Differentiable Curve🚧
備忘
0. speed: parameter 1. tangent : gradient of position 2. normal : variant of tangent 3. distance 4. area 5. curvature 6. principal curvature 7. shape operator 8. energy
curve
1. curve p(t) = (x(t), y(t)) 2. ds ‖p′(t)‖ = √x′(t)² + y′(t)² = ds 3. length s(t) = ∫₀ᵗ ‖p′(t)‖ = ∫₀ᵗ √x′(t)² + y′(t)² 4. tangent T(t) = p′(t) / ‖p′(t)‖ = normalize(dp/dt) 5. normal N(t) = p″(t) / ‖p″(t)‖ = normalize(dT/dt) 6. binormal B(t) = T(t) × N(t) 7. curvature κ(t) = ‖p′(t) × p″(t)‖ / ‖p′(t)‖³ 8. torsion τ(t) = [(p′(t) × p″(t)) ∙ p‴(s)] / ‖p′(t) × p″(t)‖²
reparameterization
arc length reparameterization (tangent normalization) x. length s(t) (strictly monotone increasing) y. inverse of s(t) t(s) z. reparameterization p(s) = p(t(s)) a. arc length ‖p(s1) - p(s2)‖ = |s1 - s2| b. ds ‖p′(s)‖ = 1 c. curvature κ(s) = ‖T′(s)‖ = ‖N(s)‖ = ‖p″(s)‖ = 1/r d. torsion τ(s) = ‖B′(s)‖ 1. tangent T(s) = p′(s) = dp/ds 2. normal (scaled) κ(s) N(s) = p″(s) = dT/ds 3. normal N(s) = p″(s) / ‖p″(s)‖ = normalize(dT/ds)
orientation
2d frenet frame
{ dT/ds = κ N
{ dN/ds = - κ T
3d frenet frame
{ dT/ds = κ N
{ dN/ds = - κ T + τ B
{ dB/ds = - τ N
Differentiable Surface🚧
donut
normal curvature
normal plane
a plane containing normal vector.
normal section
a curve from intersection of normal plane and surface.
geodesic curvature / normal curvature (scalar form)
{ κg = κ cos(γ) γ is angle between binormal B
{ κn = κ sin(γ) and surface normal n
geodesic curvature / normal curvature (vector form)
κ = κn + κg
geodesic curve
κg = 0
Meusnier's theorem
all curves on a surface passing through a given point p
and having the same tangent line at p
also have the same normal curvature at p.
principal curvatures
Euler curvature formula
κθ = κ₁ cos²(θ) + κ₂ sin²(θ)
principal curvature
{ κ₁ = H + sqrt(H² - K)
{ κ₂ = H - sqrt(H² - K)
Gaussian curvature
K = κ₁κ₂ determinant
K = Ag(p) / A(p) 加了鄰居後的面積比率
Ag(p): spherical image gauss map area
mean curvature
1 ⌠2π
H = —— | ‖x″θ(t)‖dθ 每個方向(角度0到2π)的曲線的曲率的平均
2π ⌡0
H = (κ₁+κ₂)/2 = Euler curvature formula at θ = 45°
H = -½(∇∙n⃗) = -div(n⃗)/2
Hn⃗ = ½ ∇A(p) / A(p) = grad(area)/2area 垂直移動後的面積比率
curvature special case
1. K = 0 developable surface / ruled surface in 3D
2. H = 0 minimal surface / harmonic
shape operator
directional derivative ∇vf change of f(x,y) at point p in direction v. tangent vector T(v) = ∇vp change of position p at point p in direction v. shape operator S(v) = -∇vn change of normal vector n at point p in tangent vector v. shape operator = tangent vector (if we skip scalar) 那就是排骨酥湯阿幹 eigenvalues of shape operator = principal curvatures https://math.stackexchange.com/questions/3665865/
0th-order derivative is position x 1st-order derivative is tangent vector T 2nd-order derivative is hessian matrix H that is composed of normal N and binormal B shape operator = normal change in tangent direction = tangent = 1st-order derivative transformation matrix of shape operator = derivative of shape operator = 2nd-order derivative (hessian) eigenvalues of transformation matrix of shape operator = eigenvalues of hessian = principal curvatures sum of eigenvalues of transformation matrix of shape operator = sum of eigenvalues of hessian = trace of hessian = laplacian = mean curvature * 2
