Discrete.Differential.Geometry-An.Applied.Introduction(sig2008)笔记

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Chapter 1: Introduction to Discrete Differential Geometry: The Geometry of Plane Curves

。

A better approximation than the tangent is the circle of curvature.

。

If the curve is sufficiently smooth (“【curvature-continuous at P 】”) then the circle thus approaches a definite position known as the 【circle of curvature】 or 【osculating circle】; the center and radius of the osculating circle are the 【center of curvature】 and 【radius of curvature】 associated to point P on the curve. The inverse of the radius is K, the 【curvature】 of the curve at P .

。

Informally we say that P , the tangent at P , and the osculating circle at P have one, two, and three coincident points in common with the curve, respectively. Each construction in sequence considers an additional approaching point in the neighborhood of P and the so-called 【order of approximation】

(0, 1, and 2 respectively) is identical to the number of additional points.

。

turning number /the index of rotation, denoted k

。

the integral of signed curvature over a closed curve, Ω, is dependent only on the turning number:

ʃ$_Ω$ K ds = 2πk .

。

no matter how much we wiggle and bend the curve, if we do not change its turning number we do not change its 【total signed curvature】

tsc(p) = ∑αi, tsc(r) = sup{p inscribed in r} tsc(p)

p: polygon

r: the continuous curve

.

geometric mesh size of p: h(p) = max{0≤i<n} d(V i , V i+1 ) .

parametric mesh size of p: hR(p) = max (t i+1 − t i )

.

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CH2 What Can We Measure?

for a convex, compact set in Rn there are only n+1 unique measurements if we require that the measurements be invariant under Euclidean motions (and satisfy certain “sanity” conditions)

.

A 【measure测度】 is a function μ defined on a family of subsets L of some set S, and it returns real values: μ:L→R。

L is closed under finite set union and intersection as well as that it contains the empty set, ∅ ∈ L.

Property:

(1) μ(∅) = 0

(2) μ(A ∪ B) = μ(A) + μ(B) − μ(A ∩ B) (additivity property)

我们只关心在欧式空间下不变的那些运动(rigid motion invariant motion)，e.g. translations and rotation。这使得我们的measure不依赖于坐标系原点和坐标轴的旋转(invariant measure)。相反，variant measure依赖于原点和坐标轴的旋转, 没有太大用处。

。

μ$_m$$^n$:

m:a measure on Rm

n: type of measurement

.

an 【affine subspace】of dimension k is spanned by k+1 points pi ∈ Rn (in general position), i.e., the space consists of all points q which can be written as affine combinations q = ∑αi*pi , ∑αi = 1.

Such an affine subspace is simply a linear subspace translated

。

integrals of Gaussian curvature on a triangle mesh become sums over excess angle at vertices 。

integrals of mean curvature can be identified with sums over edges of dihedral angle weighted by edge length.。

These quantities are always integrals. Consequently they do not make sense as pointwise quantities.

In the case of smooth geometry we can define quantities such as mean and Gaussian curvature as pointwise quantities.

。

Let P be a polyhedron with vertex set V and edge set E and B a ball in R3 then we can define integrated Gaussian and mean curvature measures as

- Kv = 2π − ∑j αj
is the excess angle sum at vertex v defined through all the incident
triangle angles at v,

- l(.) denotes the length,

- θe is the signed dihedral angle at e made between the incident
triangle normals. Its sign is positive

for convex edges and negative for concave edges (note that this
requires an orientation on the polyhedron).

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Chapter 3: Curvature Measures for Discrete Surfaces

。

A smooth space curve γ is often described by its orthonormal 【Frenet
frame】
(T, N, B)

For a curve γ lying on a surface M , it is often more useful to
consider the 【Darboux
frame】
(T, η, ν)

This orthonormal frame includes the tangent vector T to γ and the
normal vector ν to M . Its third element is thus η := ν × T ,
called the【
cornormal】。

The curvature vector of γ decomposes into parts tangent and normal
to M as T‘ = κN = κg η +κn ν. Here in fact, κn measures the
normal curvature of M in the direction T , and is independent of γ

。

The map Sp is called the 【shape
operator (or Weingarten map)】

The eigenvalues κ1 and κ2 of Sp are called 【principal
curvatures】

。

Gauss curvature K := κ1*κ2 as the determinant of Sp，

the mean curvature H := κ1 + κ2 as its trace.

vector mean curvature (or mean curvature vector) H := Hν =
(κ1 + κ2) ν (unit normal ν)

。

Mean curvature is certainly not intrinsic, but it has a nice
variational interpretation.

