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kdim

A collection of interesting helpers, data structures, and utility types for messing around with mathematical things in Node/JS/TS. No guarantees are made here as to the speed, efficiency, or correctness of tools provided!

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Contents

Installation

npm install kdim
#^ Or whatever package manager you use

Numerics

Constants

Exported Constants 
const U8_MAX = (1 << 8) - 1;
const U16_MAX = (1 << 16) - 1;
const U32_MAX = -1 >>> 0;
const I8_MAX = (1 << 7) - 1;
const I8_MIN = -I8_MAX - 1;
const I16_MAX = (1 << 15) - 1;
const I16_MIN = -I16_MAX - 1;
const I32_MAX = (U32_MAX + 1) / 2 - 1;
const I32_MIN = -I32_MAX - 1;

Complex

A numeric type with real and imaginary components. An instance of the class is immutable, so arithmetic operations on it will always produce a new instance.

Class Signature 

class Complex implements Number {
  constructor(real?: number, imaginary?: number);

  get real(): number;
  get imaginary(): number;

  static from<N extends Complex | Number>(init: N): Complex;

  add(addend: Number | Complex): Complex;
  sub(subtrahend: Number | Complex): Complex;
  mul(multiplicand: Number | Complex): Complex;
  div(divisor: Number | Complex): Complex;
  pow(exponent: Number): Complex;
  eq(other: Number | Complex): boolean;
  conjugate(): Complex;

  valueOf(): number;
  toFixed(fractionDigits?: number | undefined): string;
  toExponential(fractionDigits?: number | undefined): string;
  toPrecision(precision?: number | undefined): string;
  toString(radix?: number | undefined): string;
  toLocaleString(locales?: unknown, options?: unknown): string;
  toLocaleString(
    locales?: Intl.LocalesArgument,
    options?: Intl.NumberFormatOptions | undefined
  ): string;
  toLocaleString(
    locales?: string | string[] | undefined,
    options?: Intl.NumberFormatOptions | undefined
  ): string;

  [Symbol.toPrimitive](hint: string): string | number;
  get [Symbol.toStringTag](): string;
}

import { Complex } from "kdim";

const a = new Complex(-3, 5); // -3 + 5i
a.real; // -3
a.imaginary; // 5

const b = new Complex(7, -1); // 7 - i
a.add(b); // 4 + 4i
a.mul(b); // -16 + 38i
b.pow(2); // 48 - 14i

const c = Complex.from(b); // 7 - i
b.eq(c); // true

Rational

A rational number class for fraction arithmetic without loss of precision. Operations are only guaranteed where numerator and denominator are within Number.MIN_SAFE_INTEGER and Number.MAX_SAFE_INTEGER.

Class Signature 

class Rational implements Number {
  constructor(numerator: number, denominator: number = 1);

  get numerator(): number;
  get denominator(): number;

  static from(...input: RationalLike | [fraction: string]): Rational;
  static parse(fraction: string): Rational;

  recip(): Rational;
  add(...addend: RationalLike): Rational;
  sub(...subtrahend: RationalLike): Rational;
  mul(...multiplicand: RationalLike): Rational;
  div(...divisor: RationalLike): Rational;
  pow(exponent: number): Rational;
  mod(modulus: number): Rational;
  abs(): Rational;
  eq(...other: RationalLike): boolean;
  gt(...other: RationalLike): boolean;
  gte(...other: RationalLike): boolean;
  lt(...other: RationalLike): boolean;
  lte(...other: RationalLike): boolean;

  toFraction(options?: RationalFormatOptions): string;

  valueOf(): number;
  toFixed(fractionDigits?: number | undefined): string;
  toExponential(fractionDigits?: number | undefined): string;
  toPrecision(precision?: number | undefined): string;
  toString(radix?: number | undefined): string;
}

type RationalFormat = "space" | "nospace" | "unicode";

type RationalFormatOptions = {
  mixed?: boolean;
  format?: RationalFormat;
};

type RationalLike =
  | [rational: Rational]
  | [numerator: number]
  | [numerator: number, denominator: number];

import { Rational } from "kdim";

const a = new Rational(5, 31); // construct from numerator, denominator
const b = Rational.parse("3 / 9"); // parse from string (spaces are not required)

const result = a
  .add(b) // rationals as arguments
  .mul(12) // integer arguments
  .div(5, 4) // implicit rational arguments
  .toFraction(); // "116/155"

Rational.parse("16 / 24").eq("2 / 3"); // true

Rationals are immutable, so arithmetic methods always produce new values. They will always simplify to their most reduced form upon construction.

Serializing a Rational by calling the toFraction allows to specify whether it should be in mixed number or irrational format (the default), as well as whether the unicode FRACTION SLASH (\u2044) character should be used instead of a typical forward slash (/), which produces small fractions on some platforms, E.G. 3⁄4.

import { Rational } from "kdim";

Rational.from("3/4").toFraction({ format: "nospace" }); // "3/4"
Rational.from("3/4").toFraction({ format: "unicode" }); // "3⁄4"

Wrapping

A wrapping integer class, allowing a value to be constrained to an arbitrary range, and wrapping around the range when arithmetic operations cause it to overflow or underflow Wrapping numbers are immutable, so arithmetic methods always produce new values.

