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Computational Geometry

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Title: Convex Hull Author: Jiang Last modified by: Bob-Master Created Date: 12/11/2006 3:00:47 PM Document presentation format: (4:3) – PowerPoint PPT presentation

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Title: Computational Geometry


1
Computational Geometry
  • 2012/10/23

2
Computational Geometry
  • A branch of computer science that studies
    algorithms for solving geometric problems
  • Applications computer graphics, robotics, VLSI
    design, computer aided design, and statistics.

3
Intersection Point of Two Lines
The equations of the lines are Pa P1 ua ( P2
- P1 ) //P1 starting point (P2-P1)vector along
line a Pb P3 ub ( P4 - P3 ) //P3 starting
point (P4-P3)vector along line b Pa A point
on line a Pb A point on line b
4
Intersection Point of Two Lines
  • Solving for the point where Pa Pb gives the
    following two equations in two unknowns (ua and
    ub)
  • x1 ua (x2 - x1) x3 ub (x4 - x3)
  • y1 ua (y2 - y1) y3 ub (y4 - y3)

5
Solving ua and ub
6
Intersection Point of Two Lines
  • Substituting either ua or ub into the
    corresponding equation for the line gives the
    intersection point.
  • x x1 ua (x2 - x1)
  • y y1 ua (y2 - y1)

7
Intersection Point of Two Lines
  • The denominators for the equations for ua and ub
    are the same.
  • If the denominator for the equations for ua and
    ub is 0 then the two lines are parallel.
  • If the denominator and numerator for the
    equations for ua and ub are 0 then the two lines
    are coincident.

8
Intersection Point of Two Lines
  • The equations apply to lines, if the intersection
    of line segments is required then it is only
    necessary to test if ua and ub lie between 0 and
    1.
  • Whichever one lies within that range then the
    corresponding line segment contains the
    intersection point. If both lie within the range
    of 0 to 1 then the intersection point is within
    both line segments.

9
Cross Product
  • The cross product p1 p2 can be interpreted as
    the signed area of the parallelogram formed by
    the points (0, 0), p1, p2, and p1 p2 (x1
    x2, y1 y2).

10
Cross Product
  • The cross product p1 p2 can be interpreted as
    the signed area of the parallelogram formed by
    the points (0, 0), p1, p2, and p1 p2 (x1
    x2, y1 y2).
  • An equivalent definition gives the cross product
    as the determinant of a matrix

11
Cross Product
  • Actually, the cross product is a
    three-dimensional concept. It is a vector that is
    perpendicular to both p1 and p2 according to the
    right-hand rule and whose magnitude is x1 y2
    x2 y1.
  • Below, we will just treat the cross product
    simply as the value of x1 y2 x2 y1.

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15
  • QA
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