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

a man in rested rupee 800 is buying rupee 5 shares and then are selling at premium of rupee 1.15. He sells all the shares.find profit

I need help fast with my calculus work

What were the two main political parties that formed in the majority of the new nations of Latin America post independence? In what ways were they different? Which party ascended t

a triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3.what is the area of the triangle.

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

A business has the opportunity to expand by purchasing a machine at a cost of £80,000. The machine has an estimated life of 5 years and is projected to generate a cashflow of £20,0

the value of square root of 200multiplied by square root of 5=

how to curve trace? and how to know whether the equation is a circle or parabola, hyperbola ellipse?

I had just come back from a very interesting talk arranged by a Mathematics Centre, it was aimed at parents of primary school-going children. They had talked about, and demonstrate