Class Signature 

abstract class Bounded {
  abstract get value(): number;
  abstract get min(): number;
  abstract get max(): number;
}

type BoundedOptions = {
  max: number;
  min?: number;
};

class Wrapping implements Bounded, Number {
  constructor({ max, min }: BoundedOptions, value?: number);

  static from(bounded: Bounded): Wrapping;

  add<N extends Number>(n: N): Wrapping;
  sub<N extends Number>(n: N): Wrapping;

  get value(): number;
  get min(): number;
  get max(): number;

  valueOf(): number;
  toFixed(fractionDigits?: number | undefined): string;
  toExponential(fractionDigits?: number | undefined): string;
  toPrecision(precision?: number | undefined): string;
  toString(radix?: number | undefined): string;
  toLocaleString(locales?: unknown, options?: unknown): string;
  toLocaleString(
    locales?: Intl.LocalesArgument,
    options?: Intl.NumberFormatOptions | undefined
  ): string;

  [Symbol.toPrimitive](hint: string): string | number;
  get [Symbol.toStringTag](): string;
}

import { Wrapping } from "kdim";

let u16 = new Wrapping({ max: 0xffff }, 1); // default min = 0
u16 = u16.add(0xfffd); // (0xFFFE)
u16 = u16.add(1); // (0xFFFF)
u16 = u16.add(1); // (0x0000)
u16 = u16.add(1); // (0x0001)

Wrapping integers can have arbitrary min and max values (inclusive), and can be initialized with a starting value. Arithmetic operations can be performed with any numeric type, using the bounds of the reciever, and can be chained:

import { Wrapping } from "kdim";

let wk = new Wrapping({ min: 1, max: 7 }); // default value = min
let yr = new Wrapping({ min: 1, max: 365 }, 24);

wk = wk.add(yr); // (4)
yr = yr
  .add(wk) // (28)
  .sub(3) // (25)
  .sub(30); // (361)

NOTE: constructor throws a RangeError when min >= max or any of min, max, or initial have a fracitonal component, as will attempting to add() or sub() a number with a fractional component.

They can also be cloned from other Wrapping values, or other Bounded values like Saturating, copying their properties:

import { Wrapping, Saturating } from "kdim";

const vuMeter = new Saturating({ min: -20, max: 3 }, -3);
const clip = Wrapping.from(vuMeter);

Saturating

A saturating (or clamping) integer class allowing a value to be constrained to an arbitrary range, and clamping it to the bounds when arithmetic operations would cause it to overflow or underflow. Saturating numbers are immutable, so arithmetic methods always produce new values.

Class Signature 

class Saturating implements Bounded, Number {
  constructor({ max, min }: BoundedOptions, value?: number);

  static from(bounded: Bounded): Saturating;

  add<N extends Number>(n: N): Saturating;
  sub<N extends Number>(n: N): Saturating;
  mul<N extends Number>(n: N): Saturating;
  div<N extends Number>(n: N): Saturating;

  get value(): number;
  get min(): number;
  get max(): number;

  valueOf(): number;
  toFixed(fractionDigits?: number | undefined): string;
  toExponential(fractionDigits?: number | undefined): string;
  toPrecision(precision?: number | undefined): string;
  toString(radix?: number | undefined): string;
  toLocaleString(locales?: unknown, options?: unknown): string;
  toLocaleString(
    locales?: Intl.LocalesArgument,
    options?: Intl.NumberFormatOptions | undefined
  ): string;

  [Symbol.toPrimitive](hint: string): string | number;
  get [Symbol.toStringTag](): string;
}

import { Saturating } from "kdim";

let level = new Saturating({ min: 1, max: 99 });
level = level.add(50); // (51)
level = level.add(30); // (81)
level = level.add(30); // (99)

As with Wrapping, Saturating types support arithmetic operations with any numeric type, using the receiver.

import { Saturating, Wrapping } from "kdim";

const s = new Saturating({ min: -1, max: 10 }, 1);
const w = new Wrapping({ max: 7 });
s.add(7); // (8)
w.add(5); // (5)
s.add(w); // (8)

NOTE: constructor throws a RangeError when min >= max or any of min, max, or initial have a fracitonal component, as will attempting to add(), sub(), mul(), or div() a number with a fractional component or attempting to div(0).

Wrapping and Saturating values may also be used with mathematical operators for convenience, though they will be cast to a primitive in doing so, and will not be modified:

import { Saturating } from "kdim";

const s = new Saturating({ max: 50 }, 40);
const primitiveSum = s + 20; // 60
s; // Still (40)
s.add(20); // 50

Data Structures

Matrix

A concrete Matrix class for simple linear algebra, currently only supporting simple numbers, but with plans to add support for complex numbers.

