introduction to computer graphicspages.cpsc.ucalgary.ca/~sheelagh/courses/453/modeling2.pdf ·...
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Graphics IFaramarz Samavati
UNIVERSITY OF
CALGARY
Subdivision Curves
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UNIVERSITY OF
CALGARYCubic B-spline Subdivision
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UNIVERSITY OF
CALGARY Properties
Piecewise cubic polynomials
The limit curve is
Widely used in applications
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UNIVERSITY OF
CALGARY Open Curves vs Closed Curves
Closed Chaikin algorithm
Regular setting for open case
Open case with the interpolating condition for the first and thelast
The first point conditions
Symmetry the end point conditions
Graphics IFaramarz Samavati
UNIVERSITY OF
CALGARY Cubic B-spline Case
Closed Chaikin algorithm
Regular setting for open case
Open case with the interpolating condition for the first and thelast points
For the beginning
Symmetry the end point conditions
Graphics IFaramarz Samavati
UNIVERSITY OF
CALGARY Tensor product surface Using the curve subdivision rules for two independent directions
Assume that the pixel values are z-values for a mesh
each row
each column
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UNIVERSITY OFCALGARY
u-direction:use curve schemes for seven times
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
v-direction: use curveschemes separately
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
Tensor product of Cubic B-Spline Subdivision
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UNIVERSITY OFCALGARY Bezier Surface
Subdivision schemes for two directions
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY
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UNIVERSITY OFCALGARY Editing
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UNIVERSITY OFCALGARY Utah Tea Pot
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UNIVERSITY OFCALGARY Parametric Surface
A is usually a rectangular domain.
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UNIVERSITY OFCALGARY Parameter Domain
A
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UNIVERSITY OFCALGARY
Free Form Surfaces:Tensor Product Surface
Straight forward generalization of curve methods to surfaceBezier patch
Basis functions are Berenstein polynomial
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UNIVERSITY OFCALGARY Tensor Product Surface
Continuous deformation of a sheet of paper u-curves v-curves
A
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UNIVERSITY OFCALGARY Surfaces Having arbitrary Topologies
Tensor Product Surfaces
Non Tensor SurfacesWe can’t find u-curves and v-curves in general surfaces
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UNIVERSITY OFCALGARY
Coarse mesh Fine mesh
DOO-Sabin
Loop
Catmull-Clark
Butterfly
Kobbelt
General SubdivisionIt is not in the exam and assignments
Subdivision
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UNIVERSITY OFCALGARY Example of Subdivision
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UNIVERSITY OFCALGARY From Chaikin to Doo-Sabin Subdivision
3D corner cutting methodGeneralization of ChaikinEach step =2 stagesContracting and adjoining
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UNIVERSITY OFCALGARY One Step of Doo Subdivision
One Step of Subdivision (F,V) (NF,NV)
Line Face Curve Surface
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UNIVERSITY OFCALGARY Types of Faces
Three types of New Faces(NF) :
FF : Face- to - Face
FE : Face - to - Edge
FV: Face - to - vertexFFFF
FEFE
FFFF
FVFV
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UNIVERSITY OFCALGARY Edge-to-faces
New Contracted Face
New Contracted Face
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UNIVERSITY OFCALGARY Mask Details
Interior vertices
9/16 3/16
3/16 1/16
n= number of vertices on face
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UNIVERSITY OFCALGARY Doo Surface Properties
Convex hull property Large region of the new polyhedron are covered quad faces.Limit surface is
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UNIVERSITY OFCALGARY Examples
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UNIVERSITY OFCALGARY
3D Studio Max
Examples
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UNIVERSITY OFCALGARY Loop Subdivision
A triangle-based subdivision
Triangular mesh Triangular mesh
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UNIVERSITY OFCALGARY Face Splitting
F
A
D C B
Every face splits to four new facesF A,B,C,DUpdating the face structure
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UNIVERSITY OFCALGARY Mask for Edge-Vertex
edge-vertex
vertex-vertex
Edge-vertex mask
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UNIVERSITY OFCALGARY Displacement Mask
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UNIVERSITY OFCALGARY Mask for Vertex-Vertex
n: number of adjacent vertices (valences)n > s
has much more influence on than other neighbors
n=3 , or
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UNIVERSITY OFCALGARY Demo
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UNIVERSITY OFCALGARY