rlab
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Javascript scientific library like R
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JavaScript
var R, A, G;
module.exports = R = A = G = require("./statistics");
var extend = R.extend;
// G = R.G = R.Geometry = {}
// 各種 space : https://en.wikipedia.org/wiki/Space_(mathematics)#/media/File:Mathematical_implication_diagram_eng.jpg
// Space = HllbertSpace + BanachSpace + Manifold (流形)
G.Space = extend({}, G.Set);
// ============= Topology ====================
// https://en.wikipedia.org/wiki/Topological_space
// p : point
G.TopologicalSpace={ // 拓撲空間 : a Space with neighbor()
neighbor:function(p1, p2) {}, // TopologicalSpace is a Space with neighbor() function.
coarser:function(T1, T2) {}, // 更粗糙的 (coarser, weaker or smaller) 的拓樸
finer:function(T1, T2) { return !courser(T2,T1) }, // 更細緻的 (finer, stronger or larger) 的拓樸
continuous:function() {}, // if for all x in X and all neighbourhoods N of f(x) there is a neighbourhood M of x such that f(M) is subset of N.
homeomorphism:function() {}, // 同胚: 同胚是拓撲空間範疇中的同構(存在 f 雙射，連續，且 f-1 也連續) (注意和代數的 Homomorphism (同態) 不同，差一個 e)
}
extend(G.TopologicalSpace, G.Space);
// Kolmogorov classification T0, T1, T2, ....
// https://en.wikipedia.org/wiki/Separation_axiom
G.T0 = G.KolmogorovSpace = {
// every pair of distinct points of X, at least one of them has a neighborhood not containing the other
}
extend(G.T0, G.TopologicalSpace);
G.T1 = G.FrechetSpace = {}
// 任兩點都可鄰域分離者，為郝斯多夫空間。
G.T2 = G.HausdorffSpace = {}
G.T2Complete = {}
G.T2p5 = G.Urysohn = {} // T2?
G.T3 = G.RegularSpace = {
// every closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods
}
G.T3p5 = G.TychonoffSpace = {} // T3?
G.T4 = G.NormalHausdorff = {}
G.T5 = G.CompletelyNormalHausdorff = {}
G.T6 = G.PerfectlyNormalHausdorff = {}
// https://en.wikipedia.org/wiki/Discrete_space
G.DiscreteSpace = {} //
extend(G.DiscreteSpace, G.TopologicalSpace);
// 1. d(x,y)>=0, 2. d(x,y)=0 iff x=y 3. d(x,y)=d(y,x) 4. d(x,z)<=d(x,y)+d(y,z)
G.Metric = {
d:function(x,y) {},
test0:function() {
be(d(x,y)>=0);
be(d(x,x)==0);
be(d(x,y)==d(y,x));
},
test:function() {
test0();
be(d(x,z)<=d(x,y)+d(y,z));
},
}
G.UltraMetric = {
test:function() {
test0();
be(d(x,z)<=max(d(x,y),d(y,z)));
},
}
extend(G.UltraMetric, G.Metric);
// https://en.wikipedia.org/wiki/Metric_space
G.MetricSpace = { // distances between all members are defined
d:function(p1,p2) {},
test:function() {
be(d(x,y)>=0);
be(d(x,x)===0);
be(d(x,y)===d(y,x));
be(d(x,z)<=d(x,y)+d(y,z));
}
}
G.CompleteMetricSpace = { // 空間中的任何柯西序列都收斂在該空間之內。
}
extend(G.CompleteMetricSpace, G.MetricSpace);
// https://en.wikipedia.org/wiki/Complete_metric_space
G.CompleteMetricSpace = {
// M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.
}
// https://en.wikipedia.org/wiki/Uniform_space
G.UniformSpace = { // 帶有一致結構的集合，「x 鄰近於a 勝過y 鄰近於b」之類的概念，在均勻空間中是有意義的。
}
G.Rn = {
} // (R,R,....)
