1. Chi-Cheng Lin, Winona State University CS430 Computer Graphics Vectors Part III Intersections

2. 2 Topics l Intersection of Two Line Segments l Application – Excircle l Intersection of Line and Plane

3. 3 Intersection of Two Line Segments l Problem: Given two line segments AB and CD, determine whether they intersect, and if they do, find the intersection point l Cases A B C D A B C D A B C D A B C D A B D C A B D C

4. 4 Parametric Line l AB(t) = A + bt, where b = B -A z -  < t <  : parent line of AB z 0.0  t  1.0: line segment AB z 0.0  t <  : ray A B t >1 t =1 t =0 t <0

5. 5 Intersection of Two Line Segments l AB(t) = A + bt, where b = B -A CD(u) = C + du, where d = D -C l Parent lines of AB and CD intersect if A + bt = C + du Let c = C - A  bt = c + du  d bt = d c + d du  d bt = d c

6. 6 Intersection of Two Line Segments Case I: d b  0  AB and CD are not parallel (why?) t = (d c)/(d b) Similarly, (how?) u = (b c)/(d b) l Q: When do two AB and CD intersect? l A: When 0.0  t  1.0 and 0.0  u  1.0

7. 7 Intersection of Two Line Segments Case II: d b = 0  AB and CD are parallel z If (b   c) = 0, the line segments are co- linear (why?)  have to test for overlap z If (b   c)  0, the line segments are not co- linear  they do not intersect l Examples

8. 8 Application – Circle through 3 Points l Given 3 non-collinear points A, B, and C, find the circle passing through the 3 points  excircle of triangle defined by the points l The center S is the intersection of the perpendicular bisectors of AB, BC, and CA A B C A B C ? S

9. 9 Circle through 3 Points l Let a = B - A, b = C - B, c = A – C l Midpoint of A = A + a/2 (why?) Direction perpendicular to AB = a   Parametric form of perpendicular bisector of AB = A + a/2 + a  t (why?) l Parametric form of perpendicular bisector of CA = C + c/2 + c  u

10. 10 Circle through 3 Points l The intersection of AB and CA is S  A + a/2 + a  t = C + c/2 + c  u As a + b + c = 0 (why?)  a  t = b/2 + c  u (how?)  t = 0.5(b  c)/(a   c) (how?)  center radius = |S - A| =

11. 11 Intersection of Line and Plane l Line in parametric form z R(t) = A + ct where A is a known point on the line c is the direction vector of the line R is a point on the line zIf t is allowed to vary from 0 to infinity, R(t) is a ray

12. 12 Intersection of Line and Plane l Plane in point normal form z n  (P -B) = 0 where n is a normal vector to the plane B is a known point on the plane P is any other point on the plane zIf P is not on the plane, n  (P -B)  0

13. 13 Intersection of Line and Plane l If a ray R(t) intersect the plane, then n  (R(t) -B) = 0 (we want to find t)  n  (A + ct -B) = 0  n  (A -B) + n  ct = 0  t (n  c) = n  (B -A) P P

14. 14 Intersection of Line and Plane Case I: n  c  0 t = n  (B -A)/(n  c)  “Hit point” P = A + ct Case II: n  c = 0  ray and plane are parallel z If n  (B -A) = 0, the ray lies on the plane zOtherwise, there is no intersection

15. 15 Direction Ray Hits the Plane If n  c > 0   < 90 o z Ray is aimed along with the normal If n  c = 0   = 90 o z Ray is parallel to the normal If n  c 90 o z Ray is aimed counter to the normal n c A n c A along withcounter to  