H = −δ Area / δ Vol

δv Vol = ∫ V · ν dA,
δv Area = − ∫ V
· H ν dA

。

With respect to the L2 inner product <U, V > := ∫
Up · Vp dA on vectorfields, the vector mean curvature is the
negative gradient of the area functional, often called the first
variation of area:

H = Hν = −∇
Area.

(Similarly, the negative gradient of length for a curve is κN ，
κN = −∇ Length)

.

Gauss curvature K is the geometric version of second derivative for
curves,

Mean curvature H is the geometric
version of the Laplacian ∆

.

if a surface M is written locally as the graph of a height function f
over its tangent plane TpM then

H(p) = ∆f . Alternatively, we can write H = ∇$_M$ ·ν = ∆$_M$
x, (

where x is the position vector in R3, ∆$_M$ is 【Beltrami’s
surface Laplacian】,
ν is the unit normal .

.

If we flow a curve or surface to reduce its length or area, by
following these gradients κN and Hν, the resulting parabolic heat
flow is slightly nonlinear in a natural geometric way. This
so-called 【mean-curvature
flow】
has
been extensively studied as a geometric smoothing flow.

。

Gauß–Bonnet theorem for discrete surfaces

∫∫$_D$ K dA := ∑$_p$
Kp; Kp := 2PI - ∑$_i$θi

。

θi are the interior angles at p of the triangles meeting there

Kp is often known as the angle defect at p.

If D is any neighborhood of p contained in Star(p), then ∮$_∂D$
ηds = ∑ θi .

.

The mean curvature of a discrete
surface M is supported along the edges

.

The area of the discrete surface
is a function of the vertex positions

。

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Chapter 4: A Discrete Model of Thin Shells

total curvature: κ1^2 + κ2^2

。

bending energy functional: E(S) = ∫$_S$
H^2 dvol

。

The importance of isometry for simplification of energy was
previously acknowledged in the context of surface fairing and
modeling

。

If I : S → E3 denotes the embedding of the surface, the mean
curvature normal H of S can be written as the intrinsic Laplacian ∆
(induced by the Riemannian metric of S) applied to the embedding of
the surface, H = ∆I,

E(S) = ∫$_S$
<∆I, ∆I> dvol

。

Therefore we must begin with a reasonable definition of isometry,
which in the case of triangulated surfaces we take to be that

(a) bending occurs only along edges,

(b) edges may not stretch.

。

In analogy to the smooth case, a discrete IBM will be based on
Laplacians which are—by
construction—invariant under
discrete isometric deformations.（详见CH11）

。

Although bending forces are much weaker then stretching forces, their
interaction with membrane forces determines the shape of folds and
wrinkles that we associate with garments and other thin flexible
bodies.

。

========================================

Straightest Geodesics on Polyhedral Surfaces ∗

。

Geodesics on smooth surfaces can be characterized by different
equivalent properties. The generalized properties on polyhedral
surfaces will no longer be equivalent and lead to 【different
classes of discrete geodesics】.

.

κ(s)^2 = κg (s)^2 + κn (s)^2

κg : geodesic curvature

κn : normal curvature

The geodesic curvature κg of a curve γ measures the tangential
acceleration. If κg = 0 then the curve varies up to second order
only in direction of the surface normal, therefore it is a
straightest curve

on the surface. The normal curvature κn is related with the bending
of the surface itself and can be neglected from an intrinsic point of
view

Definition 1

Let M be a smooth two-dimensional surface. A smooth curve γ:I→M
with | γ' | = 1 is a geodesic if one of the equivalent properties
holds:

1. γ is a locally shortest curve.

2. γ'' is parallel to the surface normal, i.e.

γ''(s)$^{tan M}$ = 0. (表示γ''(s)在M切面上的分量)

3. γ has vanishing geodesic curvature κg = 0.

。

Definition 4

A vertex p of a polyhedral surface S with total vertex angle θ (p)
is called Euclidean, spherical, or hyperbolic if its angle excess 2 π
− θ (p) is = 0, > 0, or < 0. Respectively, interior points
of a face or of an open edge are Euclidean。

。

The isometric unfolding of sets of a polyhedral surface is a common
procedure to study the geometry.

