Solid Modeling. Solid Modeling. Polygon Meshes. Polygon Meshes. Representing Polygon Meshes. Representing Polygon Meshes

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1 Solid Modeling Foley & Van Dam, hapter 11.1 and hapter 12 Solid Modeling olygon Meshes lane quation and Normal Vectors Volume Representation Sweep Volume Spatial Occupancy numeration inary Space artition Tree onstructive Solid Geometry oundary Representation olygon Meshes polygon mesh is a collection of polygons, along with a normal vector associated to each polygon vertex : n edge connects two vertices polygon is a closed sequence of edges n edge can be shared by two adjacent polygons vertex is shared by at least two edges normal vector pointing outside is associated with each polygon vertex roperties: olygon Meshes onnectedness: mesh is connected if there is an path of edges between any two vertices Simplicity: mesh is simple if the mesh has no holes in it lanarity: mesh is planar if every face of it is a planar polygon onvexity: The mesh is convex if the line connecting any two points in the mesh belongs to the mesh Representing olygon Meshes Representing olygon Meshes xplicit (vertex list) 1 =(V 1,,V 4 ) 2 =(,V 3,V 4 ) ointers to a vertex list V=(V 1,,V 3,V 4 ) 1 =(1,2,4) 2 =(4,2,3) ointers to an edge list V=(V 1,,V 3,V 4 ) 1=(1,2, 1,) 2=(2,3, 2,) 3=(3,4, 2,) 4=(4,2, 1, 2 ) 5=(4,1,1,) 1=( 1, 4, 5 ) 2=( 2, 3, 4 ) Represents null V V 4 3 V 3 xplicit Space effective for single polygons Not very informative ointers to Vertex List Saves spaces for shared vertices asy to modify a single vertex Difficult to find polygons sharing an edge Multiple drawing and clipping of shared edges ointers to dge List Solves multiple drawing and clipping problem Still hard to find all edges sharing a vertex

2 lane quations x y x y z D 0 or [ D] 0 z N=[,,] is the plane normal ( N =1) 1 D=-N D is the plane distance from the origin. N points outside, so: x+y+z+d<0, (x,y,z) is inside N=[,,] x+y+z+d>0, (x,y,z) is outside V 1 Three non collinear points V 1,, V 3 are sufficient to find the coefficients V,, and D D 3 lane quations Given V 1,,V 3, the plane normal (or the coefficients, and ) can be computed with the cross product: S=(V 3 -V 1 )x( -V 1 ) and N=S/ S D can be found by plugging any point of the plane in the equation x+y+z+d=0 The plane equation is not unique D V 1 V 3 N=[,,] lane quations xample: Find the equation of the plane containing the points V 1 =(2,1,2), =(3,2,1) and V 3 =(1,1,3) First we compute the vectors -V 1 = (1,1,-1) and V 3 -V 1 = (-1,0,1) The, and coefficients of the equation are given by the cross product S=( -V 1 )x(v 3 -V 1 ) N = [,, ] = S/ S = (1,0,1)/ (1,0,1) = Finally we substitute V 1 in the equation to find D. To simplify the computation, we use [,,]=(1,0,1) x+y+z+d= D=0 D=-4 nd the plane equation is x+z-4=0 The equation with the unitary normal is 1 1,0, x z Volume Representation collection of techniques to represent and define volumetric objects Desired properties: Rich representation Unambiguous Unique representation ccurate ompact fficient ossible to test validity Volume Representation Volume Representation rimitive Instancing Sweep volumes Spatial Occupancy (voxels, Octree, S) onstructive Solid Geometry oundary Rep. olyhedra Free form rimitive Instancing Define a family of parameterized objects The definition is procedural (a routine defines it) Not general, must be individually defined for each family of objects xample: Wheels/Gears Diameter = 10 Teeth = 16 Hole = 3 Diameter = 8 Teeth = 24 Hole = 5

3 Sweep Volume Sweep Volume: sweeping a 2D area along a trajectory creates a new 3D object Translational Sweep: 2D area swept along a linear trajectory normal to the 2D plane Tapered Sweep: scale area while sweeping Slanted Sweep: trajectory is not normal to the 2D plane Rotational Sweep: 2D area is rotated about an axis General Sweep: object swept along any trajectory and transformed along the sweep Sweep Volume Translational and rotational sweep volumes Spatial Occupancy numeration Space is described as a regular array of cells (usually cubes). ach cell is called a Voxel 3D object is represented as a list of filled voxels Spatial Occupancy numeration ros: asy to verify if a point (a voxel) is inside or outside an object oolean operations are easy to apply ons: Memory costs are high Resolution is limited to size and shape of voxel Quadtrees Quadtree is a data structure enabling efficient storage of 2D data ompletely full or empty regions are represented by one cell; recursive subdivision is used on the others Octrees n Octree is a 3D generalization of a Quadtree ach node in an Octree has eight children rather than four Describes a recursive partitioning of a volume into cells that are completely full or empty

4 inary Operations on Quad/Octrees Notation: F Internal Node Full (artially full) Leaf Node Union: F = F = Intersection: F = = mpty Leaf Node omplement: F = = F inary Operations on Quad/Octrees F F F F F F F F F Difference: F = = inary Space artition Trees - S ach internal node represents a plane in 3D space ach node has 2 children pointers one for each side of the plane. leaf node represents a homogeneous portion of space - either in or out. asy to determine if a point is inside or outside an object (recurse down the S tree) f d c b i e h a j g k b a c e d out f g in out in out h out in i in out j k out in out onstructive Solid Geometry ombine simple primitives using oolean operations and represent as a binary tree. To generate the object the tree is processed in a depth-first pass. ons: representation is not unique - onstructive Solid Geometry n object defined by a binary SG tree - - oundary Representations closed 2D surface defines a 3D object t each point on the boundary there is an in and an out side oundary representations can be defined in two ways: rimitive based. collection of primitives forming the boundary (polygons, for example) Freeform based (splines, parametric surfaces, implicit forms)

5 oundary Representations polyhedron is a solid bounded by a set of polygons polyhedron is constructed from: Vertices V dges Faces F ach edge must connect two vertices and be shared by exactly two faces t least three edges must meet at each vertex oundary Representations simple polyhedron is one that can be deformed into a sphere (contains no holes) simple polyhedron must satisfies uler's formula: V-+F=2 V = 8 = 12 F = 6 V = 5 = 8 F = 5 V = 6 = 12 F = 8 oundary Representations uler s formula can be generalized to a polyhedron with holes and multiple components V-+F-H=2(-G) Where: H is the number of holes in the faces is the number of separate components G is the number of pass-through holes (genus if =1) V, and F are respectively vertices, edges and faces V + F H = 2 (-G) = 2 (1-1)