CS 430 Computer Graphics Vectors Part IV Polygon

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CS 430 Computer Graphics Vectors Part IV Polygon Intersection Chi-Cheng Lin, Winona State University

CS 430 Computer Graphics Vectors Part IV Polygon Intersection Chi-Cheng Lin, Winona State University

Topics Polygon Intersection Problems l Convex Polygons and Polyhedra l Ray Intersection and Clipping

Topics Polygon Intersection Problems l Convex Polygons and Polyhedra l Ray Intersection and Clipping l Cyrus-Beck Clipping Algorithm l 2

Polygon Intersection Problems Polygons are fundamental objects in both 2 D and 3 D

Polygon Intersection Problems Polygons are fundamental objects in both 2 D and 3 D graphics l A polygonal mesh can be used to model a 3 D graphics object l Polyhedron l z. A polygonal mesh which forms a closed surface that encloses some space 3

Polygon Intersection Problems Is a given point P inside or outside the object? l

Polygon Intersection Problems Is a given point P inside or outside the object? l Where does a given ray R first intersect the object? l Which part of a given line L lies inside/outside the object? l 4

Polygon Intersection Problems 5

Polygon Intersection Problems 5

Convex Polygons and Polyhedra General case of polygon/polyhedron intersection problems is complex l Convex

Convex Polygons and Polyhedra General case of polygon/polyhedron intersection problems is complex l Convex polygons/polyhedra are easier to deal with l z 2 D: a convex polygon can be completely described by a set of “bounding lines” z 3 D: a convex polyhedron can be completely described by a set of “bounding planes” Deal with bounding lines/planes 6

Convex Polygons and Polyhedra 7

Convex Polygons and Polyhedra 7

Convex Polygons and Polyhedra l Outward normal z. Every bounding line of a 2

Convex Polygons and Polyhedra l Outward normal z. Every bounding line of a 2 D convex polygon z. Every bounding plane of a 3 D convex polyhedron 8

Convex Polygons and Polyhedra l Outside half-space Inside half-space Outside half-space L 1 l

Convex Polygons and Polyhedra l Outside half-space Inside half-space Outside half-space L 1 l Polyhedron z. Intersection of all the inside half-spaces 9

Ray Intersection A Intersection problem: when does a ray enter and exit a convex

Ray Intersection A Intersection problem: when does a ray enter and exit a convex polygon? l A ray A + ct hits a convex polygon P exactly twice l z. Entering hit point: A + ctin z. Exiting hit point: A + ctout z. The ray is inside P for all t [tin, tout] A c P 10

Clipping problem: given points A and C, which part of line segment AC lies

Clipping problem: given points A and C, which part of line segment AC lies inside a convex polygon P ? l A’ = A + c max(0, tin) C’ = A + c min(tout, 1) l 0 A tout tin P 1 C 0 1 tout tin C A tin 0 1 tout A C 11

Clipping l How are tin and tout computed? z. We must find the intersection

Clipping l How are tin and tout computed? z. We must find the intersection of the ray and each bounding line in turn l Assume a bounding line is represented as {B, n}, where z. B: some point on the line zn: outward normal 12

Clipping If n c > 0, ray is exiting from P If n c

Clipping If n c > 0, ray is exiting from P If n c = 0, ray is parallel to P If n c < 0, ray is entering P l For each bounding line, find l z. Hit time of the ray with bounding line z. Whether the ray is entering or exiting n 1 A c B 1 B 2 n 2 13

Clipping l Approach z. Candidate interval of t : [tin, tout ] z. Keep

Clipping l Approach z. Candidate interval of t : [tin, tout ] z. Keep track of the maximum entering time tin z. Keep track of the minimum exit time tout z. We want to chop the interval at each iteration z. Can stop as soon as tin > tout (what does this mean? ) 14

Clipping - Algorithm Initialize [tin, tout] [0, 1] for each boundary find the hit

Clipping - Algorithm Initialize [tin, tout] [0, 1] for each boundary find the hit time thit if entering then tin = max(tin, thit) else tout = min(tout, thit) if tin > tout then no intersection stop segment from A + ctin to A + ctout lies inside P (We found: endpoints of clipped lines entering and exiting points of ray) 15

Clipping - Example Updates on tin and Line test tin 0 0 1 0

Clipping - Example Updates on tin and Line test tin 0 0 1 0 2 0 3 0 4 0. 2 5 0. 28 tout: tout 0. 83 L 0 0. 66 @1 @. 83 @. 66 0. 66 C L 1 0. 66 @3. 4 0. 66 L 2 @0 L 5 @. 28 L 4 A @-4. 7 L 3 16

Cyrus-Beck Clipping Algorithm Clip a line segment against any convex polygon l Input parameters

Cyrus-Beck Clipping Algorithm Clip a line segment against any convex polygon l Input parameters l z. Line segment z. List of bounding lines l Output parameter z. Clipped line segment l Return value z 1, if part of segment lies in P z 0, otherwise 17

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