if (Vi.y lt P.y)
           // start y lt P.y            if
  (Vi1.y gt P.y)      // an upward
  crossing                if (ccw( Vi, Vi1,
  P) gt 0)  // P left of edge                   
  wn            // have a valid up
  intersect                else // Vi.y gt
  P.y            if (Vi1.y lt P.y)     // a
  downward crossing                if (ccw( Vi,
  Vi1, P) lt 0)  // P right of
  edge                    --wn            //
  have a valid down intersect               
  return wn

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Convex hull (I)

• Given a set of points, determine the smallest
  convex set containing all the points
• A set S is convex if whenever two points P and Q
  are inside S, then the whole line segment PQ is
  also in S

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Convex hull (II)

• One simple algorithm is Graham Scan, often cited
  as the first computational geometry algorithm
• Pseudocode
• Select the lowest leftmost point P0 in S
• Sort the rest of the points in S according to the
  angle made with P0, break ties base on distance
  to P0
• Let P0..n-1 be the sorted array of points
• Create an empty stack ST
• ST.push(Pn-1) //Pn-1 must be on the hull
• ST.push(P0) //P0 must be on the hull
• For i from 2 to n 1
• Let PT1 denote the topmost point on ST
• Let PT2 denote the second topmost point on ST
• while (ccw(PT2, PT1, Pi) lt 0) ST.pop()
• ST.push(Pi)

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Other issues

• Case analysis
• Use of floating point numbers
• Overflow

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References

• http//softsurfer.com/, Dan Sunday
• Algorithms, Robert Sedgewick
• Programming Challenges, Skiena and Revilla