。

Definition 5

The 【(total)
Gauß curvature】
K(p) of a
vertex p on a polyhedral
surface S is defined as the vertex angle excess

K(p) = 2π − θ (p) = 2π − ∑θi(p)

.

之所以选择straightest
geodesic而不选择shortest
geodesics，因为后者沿不同方向有不同的解。

=============================================
CH 8  Building Your Own DEC at Home
如何离散化微分算子d和Hodge star

CH9 介绍 如何离散化速度/散度/旋度；Discrete.Differential.Geometry-An.Applied.Introduction(sig2013).pdf 里6.3. Discretization via DEC介绍如何离散化Laplacian算子

。
假设说有一个2d网格M，里面有边和顶点。有一个向量场F，现在求F对M的flux。flux是要对face做运算
的。但现在假设只能对vertex做运算，该怎么办？也就是说要考虑，原先F作用于某个区域，现在我要把这个区域用一个vertex来代替，然后让F作用
于这个vertex上。那么问题来了，我怎么把一个区域用vertex vi而不是vj来代替？
一种很直观的方法就是，如果某个区域上的点到vi的距离(|p-vi|)最近，那么就有理由用vi来代表这个区域。这就是Voronoi划分。
原先F在连续区域上做运算，现在F在离散点上做运算。这就是DEC
上面只是粗糙的比喻。实际上为了能计算出正确结果，那个连续区域也要变。变成什么？由Hodge star来计算
。
Hodge star为什么那么定义？论文里有说：
We again attempt to motivate the definition with some intuition.
When
transferring a quantity from a primal simplex to a dual cell, the
quantities must “agree” somehow. Since these are integral values, simply
setting the value on the dual to be equal to the value on the primal
does not make sense, as the domain of integration is unrelated. Instead,
we require that the integral density be equal. So, if ω denotes the
evaluation of a form on a primal k-simplex σ , then ٭ω is the value on
the dual (n − k)-cell σ ̃ such that ：
ω / Vol(σ ) = ٭ω / Vol( σ ̃ )

=========================================================

Chapter 9: Stable, Circulation-Preserving, Simplicial
Fluids(如何离散化速度/散度/旋度)

In order to work towards this goal, more work needs to be done to
further demonstrate that

this idea of forms as fundamental readily-discretizable elements of
differential equations can be successfully used in various other
contexts where predictive power is crucial.

.

Despite the simplicity of this
fractional integration, one of its consequences is excessive
numerical diffusion: advecting velocity before reprojecting onto a
divergence-free field creates significant energy loss详见：《Analysis of an exact fractional step method》

。

One can understand this seemingly
inevitable numerical diffusion through the following geometric
argument: the solutions of Euler equations are geodesic (i.e.,
shortest) paths on the manifold of all possible divergence-free
flows;详见：《Analysis of an exact fractional step method》

。

In this paper we show that a careful setup of discrete
differential quantities, designed to respect structural
relationships such as vector calculus identities, leads in a direct
manner to a numerical simulation method which respects the defining
geometric structure of the fluid equations. A
key ingredient in this approach is the location of physical
quantities on the appropriate geometric structures (i.e.,vertices,
edges, faces, or cells).This
greatly simplifies the implementation as all variables are intrinsic.

.

易于实现：

all quantities are stored
intrinsically (scalars on edges and faces) without reference to
global or local coordinate frames

The reader should now realize that
in our discretization of physical quantities,

the notion of flux that we
described is thus a primal 2-form (integrated over faces),

while its vorticity is a dual
2-form (integrated over dual faces),

and its divergence becomes a primal
3-form (integrated over tetrahedra).

。

Helmholtz-Hodge decomposition
theorem：any
vector field u can be decomposed into three components (given appropriate
boundary conditions)

u = ∇ ×φ + ∇ψ + h

。

k-form的插值牵扯到Whitney
forms [Bossavit 1998].

本文里的Voronoi
cell based generalized barycentric interpolation是把Whitney
form扩展到dual
setting

。

generalized barycentric
coordinates have linear accuracy [Barycentric coordinates for convex
sets.]

。

如何离散化物理量：

Velocity as Discrete Flux

Divergence as Net Flux on Tets

Vorticity as Flux Spin

为什么只能这么离散化？其他离散化的方式不行么？不行，因为作者说这种离散化的方式可由代数关系推导得到（section
3）

3.1

In the continuous
three-dimensional setting,

a 0-form is simply a function
on that 3D space.