// 向量空間： VectorSpace = Rn + Linearity
// V:AbelGroup(交換群), F:Field
// 1. F × V → V，(r,v)→ rv
// 2. r(x+y) = rx+ry
// 3. (r+s)x = rx+sx
// 4. (rs)x = r(sx)
// 5. 1x = x
G.VectorSpace = G.LinearSpace = {
add(x,y) { return x.add(y) },
mul(a,x) { return a.mul(x) },
bilinear(b) { // https://en.wikipedia.org/wiki/Bilinear_form
linear((u)=>b(u,w));
linear((w)=>b(u,w));
},
positiveDefinite(b) { }, // 正定
symmetric(b) {}, // 對稱
dualSpace() {}, // https://en.wikipedia.org/wiki/Dual_space
}
extend(G.VectorSpace, G.Rn);
G.FiniteVectorSpace = {} // 有限體向量空間
extend(G.FiniteVectorSpace, G.VectorSpace);
G.NormedVectorSpace={ // 賦範空間
norm:function(v) { return x.norm() },
}
extend(G.NormedVectorSpace, G.VectorSpace);
G.InnerProductSpace={ // 內積空間
dot(x,y) { return x.vdot(y) },
}
extend(G.InnerProductSpace, G.NormedVectorSpace);
// 仿射空間：a1 v1 + ... + an vn 中 sum(ai)=1
// https://en.wikipedia.org/wiki/Affine_space
G.AffineSpace = {
sub(x,y) { }, // 點與點的差是一向量
add(x,v) { }, // 點加向量得一點，但點與點不可作加法
}
// ============= Euclidean Geometry 歐氏幾何 ====================
G.EuclideanSpace = {
d(x,y) { return x.sub(y).norm() }, // v= x.sub(y); v.dot(v).sqrt()
dot(x,y) { return x.vdot(y) },
}
G.LocallyConvaxSpace={}
extend(G.LocallyConvaxSpace, G.VectorSpace);
G.HilbertSpace={} // HilbertSpace => InnerProductSpace => LocallyConvaxSpace => VectorSpace (LinearSpace)
extend(G.HilbertSpace, G.InnerProductSpace);
// BanachSpace => NormedVectorSpace => MetricSpace => TopologicalSpace
// NormedVectorSpace that Cauchy sequence of vectors always converges to a well defined limit
G.BanachSpace={}
extend(G.BanachSpace, G.NormedVectorSpace);
G.MeasureSpace = {} // 測度空間 M(Set) => R
G.ProbabilitySpace = {} // 機率空間 M(Set) = 1/3
// ============= Elliptic Geometry 橢圓幾何 ====================
G.EllipticGeometry = {} // 橢圓幾何
G.SphericalGeometry = {} // 球面幾何
G.SphericalTrigonometry = {} // 球面三角學
G.Geodesy = {} // 大地測量學
G.GreatCircleDistance = {} // 大球距離
// 圓： x^2+y^2 = 1 => (x,y)=(sin t, cos t)
// 雙曲線： x^2-y^2 = 1 => (x,y) = (sinh t, cosh t)
// sinh = (e^x-e^{-x})/2, cosh = (e^x+e^{-x})/2
// https://en.wikipedia.org/wiki/Hyperbolic_function
// sinh x = -i sin ix ; cosh x = cos ix
// sin x 泰勒展開 = x - 1/3! x^3 + 1/5! x^5 ....
// sinh x 泰勒展開 = x + 1/3! x^3 + 1/5! x^5 ....