Class Signature 

class Matrix<M extends number, N extends number>
  implements Iterable<Tuple<number, N>>
{
  constructor(data: MatrixLike<M, N>);

  static zero<N extends number>(n: N): Matrix<N, N>;
  static identity<N extends number>(n: N): Matrix<N, N>;
  static withSize<M extends number, N extends number>(
    rows: M,
    cols: N,
    fillValue?: number
  ): Matrix<M, N>;
  static fromDiagonal<N extends number>(
    diagonal: Tuple<number, N> | number[]
  ): Matrix<N, N>;
  static fromMTX<M extends number, N extends number>(
    data: string,
    options?: MTXOptions
  ): Matrix<M, N>;
  static isMatrixLike<M extends number, N extends number>(
    arg: unknown,
    ...dimensions: [m: M, n: N] | []
  ): arg is MatrixLike<M, N>;

  get rows(): M;
  get cols(): N;
  get size(): number;
  get data(): MatrixLike<M, N>;

  isSquare(): boolean;
  isOrthogonal(): boolean;

  at(i: number, j: number): number | undefined;
  row(i: number): Tuple<number, N> | undefined;
  col(j: number): Tuple<number, M> | undefined;
  clone(): Matrix<M, N>;
  submatrix<M extends number, N extends number>(
    options: SubmatrixOptions
  ): Matrix<number, number>;
  augment<O extends number, P extends number>(
    other: MatrixOperand<M, O>
  ): Matrix<M, P>;

  trace(): number;
  determinant(): number | undefined;
  inverse(tolerance?: number): Matrix<M, M> | undefined;
  transpose(): Matrix<N, M>;

  add(other: MatrixOperand<M, N>): Matrix<M, N>;
  sub(other: MatrixOperand<M, N>): Matrix<M, N>;
  mul<I extends MatrixOperand<number, number> | number>(
    other: I
  ): MatrixResult<M, N, I>;
  pow(k: number): Matrix<M, M>;
  eq(other: MatrixOperand<M, N>, tolerance?: number): boolean;
  dot(other: MatrixOperand<M, 1>): number;

  [Symbol.iterator](): Iterator<Vec<N>>;
}

import { Matrix } from "kdim";

const mtx = new Matrix<3, 3>([
  [5, 2, 10],
  [-1, 7, 7],
  [3, -3, 9],
]);

mtx.rows; // 3
mtx.cols; // 3
mtx.col(2); // [10, 7, 9]
mtx.mul(2);
// [
//   [10, 4, 20],
//   [-2, 14, 14],
//   [6, -6, 18],
// ]

mtx.mul(Matrix.identity(3)).eq(mtx); // true

BST

A simple binary search tree for time-efficient search or arbitrary data.

Class Signature 

class BSTNode<V> {
  constructor(data: V, parent?: BSTNode<V>);
  asBST(): BST<V>;
}

class BST<V> implements Iterable<V> {
  constructor(compareFn?: CompareFunction<V>);

  static fromNode<V>(node: BSTNode<V>): BST<V>;

  search(value: V): BSTNode<V> | null;
  insert(value: V): boolean;
  has(value: V): boolean;
  delete(valueOrNode: V | BSTNode<V>): boolean;
  max(node?: BSTNode<V>): BSTNode<V> | null;
  min(node?: BSTNode<V>): BSTNode<V> | null;
  successor(node: BSTNode<V>): BSTNode<V> | null;
  predecessor(node: BSTNode<V>): BSTNode<V> | null;

  asOrdered(): V[];
  asPreOrdered(): V[];
  asPostOrdered(): V[];

  *[Symbol.iterator](): IterableIterator<V>;
}

import { BST, CompareFunction } from "kdim";

type Person = { name: string; age: number };

const comp: CompareFunction<Person> = (a, b) =>
  a.name === b.name ? a.age - b.age : a > b ? 1 : -1;

const tree = new BST<Person>(comp);
tree.insert({ name: "Cillian Murphy", age: 47 });
tree.insert({ name: "Emily Blunt", age: 40 });

KDTree

A time-efficient data structure for searching higher-dimensional datasets. Inspired by Mike Pound's Computerphile Video.

Class Signature 

class Node<T> {
  parent: Node<T> | null;
  point: T;
  left: Node<T> | null;
  right: Node<T> | null;

  constructor(point: T);
}

type KDTreeOptions = {
  clone?: boolean;
};

class KDTree<K extends number> implements Iterable<Vec<K>> {
  constructor(data?: Vec<K>[], options?: KDTreeOptions);

  get dimensions(): number;
  get tree(): Node<Vec<K>> | null;