A 1-form, or line-form, is a
quantity that can be evaluated through integration over a

curve. Thus a 1-form can be
thought of as a proxy for a vector field, and its integral over a
curve becomes the circulation of this vector field.

A 2-form, or area-form, is to
be integrated over a surface, that is, it can be viewed as a proxy
for a vector perpendicular to that surface (and its evaluation
becomes the flux of that vector field through the surface);

a 3-form, or volume-form, is to
be integrated over a volume and can be viewed as a proxy for a
function.

Classic calculus and vector
calculus can then be substituted with a special calculus involving
only differential forms, called 【exterior
calculus】

就是说，原先计算微积分表达式时，最开始就让‘和坐标系相关的数据”（比如某个vector
field）参与计算了，最后求的结果。

现在借助differential
form，可以不让‘和坐标系相关的数据”参与计算的情况下来计算微积分表达式。在最后求最终结果(evaluation)的时候,再把‘和坐标系相关的数据”带入进去。这就是
exterior
calculus。就是把外围微积分表达式先计算出来，最后带入具体的‘和坐标系相关的数据”从而得出结果。

.>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

做个类比:

设f(x,
y) = x+y, g(x, y) = x+2y, 求值f(1,
2)+g(1, 2)

方法1（类比于传统微积分计算方法）：

f(1, 2)+g(1, 2)

= x+y|$_{x=1, y=2}$   +   x+2y|$_{x=1, y=2}$

= 1+2                        +   1+2*2                     
(从上一步到这一步，叫evaluation。最开始就让‘和坐标系相关的数据”（x=1，
y=2）参与计算了)

= 3 + 5 = 8;

方法2（类比于exterior
calculus计算方法）：

f(1, 2)+g(1, 2)

= f(x, y) + g(x, y) | $_{x=1,
y=2}$

= x+y    + x+2y    |$_{x=1, y=2}$

= 2x+3y              |$_{x=1, y=2}$  
(上面这两步类似于
exterior calculus。只做符号计算，x，y的具体值并没有参与计算。 方法2
在无需带入x，y的具体值的情况下就可以针对函数做一些处理，从而研究函数。类似，exterior
calculus在无需带入‘和坐标系相关的数据”的情况下就可以对n-form做一些处理,
从而研究n-form)

= 2*1+3*2                                    
(从上一步到这一步，带入‘和坐标系相关的数据”，叫evaluation)

= 8

.<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

============================================

CH 10 Conformal Equivalence of
Triangle Meshes

The problem of finding a flat
mesh that is discretely conformally equivalent to a given mesh can be
solved efficiently by minimizing a convex energy function,。。。

This method can also be used to
map a surface mesh to a parameter domain which is flat except for
isolated cone singularities, and we show how these can be placed
automatically in order to reduce the distortion of the
parameterization.

。

What corresponds to a
Riemannian metric and Gaussian curvature in an analogous theory for
triangle meshes?

When should two triangle meshes
be considered conformally equivalent?

。

In this paper we present the
first algorithm for

mesh parameterization which is
based on a definition of discrete

conformal equivalence between
triangle meshes which is satisfac-

tory in the sense that (a) it
depends only on the geometry of the

meshes and (b) defines an
equivalence relation.

。

关于共形的一些性质：

Meshes in a conformal
equivalence class are characterized by length scale factors

associated to the vertices and
by conserved quantities: the length cross ratios（下面有解释）.

。

共形等价的定义：

In smooth differential geometry
two Riemannian metrics g and g ̃ on a differentiable 2-manifold M
are said to be 【conformally
equivalent】
if

g ̃ = e^(2u) g (1)

for some smooth function u : M
→ R, which gives the logarithm of length change between g and g ̃
.

.

we define（类比于smooth
setting，且it
behaves correctly under Möbius transformations（？）
of space）：

Definition 2.1.

A 【discrete
metric】
on M is a function l on the set of edges E,
assigning to each edge eij a
positive number lij so that
the triangle inequalities are satisfied for all triangles t$_{ijk}$ ∈
T .

.

Definition 2.2.

