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Computational Geometry

- 2012/10/23

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.

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

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)

Solving ua and ub

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)

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.

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.

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).

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

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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- QA