variational geometry
curve stretching energy ∫ ‖p′(t)‖² dt (sum of squared length) curve bending energy ∫₀ˡ κ²(s) ds (euler elastica) (with reparameterization) ∫ ‖p″(t)‖² dt surface stretching energy ∫ ‖∇p(u,v)‖² dudv (laplacian = 0) surface bending energy ∫ (½(κ₁+κ₂))² dA = ∫ H² dA (bilaplacian = 0) (with reparameterization) ∫ ‖∇∇p(u,v)‖² dudv (trival!) 1. become flat surface, ∇p(u,v) = 0 2. off-diagonal of hessian matrix are gone, H = tr(∇∇p)/2 jacobian matrix p′(t) = ∇p hessian matrix x″(u,v) = ∇∇x bending energy ∫ (κ₁+κ₂)²/4 dA = ∫ H² dA Willmore energy ∫ (κ₁-κ₂)²/4 dA = ∫ (H² - K) dA
fundamental form
first fundamental form x-component of dot product of two tangent vectors [xu∙xu xu∙xv] trace = xu² + xv² = grad² ‖Ⅰ‖² = (xu + xv)² [xv∙xu xv∙xv] min ∫ trace => laplace = 0 (membrane energy) second fundamental form x-component of dot product of normal vector and tangent variant? [n∙xuu n∙xuv] ‖Ⅱ‖² = sum sigma² = k1² + k2² [n∙xvu n∙xvv] min ∫ ‖Ⅱ‖² => laplace² = 0 (thin-plate energy) dN/dp ??? fundamental form application tr(Ⅰ) = length² = ‖xu‖² + ‖xv‖² = grad² det(Ⅰ) = area = ‖xu × xv‖ tr(Ⅱ Ⅰ⁻¹) = 2H = k1 + k2 (eigenvalue = principal curvature) det(Ⅱ Ⅰ⁻¹) = K = k1k2 fundamental form application 1. shape operator S = Ⅱ Ⅰ⁻¹ 2. stretching energy ∫∫Ω ‖diag(Ⅰ'(u,v)-Ⅰ(u,v))‖² dudv 3. bending energy ∫∫Ω ‖Ⅱ'(u,v)-Ⅱ(u,v)‖² dudv
Ⅰ = dp∙dp Ⅱ = -dp∙dn Ⅲ = dn∙dn KⅠ + 2HⅡ + Ⅲ = 0
function curve
explicit function
curvature of function curve κ(t) = p″(t) / √(1 + p′(t)²)³ κ(t) ≈ p″(t) when ‖p′(t)‖ << 1 principal curvatures of function surface https://math.stackexchange.com/questions/481060/
implicit function
curvature of implicit curve / isocurve https://math.stackexchange.com/questions/27967/ principal curvatures of implicit surface / isosurface https://math.stackexchange.com/questions/383407/
curvature of isophote https://en.wikipedia.org/wiki/Isophote
Differentiable Polyline🚧
discrete curve
discrete curvature 1. dT = dN , edge = 0 , vertex = θ 轉角大小是曲率積分，角平分線是曲率方向 2. curavture: κn⃗ = 2sin(θ/2)n⃗ 3. https://zhuanlan.zhihu.com/p/72083902
Differentiable Mesh🚧
Differentiable Mesh
1. gradient 2. laplacian-beltrami
Differential Form
「微分型」。點、邊、面，之間的關聯。
Exterior Calculus
「外微積分」。微分、積分、區域、邊界，之間的關聯。
Exterior Algebra
「外代數」。梯度、散度、旋度，之間的關聯。
https://reurl.cc/odkpAl https://www.researchgate.net/profile/Marc_Gerritsma/publication/301851033/figure/fig4/AS:667721201102864@1536208548882/Physical-quantities-in-R-3-can-be-associated-with-either-inner-or-outer-oriented-points.png https://images.app.goo.gl/s2Mmvhv51BMkvMER9 http://ddg.cs.columbia.edu/SIGGRAPH06/DECApplications.pdf https://www.cs.cmu.edu/~kmcrane/Projects/DGPDEC/
point ∇ edge ∇× face ∇∙ cell scalar ---> vector ---> vector ---> scalar |⋆ |⋆ |⋆ |⋆ cell ∇∙ face ∇× edge ∇ point scalar <--- vector <--- vector <--- scalar ∆ = d⋆d⋆ + ⋆d⋆d
