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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