  [Symbol.iterator](): Iterator<Vec<K>>;

  insert(point: Vec<K>): void;
  remove(point: Vec<K>): Vec<K> | null | void;
  has(point: Vec<K>): boolean;
  nearestNeighbor(point: Vec<K>): {
    point: Vec<K> | null;
    distance: number;
  };
}

import { Vec, KDTree } from "kdim";

const data: Vec<5>[] = [
  [0, 1, 1, 0, 1],
  [1, 1, 1, 1, 1],
  [0, 0, 0, 1, 0],
];

const tree = new KDTree<5>(data);
const testPoint: Vec<5> = [1, 0, 1, 1, 1];

tree.has(testPoint); // false

const { point, distance } = tree.nearestNeighbor(testPoint);
// { point: [1, 1, 1, 1, 1], distance: 1 }

tree.insert(testPoint);
tree.has(testPoint); // true

tree.remove(testPoint);
tree.has(testPoint); // false

NOTE: by default, input datasets are shallowly copied during tree construction, and retained within the data structure. If desired, the underlying dataset may be used without copying, using the option copy: false;

BloomFilter

A probabalistic data structure used to test for membership of a large set in which storage of all set elements is unfeasible. False positives are possible, but false negatives are not, so that the structure will tell you if an element is either "probably in the set" or "definitely not in the set".

Class Signature 

type HashFunction<T> = (data: T) => Promise<string> | string;

type BloomFilterOptions<T> = {
  size?: number;
  hashFunctions?: Array<HashFunction<T>>;
};

class BloomFilter<T = any> {
  constructor(options?: BloomFilterOptions<T>);
  static DEFAULT_HASH_FUNCTIONS: HashFunction<any>[]:
  async add(element: T): void;
  async test(element: T): boolean;
}

import { BloomFilter } from "kdim";

const filter = new BloomFilter<{ name: string }>();
await filter.add({ name: "Julien Baker" });
await filter.add({ name: "Phoebe Bridgers" });
await filter.add({ name: "Lucy Dacus" });

await filter.test({ name: "Phoebe Bridgers" }); // true
await filter.test({ name: "Julia Jacklin" }); // false

NOTE: by default, BloomFilter uses a set of object hashers that considers two objects to have the same hash if all their properties are equal, therefore will consider structurally identical objects equal.

RingBuffer

A fixed-capacity FIFO queue that overwrites earliest entries when its capacity is exceeded.

Class Signature 

type Constructor<T> = { new (capacity: number): RingBuffer<T> };
class RingBuffer<T> implements Iterable<T | null> {
  constructor(capacity: number);

  static from<T>(this: Constructor<RingBuffer<T>>, data: T[]): RingBuffer<T>;

  get capacity(): number;
  get data(): (T | null)[];
  get isEmpty(): boolean;
  get isFull(): boolean;

  get [Symbol.toStringTag](): string;
  toString(): string;

  [Symbol.iterator](): IterableIterator<T | null>;

  peek(index: number = 0): T | null;
  enqueue(element: T): void;
  dequeue(): T | null;
  drain(): T[];
}

import { RingBuffer } from "kdim";

// Initialized with capacity
const buff = new RingBuffer<string>(10);

buff.enqueue("age");
buff.enqueue("quod");
buff.enqueue("agis");

buff.isFull; // false
buff.isEmpty; // false

buff.dequeue(); // "age"
buff.peek(); // "quod"
buff.drain(); // ["quod", "agis"]

buff.isEmpty; // true

// Initialized with data
const buff2 = RingBuffer.from([12, 24, 36, 48]);

buff.capacity; // 4

buff.enqueue(1);
buff.peek(); // 1
buff.peek(2); // 36
buff.dequeue(); // 1
buff.dequeue(); // 24
buff.drain(); // [36, 48];

Generators

Random

Produce random values of common numeric and other types (E.G. u8, integer, bool), sample and shuffle Arrays and Sets of values, and determine characteristics of those sets.

Class Signature 
class Random {
  static bool(): boolean;
  static natural(max?: number): number;
  static counting(max?: number): number;
  static integer(opts?: { min?: number; max?: number }): number;
  static float(opts?: { min?: number; max?: number }): number;
  static dice(sides: number): number;
  static u8(): number;
  static u16(): number;
  static u32(): number;
  static i8(): number;
  static i16(): number;
  static i32(): number;

  static unitVector<N extends number>(n: N): Vec<N>;

  static sample<T>(options: T[] | Set<T>): T | undefined;
  static take<T>(options: T[] | Set<T>): T | undefined;
  static permute<T>(array: T[]): void;
  static permutation<T>(array: T[]): T[];
  static permutationsOf(set: number | Array<unknown> | Set<unknown>): number;
  static derange<T>(array: T[]): void;
  static derangement<T>(array: T[]): T[];
  static derangementsOf(set: number | Array<unknown> | Set<unknown>): number;

  static Seedable = Mulberry32;
  static SFC32 = SFC32;
  static JSF32B = JSF32B;
  static SplitMix32 = SplitMix32;
  static Mulberry32 = Mulberry32;
  static GJRand32 = GJRand32;
}
import { Random } from "kdim";

// Generate values
const trueOrFalse = Random.bool();
const volume = Random.integer({ max: 11 });
const attackRoll = Random.dice(20);

// Sample lists and sets
const adjectives = ["harder", "better", "faster", "stronger"];
const doIt = Random.sample(adjectives);

const cookieJar = new Set(["chocolate chip", "oatmeal", "macadamia"]);
const eaten = Random.take(cookieJar); // "macadamia" maybe?
cookieJar.has(eaten); // false

// Shuffle lists
const code = [13, 17, 29, 42];
const shuf = Random.permutation(code); // [29, 17, 13, 42] maybe?

const friends = ["alice", "bob", "carlos", "dan", "erin"];
const secretSantas = Random.derangement(friends); // Shuffled with no fixed points
Random.derangementsOf(friends); // 44: ways to match 5 people for secret santa

NOTE: Generators that take numeric arguments will throw if range is invalid (E.G. min > max), values are invalid (decimal number passed to integer generator), or other.