No Free Lunch: Curvature
事情變得很複雜。一種運算有多種定義方式。
http://geometry.cs.cmu.edu/ddgshortcourse/AMSShortCourseDDG2018_Overview.pdf
No Free Lunch: Laplace–Beltrami Operator
Discrete Laplace operators: No free lunch 如果在二維空間 三個願望一次滿足 四個願望就不行了
discrete surface
discrete curvature
1. principal https://computergraphics.stackexchange.com/questions/1718
2. mean http://copyme.github.io/flower/mean-curvature/
from vertex
1. angle deficit: K = 2π - sum θᵢ 角錐展開、頂角大小
gaussian curvature: K = (2π - sum θᵢ) / (A(p)/3) 曲率積分
from cell
2. mean curvature: 2Hn⃗ = ∇A(p) / A(p) 角錐壓平、面積最小
3. mean curvature: 2Hn⃗ = -∆x 所有方向的二次微分加總
from edge
4. mean curvature: H(e) = ½β‖e‖
dihedral angle: β 相鄰三角形的法向量夾角，相鄰三角形的轉角。
(concave is positive)
5. shape operator: S(e) = βeeᵀ / ‖e‖² 3x3矩陣
{ S(e)e = βe
{ S(e)v = 0 when e ⟂ v
6. shape operator: S(v) = sum S(e) / (A(p)/3)
discrete gradient 1. hat function 2. barycentric coordinate discrete laplacian (Laplace–Beltrami Operator) 1. combinatorial laplacian (min pairwise squared distance) [from mean curvature to edge gradient] 1 ⌠2π —— | ‖x″θ(t)‖dθ ----> div(∇x) = ∆x 2π ⌡0 每個方向(角度0到2π)的曲線的曲率的平均 ---> 鄰邊梯度的微分的總和（無須平均） 所有方向的二次微分加總 2. cot laplacian (min surface area) [from area differential to mean curvature] 2Hn⃗ = ∇A(p) / A(p) ----> grad(voronoi cell) = laplace_cot(vertex) 讓角錐側面積盡量小（側面積→底面積）的頂點移動方向的反方向 https://www.slideshare.net/gpeyre/mesh-processing-course-mesh-parameterization 3. Discrete Differential-Geometry Operators for Triangulated 2-Manifolds (2003) 鈍角改用mixed cell 4. A Laplacian for Nonmanifold Triangle Meshes (2020)
normal
normal 1. length is 1 2. cross product normal type 1. vertex 2. face 3. corner normal reconstruction 1.1 face -> edge 1.2 face/edge -> vertex 2. vertex -> face normal interpolation 1. 角平分線 (normal length = 1, 等腰三角形對稱軸) 2. sphere interpolation (no need normalization)
tangent: Principal Curvature Directions
https://mathematica.stackexchange.com/questions/127975/
《Discrete Differential-Geometry Operators for Triangulated 2-Manifolds》
mean curvature
1 1 1 2(pᵢ - pⱼ)∙n⃗
H = ————— sum ( — cot(θᵢⱼ⁺) + — cot(θᵢⱼ⁻) ) ‖pᵢ - pⱼ‖² —————————————
A(pᵢ) (i,j) 8 8 ‖pᵢ - pⱼ‖²
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^
wᵢⱼ κᵢⱼ
radius of curvature
‖pᵢ - pⱼ‖²
r = 1/κᵢⱼ = ————————————— 弦(pᵢ,pⱼ)，i點法向量n⃗
2(pᵢ - pⱼ)∙n⃗ 半弦長、cos，得到斜邊r
symmetric curvature tensor
B = [ a b ] where a + c = 2H = tr(Ⅱ Ⅰ⁻¹)
[ b c ] ac - b² = K = det(Ⅱ Ⅰ⁻¹)
vᵢⱼ = normalize( (pⱼ - pᵢ) - projn⃗(pⱼ - pᵢ) )
solve vᵢⱼᵀ B vᵢⱼ = κᵢⱼ
min sum wᵢⱼ (vᵢⱼᵀ B vᵢⱼ - κᵢⱼ)²
j
energy
euler elastica
http://www.pci.tu-bs.de/aggericke/PC4e/Kap_V/Normalschwingungen.htm
discrete mean curvature H = -½(∇∙n⃗) ---> 2Hn⃗ = div(grad(x)) = ∆x
discrete stretching energy ∫ ‖∇x‖² dA (∆x = 0)
discrete bending energy ∫ ‖diag(∇∇x)‖² = ∫ ‖∇∙∇x‖² = ∫ ‖∆x‖² (∆²x = 0)
discrete Willmore energy Wᵢ = sum βᵢⱼ - 2π [Schröder05]
discrete curve bending energy
ds = (si+1 - si)
κ(s) = ‖pi+1 - pi‖ / ds (arc length reparameterization)
