Assemble the coefficient matrix and solve the linear system, Mathematics

Solve discrete harmonic mapping of a given surface patch (suppose the surface is genus-0 and with one boundary)

1. Map the boundary loop onto a unit rectangle using chord-length parameterization:

a) Pick a random starting point B0, trace the boundary loop L, compute its total length |L|;

b) Starting from B0, find vertices B1, B2, B3 on L, such that the lengths of the boundary paths from B0 to B1, B2, and B3 , are (or just passed) |L|/4, |L|/2, and 3|L|/4, respectively;

c) Map B0, B1, B2, and B3, to (0,0), (1,0), (1,1), and (0,1) respectively, map curve segments (B0, B1), (B1, B2), (B2, B3), (B3, B0) to four boundaries of the unit square accordingly using chord-length parameterization. This fixes the boundary mapping: all boundary vertices are mapped.

2. For every interior vertex, map it to the weighted average center of its one-rings. The weight is the harmonic weight, i.e., the cotangent weight defined on the original mesh.

Initially, map all interior vertices to (0.5,0.5), then use one of the following two ways to find the solution:

A easier but slower way:

Iteratively move the mapping position of each vertex:

2207_equationm.png

Here u denotes the coordinates on the plane, and we use k and k+1 to indicate the position before and after the (k+1)-th iteration.

* You may need to do a few thousand iterations until you see the movement is small.

A faster way:

Assemble the coefficient matrix and solve the linear system (check slides)

Posted Date: 4/2/2013 1:32:31 AM | Location : United States







Related Discussions:- Assemble the coefficient matrix and solve the linear system, Assignment Help, Ask Question on Assemble the coefficient matrix and solve the linear system, Get Answer, Expert's Help, Assemble the coefficient matrix and solve the linear system Discussions

Write discussion on Assemble the coefficient matrix and solve the linear system
Your posts are moderated
Related Questions
Newton's Method : If x n is an approximation a solution of f ( x ) = 0 and if given by, f ′ ( x n ) ≠ 0 the next approximation is given by

a cheeseburger cost $6.39 more than a burger of $2.29, what is the difference?

Frequency Distribution or Variance Ratio Distribution This was developed by R. A Fisher in 1924 and is normally defined in terms of the ratio of the variances of two usually d

how to find area under a curve?

Table shows the productivity for the countries Pin and Pang. 1) If the working population of Pin and Pang are both 6 million, divided equally between the two industries in

The two sides of a triangle are 17cm and 28cm long, and the length of the median drawn to the third side is equal to 19.5 cm. What is the distance from an endpoint of the median to

Definition of a Function Now we need to move into the second topic of this chapter.  Before we do that however we must look a quick definition taken care of.

What is a lattice? Which of the following graphs are lattice and why? Ans:  Let (L, ≤) be a poset. If each subset {x, y} consisting of any two elements of L, comprises a glb (I

a, b,c are in h.p prove that a/b+c-a, b/a+c-b, c/a+b-c are in h.p To prove: (b+c-a)/a; (a+c-b)/b; (a+b-c)/c are in A.P or (b+c)/a; (a+c)/b; (a+b)/c are in A.P or 1/a; 1

what states and marketing tasks?