Seedable PRNGs

In addition to the default, static random generator functions, there are a number of constructable, seedable PRNGs, each with slightly different characteristics:

  • Seedable: the default seedable PRNG, currently Mulberry32
  • Mulberry32: a fast, well-distributed PRNG with 32 bits of internal state and a reasonable period of ~4 billion
  • SFC32: a fast, chaotic PRNG with 128 bits of internal state and a large period
  • GJRand32: a chaotic PRNG with 128 bits of internal state and a massive period, with pretty good performance
  • JSF32B: yet another fast, chaotic PRNG with overall good characteristics
  • SplitMix32: the fastest PRNG here, with overall great characteristics, if relatively small seed and period
const m32 = new Random.Mulberry32(69);
const sfc = new Random.SFC32(69, 420, 0, 42);
const gjr = new Random.GJRand32(69, 420, 0, 42);
const jsf = new Random.JSF32B(69, 420, 0, 42);
const smx = new Random.SplitMix32(69);

[m32, sfc, gjr, jsf, smx].map((rng) => rng.dice(20));
// [ 16, 1, 1, 18, 10 ]

NOTE: remember to construct these PRNGs! Unlike the Random singleton itself, each of these needs an initial seed, which can be chosen as a constant if predictable results are desired, or from another source of entropy, like Math.random().

Range

Produce values over numeric ranges, including infinite ranges, descending ranges, and custom step sizes. The Range.of static method will produce an array, while the Range.lazy static method returns a Generator which can be iterated, producing values on demand.

Class Signature 

type RangeOptions = { from?: number; to: number; step?: number };

type Constructor<T, A extends Array<unknown>> = {
  new (...args: A): T;
};

type Factory<T, A extends Array<unknown>> = (...args: A) => T;

class Range {
  static of<N = number>(
    where: number | RangeOptions,
    factory?: Constructor<N, [n: number]> | Factory<N, [n: number]>
  ): N[];

  static *lazy<N = number>(
    where: number | RangeOptions,
    factory?: Constructor<N, [n: number]> | Factory<N, [n: number]>
  ): Generator<N, void, void>;
}

Simple numeric ranges

import { Range } from "kdim";

// Produce simple numeric ranges
const zeroToFive = Range.of(5); // [0, 1, 2, 3, 4]

// Produce ranges with custom bounds and step sizes
const oddNumbers = Range.of({ from: 1, to: 9, step: 2 }); // [1, 3, 5, 7, 9]
const descending = Range.of({ from: 43, to: 40 }); // [43, 42, 41, 40]

By default, from = 0 and step = 1, unless specified.

NOTE: throws an Error when given a negative step size; the sign of the step is determined by the direction of the range, E.G. Math.sign(to - from).

Ranges of custom values via factory function

// Produce a range of custom values with a factory function
type Foo = { bar: number };
const foos = Range.of<Foo>(4, (n) => ({
  bar: n ** n,
}));
// [{ bar: 0 }, { bar: 1 }, { bar: 4 }, { bar: 27 }]

Ranges of class instances

// Produce instances of a class with a constructor of type
// { new (n: number): T }
class Baz {
  n: number;
  s: number;
  constructor(n: number) {
    this.n = n;
    this.s = n * n;
  }
}

const bazzes = Range.of({ from: 2, to: 4 }, Baz);
// [Baz { n: 2, s: 4 }, Baz { n: 3, s: 9 }, Baz { n: 4, s: 16 }]

Lazy ranges

// Iterating an infinite range
const gen = Range.lazy({ from: 1, to: Infinity });
for (const i of gen) {
  if (i >= 99) break;
}
gen.next(); // 100

// Using a lazy range to defer producing values until needed
const deferred = Range.lazy({ to: 20 }, (n) => new Complex(-n, n - 5));
// Some time later...
const complexes = Array.from(deferred); // Only now are values produced

NOTE: Generators over infinite ranges will lock up resources and crash the process if you attempt to convert them to an Array via Array.from(gen), [...gen], or other means.

Noise

Generate structured mathematical noise patterns like Perlin and Simplex, in both 2D and 3D spaces. Layer patterns for fractal noise fields. Efficiently fill TypedArray and ImageData buffers for use in graphics applications. These can be parametrized with Seedable PRNGs, or use the default (unseedable) PRNG based on Math.random().

