cmsc427 points, polylines and polygons
TRANSCRIPT
CMSC427Points,polylinesandpolygons
• Intheory,smoothcurve:
• Inreality,piecewisediscreteapproxima@on:
Issue: discretization of continuous curve
Modeling with discrete approximations
Increase fidelity with more points
Points, polylines and polygons
Points Also called vertices
Polyline Continuous sequence
of line segments
Polygon Closed sequence of line segments
Polygon properties I
• Simplenoself-intersec@onsnoduplicatepoints
• Non-simple self-intersections duplicate points
Polygon properties II
• ConvexpolygonAnytwopointsinpolygoncanbeconnectedbyinsideline
• Concave polygon Not true of all point pairs inside polygon
Point in polygon problem
IsPinsideoroutsidethepolygon?
Case1 Case2
Case3
Point in polygon problem
IsPinsideoroutsidethepolygon?
Case1 Case2
Case3
1 crossing
3 crossings
2 crossings
Point in polygon problem
IsPinsideoroutsidethepolygon?Oddcrossings–insideEvencrossings–outside
Case1 Case2
Case3
1 crossing
3 crossings
2 crossings
Algorithmicefficiency?
Point in polygon problem
IsPinsideoroutsidethepolygon?
Point in polygon problem
IsPinsideoroutsidethepolygon?
• Polygoncollision• Returnyes/no
• Polygonintersec@on• Returnpolygonofintersec@on(P)
• Polygonrasteriza@on• Returnpixelsthat intersect
• Polygonwindingdirec@on• Returnclockwise(CW)orcounterclockwise(CCW)
Other polygon problems
Moral: easier with simple, convex, low count polygons
Vs
Triangular mesh
Vs
Why triangles …
Whytriangles?1.Easiestpolygontorasterize2.Polygonswithn>3canbenon-planar3.Ligh@ngcomputa@onsin3Dhappenatver@ces-morever@cesgivesmootherillumina@oneffects
Polygon triangulation
Theorem: Every simple polygon has a triangulation
• Proof by induc-on Base case: n = 3 Induc-ve case A) Pick a convex corner p. Let q and r be pred and succ ver-ces. B) If qr a diagonal, add it. By induc-on, the smaller polygon has a triangula-on. C) If qr not a diagonal, let z be the reflex vertex farthest to qr inside △pqr. D) Add diagonal pz; subpolygons on both sides have triangula-ons.
• Parametriccurves1. Modelobjectsbyequa@on2. Complexshapesfromfewvalues3. Modelingarbitraryshapecanbehard
• Polylines1. Modelobjectsbydatapoints2. Complexshapesneedaddi@onaldata3. Canmodelanyshapeapproximately
• Lookingforward• Usepolylinestocontrolgeneralparametriccurves• B-splines,NURBS
Parametric curves vs. polylines
1. Defini@onsofpolylineandpolygons2. Polylinesandpolygonsaspiecewise
discreteapproxima@onstosmoothcurves3. Defini@onsofproper@esofpolygons
(simple/non-simple,concave/convex)4. Defini@onofpoint-in-polygonproblemand
crossingsolu@on5. Trianglesaregood(simplestpolygon,
alwaysplanar,easytorasterize,moreisgood)
6. Defini@onofpolygontriangula@on(don’tneedtoknowthetheoremyet)
What you should know after today