Two discrete metrics l and l ̃
on M are 【(discretely)
conformally equivalent】
if, for some assignment of numbers ui
to the vertices vi , the
metrics are related by

l ̃ij = e^((ui+uj)/2) lij
（2）（定义λij
:= 2 log lij，则
λ ̃ij = λij + ui + uj （3））

this notion of discrete
conformal equivalence is indeed an equivalence relation (i.e., it is
reflexive, symmetric, and transitive)

We consider two meshes as
discretely conformally equivalent if they have the same abstract
triangulation（拓扑？）
and equivalent edge lengths according to Eq. (2)

。

Definition 2.3. Given a
discrete metric l, we associate with each

interior edge e i j (between t
i jk and t jim ) the 【length
cross ratio】

c ij := l im /l mj · l jk /l
ki

。

Proposition. Two meshes are
discretely conformally equivalent if and only if their length cross
ratios are the same.

。

==========================================

CH 11 Discrete Laplace
operators

.

Properties of smooth
Laplacians:

(NULL ) Δu = 0 whenever u is
constant.

(SYM ) Symmetry: (Δu, v) L2 =
(u, Δv) L2 whenever u and v are sufficiently smooth and vanish along
the boundary of S.

(LOC ) Local support: for any
pair p != q of points, Δu(p) is independent of u(q). Altering the
function value at a distant point will not affect the action of the
Laplacian locally.

(LIN ) Linear precision: Δu =
0 whenever S is part of the Euclidean plane, and u = ax + by + c is a
linear function on the plane.

(M AX ) Maximum principle:
harmonic functions (those for which Δu = 0 in the interior of S)
have no local maxima (or minima) at interior points.

(PSD) Positive
semi-definiteness: the Dirichlet energy, E$_D$(u) = ∫s
||grad u||^2 dA, is non-negative. By our choice of sign for Δ, we
obtain E$_D$(u) = (Δu, u)L2 ≥ 0 whenever u is sufficiently smooth
and vanishes along the boundary of S.

.

作者证明discrete Laplacian 无法同时满足上面的性质

。

==========================================

CH12 Discrete Geometric
Mechanics for Variational Time Integrators（解微分方程）

a geometric approach to the
problem of time integration.

.

the very essence of a
mechanical system is characterized by its symmetries and
invariants.Thus preserving these symmetries and invariants (e.g.,
certain momenta) into the discrete computational setting is of
paramount importance if one wants discrete time integration to
properly capture the underlying continuous motion

。

We will show that in fact, one
does not have to ask for either predictability or accuracy
（类比于测不准原理？）

。

an introduction to geometric
mechanics,。。。we
will introduce the notion of variational integrators as a class of
solvers specifically designed to preserve this underlying physical
structure, even for large time steps

。

Hamilton’s principle （the
least action principle）：

a dynamical system always finds
an optimal course from one position to another（“Nature

is thrifty in all its actions“）

A more formal definition will
be presented in Section 4.1, but one consequence is that we can
recast the

traditional way of thinking
about an object accelerating in response to applied forces into a
geometric viewpoint.

a geometric viewpoint：
There, the path followed by the object has
optimal geometric properties—analog
to the notion of geodesics on
curved surfaces。（这是为什么和几何有关）

这个几何观点类似于力学里的牛顿定律，但这个几何观点可适用于E&M和量子力学。

。

单摆动力学方程：

q' = v (2)

v' = −g/L sin(q) (3)

。

离散化方法：

explicit Euler method:

qk+1 = qk + h vk

vk+1 = vk − h g/L sin(qk)

implicit Euler method:

qk+1 = qk + h vk+1

vk+1 = vk − h g/L sin(qk+1)

symplectic Euler method(mix
above two methods):

vk+1 = vk − h g/L sin(qk)

qk+1 = qk + h vk+1

These three methods are called
【finite
difference methods】

。

Since q is a path, we cannot
simply take a “derivative” with respect to q; instead, we take
something called a 【variation】.
A variation of the path q is written δq, and can be thought of as an
infinitesimal perturbation to the path at
each point, with the important property that the perturbation is null
at the endpoints of the path.(变分法的思想）

。

The 【action
functional 】is
then introduced as the integral of L along a path q(t) for time t ∈
[0, T ]:

S(q) =∫{0,
T} L(q, q‘) dt

Hamilton’s principle：

the correct path of motion of a
dynamical system is such that its action has a stationary value,

。

Euler-Lagrange equations：

∂L/∂q – d/dt (∂L/∂q')
= 0

。

in the Hamiltonian formulation,
the dynamics are described in 【phase
space】,
i.e, the current state of a dynamical system is given as a pair (q,
p), where q is the state variable, while p is the momentum, defined
by p = ∂L/∂q'

。

Noether’s theorem：

each symmetry of a system leads
to a physical invariant (i.e., a conserved quantity)

。

These symmetries, if respected
in the discrete setting, will provide equivalent discrete invariants
in time integrators! In fact, we will see that these
invariants can be preserved in time integrators at no extra
computational cost by simply respecting the geometric, variational
nature of dynamics.

