1 PA1 Supplementary notes Programming assignment You need to implement the following: 1. Display basic mesh Information Find the number of vertices, edges, faces, boundaries, and compute Euler characteristic and number of genus Corresponding to function Mesh::DisplayMeshInfo() 2. Compute normal at each vertex Corresponding to function Mesh::ComputeVertexNormals() 3. Compute mean curvature at each vertex Corresponding to function Mesh::ComputeVertexCurvatures()
Transcript
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PA1 Supplementary notes
Programming assignment
You need to implement the following:
1. Display basic mesh Information Find the number of vertices, edges, faces, boundaries,
and compute Euler characteristic and number of genus Corresponding to function Mesh::DisplayMeshInfo()
2. Compute normal at each vertex Corresponding to function
Mesh::ComputeVertexNormals()
3. Compute mean curvature at each vertex Corresponding to function
Mesh::ComputeVertexCurvatures()
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PA1 Supplementary notes
Programming assignment
4. Explicit umbrella smoothing Corresponding to function
Mesh::UmbrellaSmooth ()
Implicit umbrella smoothing Corresponding to function
Mesh::ImplicitUmbrellaSmooth ()
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PA1 Supplementary notes
Wavefront OBJ file format comment
three vertices of each face
coordinates of vertex 1
v1
v2
v3
v4
counter-clockwise order as you look at the face from outside
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PA1 Supplementary notes
How to modify program arguments
1) Select the menu item “Project Property”
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PA1 Supplementary notes
How to modify program arguments
2) Select the “Debugging” tag
3) Type your arguments here
4) Press enter to comfirm
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PA1 Supplementary notes
Normal at a vertex
From Siggraph 2000 subdivision course notes, the normal vector at an interior vertex of valence k can be computed as t1 x t2, where ti are tangent vectors computed as:
Example: for valence four, the masks are
[1, 0, –1, 0] and [0, 1, 0, –1].
1
02
1
01
2sin
2cos
k
ii
k
ii
k
i
k
i
pt
pt
1 -1
0
0
0 0
-1
1
t1
t2
Masks for valence four
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PA1 Supplementary notes
Normal at a vertex
At a boundary vertex p with valence k, the normal is computed as talong x tacross computed as follows:
p0
p1
pk-1
p
)1/(
4sin)2cos2()(sin
3
22
2
110
1
10
10
kwhere
ki
k
k
k
iik
across
kalong
ppp
pp
ppp
t
ppt
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PA1 Supplementary notes
Mean curvature at a vertex
From Siggraph99 Desbrun et al , the discrete mean curvature at an interior vertex p with valence k can be computed as the L2-norm of
where A is the sum of areas of all the triangles sharing the vertex p
1
0
))(cot(cot4
1 k
jjjjAppn
p
pj
p
pj-1
pj
Pj+1
one term of the summation, corresponding to edge p pj
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PA1 Supplementary notes
Explicit Umbrella Smoothing
Fairing operator
In matrix form
In your implementation you do not need to build the matrix
1
0
1 k
jiji xx
kx -
iinewi xxx
)L(XXX tt1t
t1t L)X(IX λ
Drawbacks: small time step for large mesh slow
Where t is time stamp
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PA1 Supplementary notes
Implicit Umbrella Smoothing Explicit updating
Allows large time step In your implementation setting to 1 is okay You can use biconjugate gradient (BCG)
method to solve the linear system
)L(XXX tt1t 1
t1t XL)X(I λ
t1t XL)(IX 1 λ
λ
You need to solve a sparse linear system
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PA1 Supplementary notes
Sparse Linear System
You need to build the sparse linear system by create a Matrix object Use Matrix::AddElement to add an element Use Matrix::SortMatrix to sort the elements after
adding all of them
You need to implement the function Matrix::BCG() to solve the linear system Use Matrix::Multiply to compute b = Ax Use Matrix::PreMultiply to compute bT = xTA
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PA1 Supplementary notes
Using Boundary Half-edge
For open meshes, you can also maintain boundary half-edges such that the searching and updating will be much more easy. (every things are circular list, also no need to handle NULL pointers)
Here the boundary half-edges are shown in red as the triangles are being loaded.
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PA1 Supplementary notes
Useful Classes and Functions Classes OneRingHEdge and OneRingVertex
provide an interface to access the one-ring neighboring half-edges and vertices of a given vertex.
Function Vertex::Valence() let you know the valence (# of neighboring vertices) of a vertex.