Abstract Class Signature 

abstract class NoiseGenerator {
  abstract seed(seed: number): this;
  abstract xy(x: number, y: number): number;
  abstract xyz(x: number, y: number, z: number): number;
  abstract fill(target: NoiseTarget, options?: NoiseFillOptions): void;
}

type NoiseTarget = ImageData | number[][] | TypedArrayNoiseTarget;

type TypedArrayNoiseTarget = {
  data: Uint8ClampedArray;
  width: number;
  stride: number;
};

type NoiseFillOptions = {
  freq?: number;
  set?: (cell: { x: number; y: number; z: number; v: number }) => void;
} & (Noise2DFillOptions | Noise3DFillOptions);

type Noise2DFillOptions = {
  z?: never;
};

type Noise3DFillOptions = {
  z: number;
};

Type Description Image
Perlin freq = 5 Animated GIF of Perlin noise
Simplex freq = 5 Animated GIF of Simplex noise
Compound noise Perlin of freq = 5 added pixel-wise with Simplex of freq = 40 contribution scaled by 0.25 Animated GIF of Perlin noise

The Noise module contains various classes implementing NoiseGenerator:

Perlin

Perlin noise is a type of gradient noise with smoothly-varying texture in 2 and 3 dimensions.

import { Noise } from "kdim";

const perlin = new Noise.Perlin();

// Iteratively create noise over a 2D plane
for (let x = 0; x < 100; x++) {
  for (let y = 0; y < 100; y++) {
    // Generates a number between [-1, 1] that smoothly varies with x and y.
    // Since we scale the coordinates down to [0, 1], the final texture
    // will have a frequency of 1 (one "cell" of a pattern).
    const value = perlin.xy(x / 100, y / 100);
    doSomething(value);
  }
}

// Fill a canvas with Perlin noise, animating smoothly as we take
// different slices of the 3D volume.
const canvas = document.querySelector("canvas");
const ctx = canvas.getContext("2d");
const img = ctx.createImageData(canvas.width, canvas.height);

let z = 0;
(function loop() {
  // Fill the underlying buffer with a 2D slice of 3D noise, which
  // has a frequncy of 5x5 "cells". The `fill` method will detect
  // the buffer size and stride automatically.
  perlin.fill(img, { freq: 5, z });
  ctx.putImageData(img, 0, 0);
  z += 0.01;

  requestAnimationFrame(loop);
})();

Simplex

Simplex noise is a type of gradient noise with with fewer dimensional artifacts than Perlin noise. It is visually more isotropic, and less computationally expensive in higher dimensions.

import { Noise } from "kdim";

const simplex = new Noise.Simplex();

// Iteratively create noise over a 2D plane
for (let x = 0; x < 100; x++) {
  for (let y = 0; y < 100; y++) {
    // Generates a number between [-1, 1] that smoothly varies with x and y.
    // Since we scale the coordinates down to [0, 1], the final texture
    // will have a frequency of 1 (one "cell" of a pattern).
    const value = simplex.xy(x / 100, y / 100);
    doSomething(value);
  }
}

// Fill a canvas with Simplex noise, animating smoothly as we take
// different slices of the 3D volume.
const canvas = document.querySelector("canvas");
const ctx = canvas.getContext("2d");
const img = ctx.createImageData(canvas.width, canvas.height);

let z = 0;
(function loop() {
  // Fill the underlying buffer with a 2D slice of 3D noise, which
  // has a frequncy of 5x5 "cells". The `fill` method will detect
  // the buffer size and stride automatically.
  simplex.fill(img, { freq: 5, z });
  ctx.putImageData(img, 0, 0);
  z += 0.01;

  requestAnimationFrame(loop);
})();

Probability

Sample weighted collections of elements to emulate complex probabilities, unfair dice, etc. These can be parametrized with Seedable PRNGs, or use the default (unseedable) PRNG based on Math.random().

Class Signature 

class Probability<T> {
  constructor(events: ProbabilityEvent<T>[] = [], rng?: PRNG | null);
  event(event: ProbabilityEvent<T>): this;
  sample(): ProbabilityEvent<T>;
  take(): ProbabilityEvent<T>;
}

type ProbabilityEvent<T> = {
  value: T;
  p?: number;
};

import { Probability } from "kdim";

const events = [
  { value: "common", p: 0.65 },
  { value: "uncommon", p: 0.25 },
  { value: "rare", p: 0.1 },
];

const d1 = new Probability(events);
d1.sample().value; // "common", probably?

const rng = new Random.Mulberry32(42);
const d2 = new Probability(events, new Random.Seedable(42));
d2.sample().value; // "uncommon", definitely.

Analysis

Comparator

A class used to determine ordering and equality of values of arbitrary types. A Comparator implements custom equality (eq) and ordering (gt, gte, lt, lte) for values based on the provided CompareFunction. If no function is provided, it defaults to the same lexical comparison used by Array.prototype.sort.

