Computer Aided Geometric Design The Bzier curvecurve intersection

Computer Aided Geometric Design The Bézier curve-curve intersection problem l How the geometric properties of Bézier curves are used in CCI solving 1

The convex hull property Recall: a Bézier curve is contained inside the convex hull of its control mesh 2

The convex hull property Recall: a Bézier curve is contained inside the convex hull of its control mesh 3

The convex hull property l 4

The convex hull property l 5

Testing for multiple intersections 6

Testing for multiple intersections 7

Testing for multiple intersections 8

Testing for multiple intersections 9

Testing for multiple intersections 10

Intermediate summary Convex hull test for discriminating between cases where the curves have no intersections vs. may have intersections. l Cones of tangent directions test for discriminating between cases where the curves have at most one intersection vs. may any number of intersections. l 11

CCI solution complete algorithm Convex hull test May intersect No intersections Tangent cones Stop (no intersections) test Any number of intersections Subdivide Fail At most one intersection Try solving numerically Succeed Stop and return result 12