Ebend = ∫₀ˡ κ²(s) ds = sum { ‖pi+1 - pi‖² / (si+1 - si) }
Surface Field🚧
field
《Directional Field Synthesis, Design, and Processing》
https://avaxman.github.io/Directional/tutorial/
field 多變量函數用在曲面：函數輸入是曲面的參數座標。 directional field 1. n-vector field (usually 2 tangent vectors) 2. parallel transport: f1:(c1,c2) dot e = f2:(c1,c2) dot e c1 = v dot t1 3. connection 連成線 4. principal matching: [see panozzo slide] [see directional tutorial] 5. interpolation singularity 1. umbilic / umbilicus (k1 = k2) 2. wedge 3. trisector singularity by gauss map 以二維向量場為例 待測點周圍逆時針繞一圈，看看向量的輻角變化，轉了幾周，不等於零的都是奇異點 1. 平流 0周 2. source放射/sink內聚 +1周=360度 3. saddle十字流出入 -1周=-360度 4. 轉圈 +1或-1 hairy ball theorem / Poincaré–Hopf theorem 球面上，連續切向量場一定有零(源匯)
tangent vector field
mapping X -> Y
fiber 反對應，y對應到的所有x，毛囊y的一堆毛x
(fiber) bundle (parameter,field) -> parameter 視作一種投影
萬物->核心。比方說世界萬物->土水風火Doodle God
section parameter -> (parameter,field) 取得其中一個元素
得到一個超級向量場，維度包含了參數和場
vector bundle (parameter,vector field) -> parameter
tangent bundle (parameter,tangent plane) -> parameter
輸入：所有切面。輸出：切點（或者是X軸方向）
可以視作一個4維流形。
section of tangent bundle = tangent vector field
parameter -> (parameter,tangent plane) 取得其中一個元素
切面以及切點（或者是X軸方向）
cotangent bundle 外代數的對偶（不是垂直補集）
切點參數和切線參數互調（也有可能是轉置？）
connection
directional derivative 純量場對向量微分
各個地點，純量對向量做微分，純量對方向微分
方向微分就是梯度在該方向上的投影（點積）
covariant derivative 向量場對向量場微分（或者，純量場對向量場微分）
各個地點，向量對向量做微分，向量對方向微分
X值、Y值分別對方向微分，總共兩次
方向微分就是梯度在該方向上的投影（點積）
lie derivative 張量場/向量場/純量場對向量場微分
connection vector bundle裝備上covariant derivative
簡單來說，section of vector bundle的方向微分運算子
(超級向量場,超級向量場) -> 超級向量場
切點的參數(u,v)可以重新想成是路徑的起點與方向？
順帶一提，多變量函數的微分就是變換矩陣
超級向量場之中，元素套用transformation得到鄰居
connection Fij ∘ Fjk = Fijk
切面(的向量)實施變換，得到隔壁切面(的向量)
可以沿著路徑複合。不同路徑可能結果不同。
affine connection 切面(的向量)實施仿射變換，得到隔壁切面(的向量)
holonomy angle 繞一圈之後的相位差
跟面積成正比、跟高斯曲率總和成正比
trivial connection 任意一個向量場都可以作為connection
相鄰向量差異大小，當作微分算子的計算結果（變換矩陣）
優點：各種路徑結果均相同
flat connection 曲率為零。向量全部指向同一方向？
holonomy angle = 0
levi-civita connection
riemannian metric 切面集增加副屬性，引入了距離
每個切面的X軸Y軸訂好位置之後
每個切面的對應點之間，形成一個點集合，定義一個距離函數
換句話說
切面上的每一個點，各有一個專屬的距離函數
所有距離函數總共R^2個，可以看成一個距離函數場
一個距離函數的輸入，是兩個切面（的兩個對應點）
可以想成二維座標距離（切面由切點決定，切點共有R^2個）
killing vector field 距離函數場對機靈向量場微分＝０
A-orthogonal 所以是 isometry geometric flow
levi-civita connection 可微、保距、torsion-free
向量與切線夾角是定值
{ Ri(Ti) = Ri(Ti+1) where R is rotation
{ Ri(Bi) = 0
切線沿著路徑亂走，切線長度與幅角不變化
parallel transport
parallel transport 表面攤平、曲線拉直
切面(的向量)實施變換，移動到隔壁切面
parallel transport 向量場對向量場微分＝０，路徑微分版本
微分＝０的意義：以路徑方向為準，向路徑看齊
流形表面選出一條路徑（向量場，路徑的方向）
流形表面選出一個切線（向量場，切線r,theta），有初始方向
極座標r,theta可以換成二維座標c1,c2
離散版本的話，就要看每個切面的X軸Y軸座標系統怎麼訂
Mesh Field🚧
field
Trivial Connections on Discrete Surfaces (2010) 計算fundamental group的holonomy angle，最小平方解。 Globally Optimal Direction Fields (2013) n-vector alignment dihedral angle 相鄰三角形的法向量夾角，相鄰三角形的轉角。
Surface Transformation🚧
Surface Transformation（Geometric Flow）
isometric mapping => gaussian curvature is the same (Theorema Egregium) harmonic mapping => ???