Class Signature 

type CompareFunction<V> = (a: V, b: V) => number;

class Comparator<V> {
  constructor(compareFn?: CompareFunction<V>);

  static lexicalCompare<V>(a: V, b: V): 0 | 1 | -1;
  static reverseLexicalCompare<V>(a: V, b: V): 0 | 1 | -1;
  static numericCompare<V extends Number = number>(a: V, b: V): number;
  static reverseNumericCompare<V extends Number = number>(a: V, b: V): number;

  eq(a: V, b: V): boolean;
  gt(a: V, b: V): boolean;
  gte(a: V, b: V): boolean;
  lt(a: V, b: V): boolean;
  lte(a: V, b: V): boolean;
}

CompareFunction

The CompareFunction type is similar to the callback passed to Array.prototype.sort, in that it is called with two arguments a and b, and returns a number determining if a should be sorted before b (negative number), after b (positive number), or equal to b (0). The Comparator class has a number of built-in static CompareFunctions for ordering primitives.

import { Comparator } from "kdim";

// Using built-in CompareFunctions for sorting
const ns = [12, 5, 80, 3, -1];
const descendingOrder = ns.sort(Comparator.reverseNumericCompare);
// [80, 12, 5, 3, -1]

const words = ["hi", "there", "friends"];
const descendingWordOrder = words.sort(Comparator.reverseLexicalCompare);
// ["there", "friends", "hi"]

Comparator of a complex type

// Compare non-primitive values by comparing their primitive properties
const quzComp = new Comparator<{ quz: number }>((a, b) => {
  return Comparator.numericCompare(a.quz, b.quz);
});

quzComp.eq({ quz: 7 }, { quz: 7 }); // true
quzComp.lt({ quz: 7 }, { quz: 3 }); // true
quzComp.gte({ quz: 1 }, { quz: -20 }); // false

Comparator for values of multiple types

type A = { id: number; tag: string };
type B = { id: number; foo: string };

// As and Bs are considered equal if their ids and string part match,
// otherwise they are ordered first by id then by string part.
const abComp = new Comparator<A | B>((first, second) => {
  if (first.id === second.id) {
    const firstStr = (first as A).tag || (first as B).foo;
    const secondStr = (second as A).tag || (second as B).foo;
    return Comparator.lexicalCompare(firstStr, secondStr);
  } else {
    return first.id - second.id;
  }
});

abComp.eq({ id: 1, tag: "cool" }, { id: 1, foo: "bad" }); // false
abComp.eq({ id: 1, tag: "cool" }, { id: 1, foo: "cool" }); // true
abComp.gt({ id: 1, tag: "cool" }, { id: 1, tag: "neat" }); // true
abComp.lte({ id: 5, tag: "ok" }, { id: 99, tag: "neat" }); // false

Statistics

Perform common statistical analyses on discrete numeric data. These data are typically number[], but can also be object number types like Complex and Rational, or your own custom number types that implement the ArithmeticObject interface listed in the class signature below.

Class Signature 

class Statistics {
  static min<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static max<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static mean<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static median<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static mode<T extends Number & Arithmetic<T>>(
    data: T[]
  ): number[] | undefined;
  static variance<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static sd<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static sem<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static range<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static iqr<T extends Number & Arithmetic<T>>(
    data: T[],
    options?: SummaryOptions
  ): T | undefined;
  static mad<T extends Number & Arithmetic<T>>(data: T[]): T | undefined;
  static percentiles<T extends Number & Arithmetic<T>>(
    data: T[],
    options?: PercentileOptions | number[]
  ): T[] | undefined;
  static summary<T extends Number & Arithmetic<T>>(
    data: T[],
    options?: SummaryOptions
  ): FiveNumberSummary<T> | undefined;
}

type ArithmethicObject<T extends Number> = Add<[T] | [number], T> &
  Sub<[T] | [number], T> &
  Mul<[T] | [number], T> &
  Div<[T] | [number], T> &
  Pow<[T] | [number], T> &
  Eq<[T] | [number]> &
  Abs<T>;

type Arithmetic<T extends Number> = number | ArithmethicObject<T>;

type FiveNumberSummary<T extends Number> = {
  q0: T;
  q1: T;
  q2: T;
  q3: T;
  q4: T;
};

type InterpolationMethod =
  | "midpoint"
  | "nearest"
  | "hrank"
  | "lrank"
  | "weighted"
  | "outer";

type SummaryOptions = {
  method?: InterpolationMethod;
};

type PercentileOptions = SummaryOptions & {
  p: number[];
};

const QUARTILES = [0, 0.25, 0.5, 0.75, 1];

Central tendency

Calculate common measures of centrality, including mean, median, mode.

import { Statistics } from "kdim";

const data = [3, 2, 2, 9, 4, 7, 1]; // data does not need to be pre-sorted

Statistics.mean(data); // 6
Statistics.median(data); // 5
Statistics.mode(data); // [2] (returned as an array for cases of multimodal data)

Dispersion

Calculate common measures of statistical dispersion.

import { Statistics } from "kdim";

const data = [2, 4, 4, 4, 5, 5, 7, 9];

