Vector spaces
(See linear algebra ref). Vector and scalar product of 2 vectors; effect of Left vs right handedness of coordinate system.
High dimensional objects
Get vector equations from geometric properties. Use linear transformations like scaling, rotation, projection to describe effects.
Hyperplane
Hyperplane \(\perp w\) through 0 : x such that \(w^{T}x = 0\); shift c from 0: \(w^{T}(x-c) = 0\). For halfspace, replace ‘=’ with \(\leq\).
Polyhedron
\(\set{x: Ax \leq b}\): The intersection of halfspaces.
Simplex/ hypertriangle
n-d triagle. Construct from (d+1) hyperplanes with linearly independent w’s.
Hypersphere surface
\(x^{}x = r\). Sift from 0: \((x-c)^{}(x-c) = x^{}x - 2c^{}x = r’\).
Hyper-ellipse surface E
Aligned with std basis
Skewed norm-ball form
E aligned with the standard basis: diagonal \(\SW \succeq 0\). \(\set{x: x^{}\SW x = r^{2}}\) is hyper-ellipse aligned with the axes, skewed as per \(\SW\). After rescaling: \(\set{x: x^{}\SW x = 1}\).
Matrix image form
Take \(\SW^{1/2} x = y\), assume \(\SW \succ 0\). This is \(\equiv\) \(E = \set{M y: \norm{y} = 1}\), where \(\SW^{-1/2} = M \succ 0\).
Radii along major axes
\(\set{\sw_i^{-1/2}e_i} \subset E\). So, radii are:\ \(\set{\sw_i^{-1/2}} = \set{\sw_i(M)}\).
Aligned with arbitrary basis
Rotate previous ellipse. Take orthogonal rotator U and apply it to previous ellipses (do \(y = U^{*}x\)): major axes of E will then be aligned with U’s columns.
Radii along major axes remains the same.
Rotated Skewed norm-ball form
\(\set{x: x^{}U\SW U^{}x = 1}\).
Matrix image form
Take: \(E = \set{M’ y: \norm{y} = 1}\), rotate to get: \(M = M’ U^{*} \succ 0\). \(E = \set{M y: \norm{y} = 1}\)
Shifted away from 0
Just use y = x-c.
Radii along major axes remains the same.
Shifted Rotated Skewed norm-ball form
\(\set{x: (x-c)^{}U\SW U^{}(x-c) = 1}\).
Using unscaled \(M’=U\SW U^{}\), the equation is : \((x-c)^{}M’(x-c) - r^{2} = x^{}M’x - r^{2} - 2x^{}M’c = 0\).
Shifted Matrix image form
\(\set{c + My | \norm{y} = 1}\) for \(M \succ 0\). This can be reparametrized as: \(\set{x | \norm{M^{-1}x - M^{-1}c} = 1} = \set{x: \norm{Ax + b } = 1}\).
Volume
Take the general expression for E: \(\set{c + My | \norm{y} = 1}\). \(vol(E) \propto \prod r_i\), where \(r_i = \sw_i(M)\) are the radii along the major axes of the ellipsoid E. \why
So, \(vol(E) \propto det(M^{-1}) = det(A^{-1})\).
For 2-D ellipsoids: \(vol(E) = \pi\prod r_i\).
Other coordinate systems
Cylindrical and spherical coordinates: \(x=r \cos \theta\).
Graph drawing
Find axis meeting points, maxima/ minima, \ inflection points.
For visualization of functionals over vector spaces, their gradients: see linear algebra ref.
Manifold
Take any small enough area in a manifold: it resembles a euclidian space of a certain dimension, aka the manifold’s dimension. 0 dim manifold: A point. 1 dim manifold: line, arc. 2 dim manifold: sphere surface.