Graphics Subdivision Surfaces cgvr korea ac kr Graphics
Graphics Subdivision Surfaces 고려대학교 컴퓨터 그래픽스 연구실 cgvr. korea. ac. kr Graphics Lab @ Korea University
Subdivision n CGVR How do You Make a Smooth Curve? cgvr. korea. ac. kr Graphics Lab @ Korea University
Subdivision Surfaces n CGVR Coarse Mesh & Subdivision Rule n Define smooth surface as limit of sequence of refinements cgvr. korea. ac. kr Graphics Lab @ Korea University
Key Questions n How Refine Mesh? n n CGVR Aim for properties like smoothness How Store Mesh? n Aim for efficiency for implementing subdivision rules cgvr. korea. ac. kr Graphics Lab @ Korea University
Loop Subdivision Scheme n CGVR How Refine Mesh? n Refine each triangle into 4 triangles by splitting each edge and connecting new vertices cgvr. korea. ac. kr Graphics Lab @ Korea University
Loop Subdivision Scheme n CGVR How Position New Vertices? n Choose locations for new vertices as weighted average of original vertices in local neighborhood cgvr. korea. ac. kr Graphics Lab @ Korea University
Loop Subdivision Scheme n CGVR Different Rules for Boundaries: cgvr. korea. ac. kr Graphics Lab @ Korea University
Loop Subdivision Scheme n CGVR Limit Surface Has Provable Smoothness Properties cgvr. korea. ac. kr Graphics Lab @ Korea University
Subdivision Schemes n CGVR There are different subdivision schemes Different methods for refining topology n Different rules for positioning vertices n o Interpolating versus approximating Face Split Vertex Split Triangular Meshes Quad. Meshes Approximating Loop (C 2) Catmull-Clark (C 2) Doo-Sabin, Midege (C 1) Biquartic (C 2) Interpolating Mod. Butterfly (C 1) Kobbelt (C 1) cgvr. korea. ac. kr Graphics Lab @ Korea University
Subdivision Schemes cgvr. korea. ac. kr Loop Butterfly Catmull-Clark Doo-Sabin CGVR Graphics Lab @ Korea University
Subdivision Schemes Loop cgvr. korea. ac. kr Butterfly Catmull-Clark CGVR Doo-Sabin Graphics Lab @ Korea University
Key Questions n How Refine Mesh? n n CGVR Aim for properties like smoothness How Store Mesh? n Aim for efficiency for implementing subdivision rules cgvr. korea. ac. kr Graphics Lab @ Korea University
Polygon Meshes n CGVR Mesh Representations Independent faces n Vertex and face tables n Adjacency lists n Winged-Edge n cgvr. korea. ac. kr Graphics Lab @ Korea University
Independent Faces n CGVR Each Face Lists Vertex Coordinates Redundant vertices n No topology information n (x 4, y 4, z 4) (x 3, y 3, z 3) F 1 (x 1, y 1, z 1) F 2 F 3 (x 2, y 2, z 2) (x 5, y 5, z 5) Face Table F 1 (x 1, y 1, z 1) (x 2, y 2, z 2) (x 3, y 3, z 3) F 2 (x 2, y 2, z 2) (x 4, y 4, z 4) (x 3, y 3, z 3) F 3 (x 2, y 2, z 2) (x 5, y 5, z 5) (x 4, y 4, z 4) cgvr. korea. ac. kr Graphics Lab @ Korea University
Vertex & Face Tables n CGVR Each Face Lists Vertex References Shared vertices (x 3, y 3, z 3) (x 4, y 4, z 4) n Still no topology information F 2 F 3 F 1 n (x 1, y 1, z 1) cgvr. korea. ac. kr (x 2, y 2, z 2) (x 5, y 5, z 5) Vertex Table Face Table V 1 V 2 V 3 V 4 V 5 F 1 V 2 V 3 F 2 V 4 V 3 F 3 V 2 V 5 V 4 x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 x 5 y 5 z 5 Graphics Lab @ Korea University
Adjacency Lists n CGVR Store all Vertex, Edge, and Face Adjacencies Efficient topology traversal n Extra storage V 2 V 3 n E 1 E 4 E 2 E 5 E 6 F 1 F 2 E 1 V 2 V 4 V 3 E 5 E 4 E 3 E 4 V 3 E 3 F 1 E 2 F 3 F 1 F 2 E 5 F 3 V 2 E 6 E 7 V 5 V 4 V 3 V 1 E 6 E 5 E 3 E 2 F 3 F 2 F 1 cgvr. korea. ac. kr Graphics Lab @ Korea University
Winged Edge n CGVR Adjacency Encoded in Edges All adjacencies in O(1) time n Little extra storage (fixed records) n Arbitrary polygons n 4 v 2 {Ei} {Vi} cgvr. korea. ac. kr f 2 e 21 2 2 1 e 22 1 f 1 e 12 e 11 v 1 {Fi} Graphics Lab @ Korea University
Winged Edge n CGVR Example F 1 (x 1, y 1, z 1) Vertex Table V 1 V 2 V 3 V 1 V 2 x 1, y 1, z 1 x 2, y 2, z 2 x 3, y 3, z 3 x 4, y 4, z 4 x 5, y 5, z 5 cgvr. korea. ac. kr (x 4, y 4, z 4) (x 3, y 3, z 3) F 2 (x 2, y 2, z 2) Edge Table E 1 E 6 E 3 E 5 E 6 E 1 E 2 E 3 E 4 E 5 E 6 E 7 F 3 V 1 V 2 V 3 V 4 V 2 V 5 V 4 V 5 (x 5, y 5, z 5) 11 12 21 22 E 2 E 4 E 3 E 1 E 3 E 6 E 2 E 5 E 1 E 4 E 1 E 3 E 7 E 5 E 3 E 6 E 4 E 7 E 5 E 2 E 7 E 4 E 5 E 6 Face Table F 1 e 1 F 2 e 3 F 3 e 5 Graphics Lab @ Korea University
Summary n Advantages: n n n CGVR Simple method for describing complex surfaces Relatively easy to implement Arbitrary topology Smoothness guarantees Multiresolution Difficulties: Intuitive specification n Parameterization n Intersections n cgvr. korea. ac. kr Graphics Lab @ Korea University
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