应该积分这些invariant，而不是积分速度/加速度之类的量。

。

Why did some of the phase
portraits look better than others? How can we preserve the closedness
of the orbits without making the time integrator more complicated?

。

if one designs a discrete
equivalent of the Lagrangian, then discrete equations of motion can
be easily

derived from it by paralleling
the derivations followed in continuous case. In essence, good
numerical methods will come from discrete analogs to the
Euler-Lagrange equations—equations that truly derive from a
variational principle

.

Discrete Lagrangian:

The main idea is to discretize
the least action principle directly rather than discretizing
Euler-Lagrange equations.

discrete Lagrangian: Ld (qk,
qk+1, h) ≈∫{tk,
tk+1} L(q, q')dt

Ld
(qk, qk+1, h) = h L( (1−α)qk + αqk+1, (qk+1 − qk)/h ), where
α ∈ [0, 1].

For
α = 1/2, the quadrature is second-order accurate, while any other
value leads to linear accuracy. 这意味着一种对称！但这个对称又意味着什么？

。

discrete
Euler-Lagrange (DEL) equations：

D1
Ld(qk, qk+1) + D2 Ld(qk−1, qk) = 0 (7)

D1
Ld表示Ld对第一个参数求偏导；

D2
Ld表示Ld对第二个参数求偏导

。

define
the momentum at time tk to be:

pk
:= D2 Ld(qk−1, qk ) = −D1 Ld(qk , qk+1)

方法1：

pk
= −D1 Ld (qk , qk+1) ,

pk+1
= D2 Ld (qk , qk+1)

给定(qk
, pk)，由D1解出qk+1,
再由D2解出pk+1

。>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

方法2：

pk
= D2 Ld(qk−1, qk )

pk-1
= −D1 Ld (qk-1 , qk)

给定(qk−1,
pk-1), 由D1解出qk,
再由D2解出pk.
和上面的是一样的！（只能沿时间正向求解，比如用k-1时刻的量去求k时刻的量，
或用k时刻的量去求k+1时刻的量）

方法3：

pk
= D2 Ld(qk−1, qk )

pk
= −D1 Ld (qk , qk+1)

给定(qk−1,
pk-1), (qk, pk)，只能求出qk+1, 但是pk+1怎么求？只能再增加一个方程：

pk+1
= D2 Ld(qk, qk+1)

所以方法3远不如方法1好。因为不仅要用到k-1和k时刻的数据，还需要计算3个方程。

。<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

Variational
integrators 。。。has
two important consequences.

First,
the integrators are guaranteed to be symplectic, which in practice
will result in excellent energy behavior, rather than perpetual
damping or blowing up.

Second,
they are also guaranteed to preserve discrete momenta of the system
(via a discrete version of Noether’s theorem)

--------------------------------------

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mesh parameterization：
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Sheffer, A., Praun, E., and Rose, K. 2006. Mesh Parameterization Methods and their Applications

-----------------------吐糟分割线----------------------------
stam的stable fulid是一个巨大的进步，后来很多人在这个基础上针对各种不足各种擦屁股。
图形学上那些引以为傲的数据结构（空间划分，adaptive划分等等）实际上都是在没有找到更准确的数学结构的情况下而采用的变通方法。说白了就是擦屁股，难怪搞数学的人瞧不起搞计算机的人。

什么是计算数学？
搞数学的M发现新出来的一些数学概念是很不错的东东，如果能编程实现的话也许能促进其他学科发展。于是M去找搞计算机的C，希望C能把它实现出来。C看了看这些新概念，想不出这些东东在计算机学科里能有神马用，最终拒绝了M。M一气之下自己来搞。慢慢地这竟然形成了数学的一个分支----计算数学。