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 P a = P1 + u a ( P2 - P1 ) //P1: starting point; (P2-P1):vector along line a P b = P3 + u b ( P4 - P3 ) //P3: starting point; (P4-P3):vector along line b P a : A point on line a P b : A point on line b

4. Intersection Point of Two Lines Solving for the point where P a = P b gives the following two equations in two unknowns (u a and u b )  x1 + u a (x2 - x1) = x3 + u b (x4 - x3)  y1 + u a (y2 - y1) = y3 + u b (y4 - y3)

5. Solving u a and u b

6. Intersection Point of Two Lines Substituting either u a or u b into the corresponding equation for the line gives the intersection point.  x = x1 + u a (x2 - x1)  y = y1 + u a (y2 - y1)

7. Intersection Point of Two Lines The denominators for the equations for u a and u b are the same. If the denominator for the equations for u a and u b is 0 then the two lines are parallel. If the denominator and numerator for the equations for u a and u b 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 u a and u b 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 p 1 × p 2 can be interpreted as the signed area of the parallelogram formed by the points (0, 0), p 1, p 2, and p 1 + p 2 = (x 1 + x 2, y 1 + y 2 ).

10. Cross Product The cross product p 1 × p 2 can be interpreted as the signed area of the parallelogram formed by the points (0, 0), p 1, p 2, and p 1 + p 2 = (x 1 + x 2, y 1 + y 2 ). 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 p 1 and p 2 according to the “right-hand rule” and whose magnitude is |x 1 y 2 – x 2 y 1 |. Below, we will just treat the cross product simply as the value of x 1 y 2 – x 2 y 1.

15. Q&A