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1 Planes, Polygons and Objects ©Anthony Steed 1999-2005

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

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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

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4 Polygonal Meshes

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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

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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

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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.

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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

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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

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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

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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...

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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

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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!

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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...

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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

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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)

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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

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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…

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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.

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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)

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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

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Vertex – A point at which two or more edges meet Edge – A line segment at which two faces intersect Face – A flat surface Vertices, Edges, Faces.

Vertex – A point at which two or more edges meet Edge – A line segment at which two faces intersect Face – A flat surface Vertices, Edges, Faces.

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