1 Planes, Polygons and Objects ©Anthony Steed

2 Overview n Polygons n Planes n Creating an object from polygons

3 No More Spheres n Most things in computer graphics are not described with spheres! n Polygonal meshes are the most common representation n Look at how polygons can be described and how they can used in ray-casting

4 Polygonal Meshes

5 Polygons n A polygon (face) Q is defined by a series of points n The points are must be co-planar n 3 points define a plane, but a 4th point need not lie on that plane

6 Convex, Concave n Convex n Concave CG people dislike concave polygons CG people would prefer triangles!! –Easy to break convex object into triangles, hard for concave

7 Equation of a Plane n a,b,c and d are constants that define a unique plane and x,y and z form a vector P.

8 Deriving a,b,c & d (1) p 0 p 2 p 1 p n The cross product defines a normal to the plane n There are two normals (they are opposite) n Vectors in the plane are all orthogonal to the plane normal vector

9 Deriving a,b,c & d (2) n So p-p 0 is normal to n therefore n But if n = (n 1,n 2,n 3 ) a= n 1 b= n 2 c= n 3 (n.p) d = n.p0 = n 1 *x 0 + n 2 *y 0 + n 3* z 0

10 Half-Space n A plane cuts space into 2 half-spaces n Define n If l(p) =0 point on plane n If l(p) > 0 point in positive half-space n If l(p) <0 point in negative half-space

11 Polyhedra n Polygons are often grouped together to form polyhedra Each edge connects 2 vertices and is the join between two polygons Each vertex joins 3 edges No faces intersect n V-E+F=2 For cubes, tetrahedra, cows etc...

12 Example Polhedron vo v1 v2 v3 v4 v5 e1 e2 e3 e4 e5 e6 e7 e9 e8 n F0=v0v1v4 n F1=v5v3v2 n F2=v1v2v3v4 n F3=v0v4v3v5 n F4=v0v5v2v1 n V=6,F=5, E=9 n V-E+F=2

13 Representing Polyhedron (1) n Exhaustive (array of vertex lists) faces[1] = (x0,y0,z0),(x1,y1,z1),(x3,y3,z3) faces[2] = (x2,y2,z2),(x0,y0,z0),(x3,y3,z3) etc …. n Very wasteful since same vertex appears at 3(or more) points in the list Is used a lot though!

14 Representing Polyhedron (2) n Indexed Face set n Vertex array vertices[0] = (x0,y0,z0) vertices[1]=(x1,y1,z1) etc … n Face array (list of indices into vertex array) faces[0] = 0,2,1 faces[1]=2,3,1 etc...

15 Vertex order matters n Polygon v0,v1,v4 is NOT equal to v0,v4,v1 n The normal point in different directions n Usually a polygon is only visible from points in its positive half-space n This is known as back- face culling vo v1 v2 v3 v4 v 5

16 Representing Polyhedron (3) n Even Indexed face set wastes space Each face edge is represented twice n Winged edge data structure solves this vertex list edge list (vertex pairs) face list (edge lists)

17 The Edge List Structure n Edge contains Next edge CW Next edge CCW Prev edge CW Prev edge CCW Next face Prev face Next vertex Prev vertex

18 Advantages of Winged Edge n Simple searches are rapid find all edges find all faces of a vertex etc… n Complex operations polygon splitting is easy (LOD) silhouette finding potentially efficient for hardware etc…

19 Building the WE n Build indexed face set n Traverse each face in CCW order building edges label p and n vertices, p and n faces and link previous CCW edge –we fill in next CCW on next edge in this face –we fill in next CW and prev CW when traversing the adjacent face.

20 Exercises n Make some objects using index face set structure n Verify that V-E+F=2 for some polyhedra n Think about testing for intersection between a ray and a polygon (or triangle)

21 Recap n We have seen definition of planes and polygons and their use in approximating general shapes n We have looked at two data structures for storing shapes Indexed face sets Winged edge data sets n The former is easy to implement and fast for rendering n The latter is more complex, but makes complex queries much faster n It is possible, though we haven’t shown how, to convert between the two