Cliff Notes#

Condensed Cheatsheet/mnemonics of stuff I forget

Differential Geometry#

• Inner Product: Angle
• Norm: Length
• Metric: Distance
• Measure: Size
• $$L_{p}$$ norm: Max() component
• $$L_{0}$$ norm: Counting norm
• Jacobian: how a differential patch area is skewed under a (linear only?) transformation
• Divergence: sink vs source aka volume density of outward flux
• Manifold: fancy name of a curved space
• Reimannian manifold: manifold with geodesic metric (Reimannian metric)
• Group: closed under multiplication, commutative, identity function, inverse
• Lie Group: curved space with a group structure i.e. a group that is a manifold where multiplication is smooth/infinitely differentiable
• Lie Algebra: tangent space of Lie group
• Tangent space: linear approximation of a curved space
• Non-abelian group: non-commutative group i.e. $$a*b \neq b*a$$ (e.g. SO(3) rotation group)
• Dual number: convenient for computation of Lie algebra
• Banach Space (norm+completeness) ⊇ Hilbert Space (inner-product norm) ⊇ Sobolev Space ("nice" derivatives up to order S)
• Functionals: functions that take functions as inputs (derivative/integral operators)
• Laplacian: difference average of neighborhood at a point - value at point
• Laplacian intuition from CMU Discrete Differential Geometry Course: Lecture 18

• Maximal smoothness/mean curvature is zero

• Poisson equation: $$\Delta u$$ = 0
• Think of boundary condition being a wire and a soap film covering the wire
• That's a $$\Delta u(x,y) = 0$$
• Another interpretation is equilibrium state. Think of temperature
• Another interpretation is that there are no bumps or local minimas in that surface

Spectral Theory#

Legendre polynomial#

The nth Legendre polynomial, $$\boldsymbol{L}_{n}$$, is orthogonal to every polynomial with degrees less than n i.e.

• $$\boldsymbol{L}_{n} \perp \boldsymbol{P}_{i}, \ \forall i\in [0..n-1]$$
• ex: $$\boldsymbol{L}_{n} \perp x^{3}$$

$$\boldsymbol{L}_{n}$$ has n real roots and they are all $$\in [-1,1]$$

Harmonic functions => $$\Delta u(x) = 0$$

Homogenous function => $$f : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}, \ f(\lambda \mathbf{v})=\lambda^{k} f(\mathbf{v})$$ where $$k,\lambda \in \mathbb{R}$$

General form of Newton's divided-difference polynomial interpolation:

\begin{aligned} f_{n}(x)=& f\left(x_{0}\right)+\left(x-x_{0}\right) f\left[x_{1}, x_{0}\right]+\left(x-x_{0}\right)\left(x-x_{1}\right) f\left[x_{2}, x_{1}, x_{0}\right] \\ &+\cdots+\left(x-x_{0}\right)\left(x-x_{1}\right) \cdots\left(x-x_{n-1}\right) f\left[x_{n}, x_{n-1}, \ldots, x_{0}\right] \end{aligned}

Lagrange interpolating polynomial scheme is just a reformulation of Newton scheme that avoids computation of divided differences

$f_{n}(x)=\sum_{i=0}^{n} L_{i}(x) f\left(x_{i}\right)\newline L_{i}(x)=\prod_{j=0 \atop j \neq i}^{n} \frac{x-x_{j}}{x_{i}-x_{j}}$

• allows for accurately approximating functions where $$f(x) \in P_{2n-1}$$ with only n coefficients

Approximation Schemes#

• Regression Schemes: (Linear or nonlinear)
• Curves do not necessarily go through sample points so error at said points might be large
• Round-off error becomes pronounced for higher order versions and ill-conditioned matrices are a problem
• Orthogonal polynomials do not necessarily suffer from this
• Interpolation Schemes: (splines, lagrangian/newtonian, etc)
• Curves must go through sample points so error at said points is small
• Not ill conditioned

Thin plate splines#

• construction is based on choosing a function that minimizes an integral that represents the bending energy of a surface
• the idea of thin-plate splines is to choose a function f(x) that exactly interpolates the datapoints (xi,yi), say,yi=f(xi), and that minimizes the bending energy $$E[f]=\int_{\mathbf{R}^{n}}\left|D^{2} f\right|^{2} d X$$
• Can also choose function that doesn't exactly interpolate all control points by using smoothing parameter for regularization $$E[f]=\sum_{i=1}^{m}\left|f\left(\mathbf{x}_{i}\right)-y_{i}\right|^{2}+\lambda \int_{\mathbb{R}^{n}}\left|D^{2} f\right|^{2} d X$$

Spherical Basis Splines#

• Gross reduction summary: b-splines with slerp instead of lerp between control points

Spherical Splines#

Last update: 2023-03-05