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###### 日期：2020-03-08 10:15

EE575: COMPUTATIONAL DIFFERENTIAL GEOMETRY FOR ENGINEERS:

appropriate). Include all of the codes at the end in an appendix. You can refer to all of the class materials

and reference books from the syllabus if needed, but I suggest focusing on HWs and codes posted. Please

post all comments or questions on blackboard or email me. No office hours or meeting with me during

midterm. Do one of problem 7 or 8. Please include screenshots/graphics that will help in assessing if your

code works or not.

Problem 1. (10pt) A fundamental solution is the solution of an equation corresponding to the initial

condition of a delta function at a known position. Consider the heat equation.

(??tu(x, t) ? k?2?x2 u(x, t) = 0 (x, t) ∈ R × (0, ∞)

u(x, 0) = δ(x)

Show that a solution to this problem is the fundamental solution

Φ(x, t) = 14πkt exp ?x24kt.

Show all the steps. Show that it is unique, i.e. there is no other solution satisfying heat equation and the

given initial condition. (hint: you do not need to derive second equation from the first one).

Problem 2. (15pt) Prove that the stereographic projection is a conformal map (hint: can use properties of

the first fundamental form).

Problem 3. (10pt) What are the principal curvatures, the principal directions, mean and Gaussian curvatures

for (1) Sphere, (2) Torus, (3) Catenoid, (4) Helicoid, (5) Cone?

Problem 4. (20pt) Compute the First Fundamental form {E, F, G} for the brain surface using its u, v

coordinates that I have shared. Color code E F G by RGB colors to plot it on the square.

(Hint: One way is to start by interpolating 3D surface coordinates into u, v space onto a regular grid

using MATLAB’s griddata, to generate 3 square images of 3D coordinates.)

Problem 5. (20pt) Implement mean curvature flow to smooth surfaces. You can use patchcurvature function

to compute the mean curvature.

Apply mean curvature flow for any mesh from http://graphics.stanford.edu/data/3Dscanrep/. Show that

the flow converges making the mesh spherical first and then a point. No need to iterate till it becomes a

point, just show upto the mesh becomes approximately sphere. (hint: if you get instability, reduce the step

size.)

Problem 6. (15pt) Surface smoothing: In this problem, we will develop a method for surface smoothing.

Let Xo, Yo, Zo represent the 3D coordinates of the vertices in the original surface mesh. We will minimize

C(Xs) = kXs ? Xok2 + α k?Xsk2C(Ys) = kYs ? Yok2 + α k?Ysk2C(Zs) = kZs ? Zok2 + α k?Zsk2

where α > 0 is a smoothing parameter and ? represents surface Laplacian operator. You can use the surface

Laplacian function that I shared on the blackboard. Minimize the three cost functions to get new smoothed

coordinates of the surface. Show original and smoothed surfaces. Apply this smoothing to the cube you

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EE575: COMPUTATIONAL DIFFERENTIAL GEOMETRY FOR ENGINEERS: MIDTERM (DUE 03/12/2020, 11:59 PM) 2

generated, and any mesh from http://graphics.stanford.edu/data/3Dscanrep/ for 3 different appropriately

chosen values of α to show different levels of smoothing. Use the conjugate gradient method for minimization

(you wrote this for HW).

Problem 7. (10pt) Let α : I → R

3 be a regular parameterized curve, and s(t) its arc length. Then show

that the inverse function t(s) exists, and β(s) = α(t(s)) is parameterized by arc-length. (Hint: need to show|β0(s)| = 1.)OR

Problem 8. (10 pt) Use any mesh from http://graphics.stanford.edu/data/3Dscanrep/. Choose two different

points on the mesh (e.g. one point on head and one point on tail of bunny). Compute and display

geodesic on the surface mesh. (hint: use Dijkstra algorithm) (5pt). Compute mean and Gaussian curvature

of the mesh (hint: use patchcurvature.m on blackboard). Along the curve, plot curve normal and surface

normal. Show that they are collinear visually. (hint: applying some smoothing to the mesh may give you

smoothly varying normals, and might help denoising the curve and normal).