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CS133
Computational Geometry
Computational Geometry Primitives
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Point Representation
A point in the 2D Cartesian space is represented as a vector from the origin to the point π = π¦, π§
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p p x y O
CS133 Computational Geometry Computational Geometry Primitives 1 - - PowerPoint PPT Presentation
CS133 Computational Geometry Computational Geometry Primitives 1 - - PowerPoint PPT Presentation
CS133 Computational Geometry Computational Geometry Primitives 1 Point Representation A point in the 2D Cartesian space is represented as a vector from the origin to the point = , p p y O x 2 Line Segment Representation
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Line Segment Representation
A line segment is represented by its two end points Length= π β π Straight lines are usually represented by two points on it
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p2 p1
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Application: CCW sort
How to sort a list of points in a CCW order around the origin? NaΓ―ve solution: Calculate all the angles and sort
Advantages: Easy and can reuse an existing sort algorithm as-is Disadvantages: arctan calculation is complex and might be inaccurate
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CCW Sort
What is the cross/dot product of the vectors
- f two points?
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CCW Comparator
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π 1 π 2 π 3 π 4
A point in π 2 or π 3 precedes a point in π 4 or π 1 Two points in the same half are
- rdered based on
their angle Two points with the same angle are assumed to be equal
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CCW Comparator
compare(π1 = π¦1, π§1 , π2 = (π¦2, π§2))
If (π¦1 < 0 and π¦2 > 0) OR (π¦1 > 0 and π¦2 < 0)
Return π¦1 < 0? -1 : +1
cp = π1 Γ π2 If ππ < 0
Return -1
If ππ > 0
Return +1
// cp = 0 If (π§1 < 0 and π§2 > 0) OR (π§1 > 0 and π§2 < 0)
Return π§1 < 0?-1 : +1
Return 0
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- 1: π1 precedes π2
+1: π2 precedes π1 0: On the same angle
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CCW Sort
How to sort a set of points around their geometric center? First, compute the geometric center Then, translate the points to make the origin at the center
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Collinearity
Given three points, check if they are on the same straight line Collinear(π1, π2, π)
Return π1π2 Γ π1π == 0
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π1 π2 π π
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Missing Square Problem
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Hint: Test the collinearity of three points on the figure
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Direction
Given a straight line (ray) and a point, find whether the point is to the right or left of the line Direction(π1, π2, π)
ππ = π1π2 Γ π1π If ππ < 0
Return βrightβ
If ππ > 0
Return βleftβ
Return βon-the-lineβ
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π1 π2 π π π
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Point-line Relationship
Given a line segment and a point, find whether the point is on the line segment or not Coincident(π1(π¦1, π§1), π2(π¦2, π§2), π(π¦, π§))
If NOT Collinear(π1, π2, π)
Return false
Return π¦ β min π¦1, π¦2 , max π¦1, π¦2 AND (π§ β min π§1, π§2 , max π§1, π§2
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Line-line Relationship
Given two straight lines, find whether they intersect or not IsIntersected(π1, π2, π3, π4)
If π1π2 Γ π3π4 β 0
Return true // intersected in a point
// The two lines are parallel If Collinear(π1, π2, π3)
Return true // Lines are coincident
Return false // Parallel ant not coincident
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Line-line Intersection
Given two straight lines, find their intersection Solve the two linear equations that represent the two lines One solution β The lines intersect in a point Infinite solutions β The lines are coincident No solutions β The lines are disjoint parallel
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Line-line Intersection
π1, π2 : First line (input) π3, π4 : Second line (input) π0: Intersection point (output) π0 must be collinear with π1π2 and π3π4 π1π2 Γ π1π0 = 0 π3π4 Γ π3π0 = 0 See the rest of the derivation in the notes
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π1 π2 π3 π4 π0
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Line Segment Intersection
Given two line segments, find whether they intersect or not. If they intersect, find their intersection point (Homework)
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