AMS 345/CSE 355 Computational Geometry Lecture: Geometric Primitives Joe Mitchell Code/figures from [O’Rourke]: Computational Geometry in C: Chap 1

Cross Products c b a

Cross Products Example: ab × ac = [2,1] × [2,2] = (2*2-2*1) = 1 ab × ad = [2,1] × [0,-1] = (2*(-1)-0*1) = -2 c=(2,2) b=(2,1) a d=(0,-1)

Determinant Form

Area of Convex Polygon

Area of Simple Polygon

Point Type

Vertex Structure

NEW and ADD

b c a

Proper Segment Intersection Fact: ab and cd properly intersect if and only if (1) points a and b are on opposite sides of line cd, AND (2) c and d are on opposite sides of line ab

Improper Segment Intersection

Between b c a b c a

Segment Intersection

Diagonalie Test if segment ab, joining vertices a and b, forms a “diagonalie” = an interior or exterior diagonal for P. Check: does the segment ab avoid intersecting all edges of P, except those that share endpoints with ab? a a b a b b

InCone

InCone b a0 a1 a b a a1 a0

Diagonal

Additional Primitives ConvexVertex(a,P) – returns T iff a is a convex vertex of P RaySegIntersectProp RayRayIntersectProp RaySegIntersect, RayRayIntersect SegSegShareEndptOnly SegSegT-Intersect PointInTriangle SegInPolygon Etc, etc

From Practice Midterm

From Practice Midterm