isoperimetric inequality 梯度平方和－梯度差平方＝面積的兩倍 (‖Fx‖² + ‖Fy‖²) - ‖Fy - ±Fx⟂‖² = ±2(Fx × Fy) local Dirichlet energy - isogonal energy = 2 ⋅ area global
surface transformation
1. isometric: J = Q, Ⅰ = I
2. conformal: J = sQ,Ⅰ = sI
3. equiareal: det(J) = ±1, det(Ⅰ) = 1
surface transformation
1. conformal first fundamental form E = G, F = 0
https://math.stackexchange.com/questions/1514312/
https://math.stackexchange.com/questions/940456/
2. harmonic first fundamental form minimize trace
Minimal Surface
rigid motion of object
2d curve: same speed, curvature
3d curve: same speed, curvature, torsion
3d surface: same first/second fundamental form
3d space: same first fundamental form
mobius transformation
位移 旋轉 縮放 反演 保長 保距 保角 和諧
http://reedbeta.com/blog/conformal-texture-mapping/ http://www.cs.jhu.edu/~misha/Fall09/16-conformalmetrics.pdf
conformal transformation
Conformal Equivalence of Triangle Meshes (2008) [continuous] g' = e^2u g g is metric on reimann manifold [discrete] lij' = lij + ui + uj lij = 2 log wij wij is metric on mesh (every face ijk satisfies triangle inequality) 不必是弧線，可以是直線...
Mesh Transformation🚧
piecewise affine transformation
piecewise affine alignment
三角形任取兩鄰邊，再取法向量，補足線性變換的基底。
兩鄰邊叉積求得法向量。叉積是面積量不是長度量。調整法向量長度時，分母開根號，成為長度量。
solve A(pᵢ - t) = (p̕ᵢ - t̕) for i = 1,2,3,4
solve A(pᵢ - p₁) = (p̕ᵢ - p̕₁) for i = 2,3,4
[ | | | ] [ | | | ]
V = [ p₂-p₁ p₃-p₁ p₄-p₁ ] V̕ = [ p̕₂-p̕₁ p̕₃-p̕₁ p̕₄-p̕₁ ]
[ | | | ] [ | | | ]
AV = V̕
A = V̕V⁻¹
normal vector of triangle (p₁,p₂,p₃):
(p₂ - p₁) × (p₃ - p₁)
p₄ - p₁ = —————————————————————————————
sqrt(‖(p₂ - p₁) × (p₃ - p₁)‖)
normal vector of triangle (p̕₁,p̕₂,p̕₃):
(p̕₂ - p̕₁) × (p̕₃ - p̕₁)
p̕₄ - p̕₁ = —————————————————————————————
sqrt(‖(p̕₂ - p̕₁) × (p̕₃ - p̕₁)‖)
Surface Mapping🚧
Shape Approximation
D(M₁, M₂) = ∫ₓ (n̕ᵢᵀx̕ᵢ - nᵢᵀxᵢ)² dA 點到切平面距離，平方誤差總和 D(M₁, M₂) = ∫ₓ ‖n̕ᵢ - nᵢ‖² dA 法向量，平方誤差總和
Shape Metric
《Functional and shape data analysis》
https://di.ku.dk/forskning/research_school/phd_courses/upcoming/international-phd-course-in-nonlinear-statistics/files/ https://di.ku.dk/forskning/research_school/phd_courses/upcoming/international-phd-course-in-nonlinear-statistics/schedule/ Square Root Velocity Function 微分開根號 兩個函數一起重新參數化，SRVF的L2距離保持不變 => REGISTRATION = PARAMETRIZATION 改一改參數化函數就行了 sqrt(r'(t)) => Fisher–Rao Metric 但是那個r(t)要怎麼窮舉咧...要怎麼離散化咧...