Alec jacobson thesis analysis

chapter 2

Q1: what is an Euler-Lagrange equation? Here why the Euler-Lagrange equation is the Laplace equation?

A1: refer to http://blog.youkuaiyun.com/seamanj/article/details/50898781

Q2: How can we get equation (2.10) from equation (2.9)?

A2: refer to http://blog.youkuaiyun.com/seamanj/article/details/50899295

Q3: what is a hat funtion?

The shape of hat function is just like a hat, notice that it's a continuous function not a discrete one.  Every vertex in mesh has a hat function, then we sum up all these hat functions as a approximate function of the real mesh surface.

say the black one is the real surface.

Then, we use the formulation(2.11) to approximate the real surface

see the blue line is coincident with the black one exactly.

the matlab source code as follows:

x1 = linspace(0,1,100);
y11 = 1 - x1;
y12 = x1;
x2 = linspace(1,2,100);
y21 = 2-x2;
y22 = x2-1;
x3 = linspace(2,3,100);
y31 = 3-x3;
y32 = x3-2;
x4 = linspace(3,4,100);
y41 = 4-x4;
y42 = x4-3;
hold on
plot(x1,y11,'.c')
plot(x1,y12,'.y')
plot(x2,y21,'.y')
plot(x2,y22,'.r')
plot(x3,y31,'.r')
plot(x3,y32,'.g')
plot(x4,y41,'.g')
plot(x4,y42,'.m')
x = [x1 x2 x3 x4];
y_real = [0.3+0.2*x1 0.6-0.1*x2 0.2*x3 1.8-0.4*x4];
y = [0.3*y11+0.5*y12 0.5*y21+0.4*y22 0.4*y31+0.6*y32 0.6*y41+0.2*y42];
plot(x,y_real,'.k')
plot(x,y,'-b')

N4: explain about elliptic PDE

Q5: what is the gradient of the hat funciton? how to solve?