// int sinh cx dx = 1/c cosh cx + C
// ============= Hyperbolic Geometry 雙曲幾何 ====================
G.HyperbolicGeometry = {}
// ============= Riemannian Geometry ====================
// https://en.wikipedia.org/wiki/Riemannian_geometry
G.RiemannianGeometry = {} // 黎曼幾何
G.RiemannianMetrics = {} // 黎曼度規
G.MetricTensor = {} // 度規張量
G.GaussBonnetTheorem = {} // 高斯博內定理
// ============= Manifold : 流形 ====================
G.Manifold={}
G.C0 = {}
G.Coo = {}
// https://en.wikipedia.org/wiki/Topological_vector_space
G.TopologicalVectorSpace = {}
// https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space
G.LocallyConvexSpace = {}
extend(G.LocallyConvexSpace, G.TopologicalVectorSpace);
extend(G.LocallyConvexSpace, G.NormedVectorSpace);
// m 維拓撲流形
G.TopologicalManifold={
// M是豪斯多夫空間，x in M 有鄰域 U 同胚於 m 維歐幾里得空間 R^{m}的一個開集
}
G.BanachManifold = {}
// =========== Topological Ring ==============
G.TopologicalRing = extend({}, A.Ring);
G.TopologicalField = extend({}, A.Field);
// ref : https://en.wikipedia.org/wiki/Group_homomorphism
// https://en.wikipedia.org/wiki/Fundamental_theorem_on_homomorphisms
// 同態：h(a • b) = h(a) x h(b)
A.homomorphism=function(h, g1, g2) {
var a=g1.random(), b=g2.random();
return eq(h(group1.op(a,b)), group2.op(h(a), h(b)))
}
// ref : https://en.wikipedia.org/wiki/Isomorphism
// https://en.wikipedia.org/wiki/Isomorphism_theorem
// 同構：h(a • b) = h(a) • h(b)
A.isomorphism=function(h1, h2, g1, g2) {
var a1=g1.random(), b1=g2.random();
var a2=g1.random(), b2=g2.random();
return homorphism(h1,g1,g2)&&homorphism(h2,g2,g1);
}
// 多邊形：Polygon
// 多面體：Polyhedron : V-E+F = 2
// 多胞形：Polytope
// ============= Spherical Geometry ====================
G.SphericalGeometry = {}
G.MandelbrotSet = {}
/*
For each pixel (Px, Py) on the screen, do:
{
x0 = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
y0 = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-1, 1))
x = 0.0
y = 0.0
iteration = 0
max_iteration = 1000
// Here N=2^8 is chosen as a reasonable bailout radius.
while ( x*x + y*y < (1 << 16) AND iteration < max_iteration ) {
xtemp = x*x - y*y + x0
y = 2*x*y + y0
x = xtemp
iteration = iteration + 1
}
// Used to avoid floating point issues with points inside the set.
if ( iteration < max_iteration ) {
// sqrt of inner term removed using log simplification rules.
log_zn = log( x*x + y*y ) / 2
nu = log( log_zn / log(2) ) / log(2)
// Rearranging the potential function.
// Dividing log_zn by log(2) instead of log(N = 1<<8)
// because we want the entire palette to range from the
// center to radius 2, NOT our bailout radius.
iteration = iteration + 1 - nu
}
color1 = palette[floor(iteration)]
color2 = palette[floor(iteration) + 1]
// iteration % 1 = fractional part of iteration.
color = linear_interpolate(color1, color2, iteration % 1)
plot(Px, Py, color)
}
*/
// 碎形幾何 : Fractal
// http://andrew-hoyer.com/
// http://andrew-hoyer.com/experiments/fractals/
// http://flam3.com/flame.pdf
// https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
// http://rembound.com/articles/drawing-mandelbrot-fractals-with-html5-canvas-and-javascript
// https://github.com/rembound/Mandelbrot-Fractal-HTML5/blob/master/mandelbrot-fractal.js
// 操控繪圖
// http://fabricjs.com/ (讚！)
// https://github.com/kangax/fabric.js
// 3D 動畫
// https://threejs.org/
// http://haptic-data.com/toxiclibsjs/
// 地理資訊
// ArcGIS : https://developers.arcgis.com/javascript/3/
// 動畫
// http://paperjs.org/features/
// https://processing.org/
// 向量
// http://victorjs.org/
// 3d: https://evanw.github.io/lightgl.js/docs/vector.html