Statistics.sd(data); // Standard Deviation = 2
Statistics.variance(data); // Variance = 4
Statistics.sem(data); // Standard Error of Mean = 0.7071067811865475
Statistics.range(data); // Range = 7
Statistics.mad(data); // Mean Absolute Deviation = 0.5
Statistics.iqr(data); // Inter-Quartile Range = 2

Interpolation methods

Many of the statistical methods offer multiple InterpolationMethod strategies for interpolating between discrete data points.

import { Statistics } from "kdim";

const data = [6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49];

// Five-Number Summary using different interpolation strategies

Statistics.summary(data); // default "midpoint"
// { q0: 6, q1: 25.5, q2: 40, q3: 42.5, q4: 49 }
Statistics.summary(data, { method: "nearest" });
// { q0: 6, q1: 36, q2: 40, q3: 43, q4: 49 }
Statistics.summary(data, { method: "lrank" });
// { q0: 6, q1: 15, q2: 40, q3: 42, q4: 49 }

Fourier

Perform Fourier analysis on discrete numeric data.

Class Signature 

class Fourier {
  static dft(input: (Number | Complex)[]): Complex[];
}

import { Fourier } from "kdim";

const samples = [1, 1, 0, 0];
const d = Fourier.dft(sample);
// [
//   Complex { real: 2, imaginary: 0 },
//   Complex { real: 1, imaginary: -1 },
//   Complex { real: 0, imaginary: 0 },
//   Complex { real: 1, imaginary: 1 },
// ]

Utilities

clamp

Constrain a value to within a given range [min, max].

Function Signature 

function clamp(min: number, max: number, value: number): number;

import { clamp } from "kdim";

// Clamp values to [0, 255]
const a = 32;
const b = 300;

clamp(0, 255, a); // 32
clamp(0, 255, b); // 255

NOTE: throws a RangeError when the range is invalid, E.G. min > max.

lerp

Linear interpolation of a value in the range [0, 1] to a value in the range [from, to].

Function Signature 

function lerp(from: number, to: number, value: number): number;

import { lerp } from "kdim";

// Interpolate 0.4 in range [0,1] to range [1,99]
const value = 0.4;
const interpolated = lerp(1, 99, value); // 40.2

NOTE: throws a RangeError when the value is outside of [0, 1]

gcf

Find the Greatest Common Factor of two integers.

Function Signature 

function gcf(a: number, b: number): number;

import { gcf } from "kdim";

gcf(45, 420); // 15

NOTE: throws a RangeError when a or b are non-integral.

lcm

Find the Least Common Multiple of two integers.

Function Signature 

function lcm(a: number, b: number): number;

import { lcm } from "kdim";

lcm(6, 20); // 60

NOTE: throws a RangeError when a or b are non-integral.

trailingZeros

Compute the number of trailing zeros in a number's 32-bit representation, equivalent to its largest power-of-two divisor.

Function Signature 

function trailingZeros(n: number): number;

import { trailingZeros } from "kdim";

trailingZeros(24); // 3

NOTE: throws a RangeError when n is non-integral.

objectHash

A hashing function for arbitrary objects and primitives that digests the value into a pseudo-unique base-16 hash string. Uses structural hashing, such that objects of identical structure will produce the same hash.

Function Signature 

type ObjectHashAlgorithm = "SHA-1" | "SHA-256" | "SHA-384" | "SHA-512";

type ObjectHashOptions = {
  algorithm?: ObjectHashAlgorithm;
};

async function objectHash<T>(
  obj: T,
  options?: ObjectHashOptions
): Promise<string>;

import { objectHash } from "kdim";

const hash = await objectHash({ foo: 7, bar: [] }, { algorithm: "SHA-1" });
// "1448bf86764e7ff7f9df0cb61b2d77c946ba854"

Assertions

castInteger

Assert that a number (or object convertible to number) has no fractional part and is within the safe integer range, and cast it to a primitive number.

Function Signature 

function castInteger<N extends Number>(n: N): number;

import { castInteger, Wrapping } from "kdim";

const i = 300;
const u8 = new Wrapping({ max: 7 }, 5);
const f = 29.7;

const safeI = castInteger(i); // 300
const safeU8 = castInteger(u8); // 5
const errF = castInteger(f); // Error: Values must be safe integers

NOTE: throws a RangeError if n has a fractional part, is not in safe integer range, or cannot be coerced from/to a number via Number(n) and n.valueOf().

assertInteger

Assert that number(s) have no fractional part and are within the safe integer range.

Function Signature 

function assertInteger(...numbers: number[]): void;

import { assertInteger } from "kdim";

assertInteger(-300); // ok
assertInteger(1, 2, 10); // ok
assertInteger(7.89); // Error: Arguments must be integers

NOTE: throws a RangeError if any number has a fractional part, or is not in safe integer range.

assertNatural

Assert that number(s) have no fractional part and are zero or greater.

Function Signature 

function assertNatural(...numbers: number[]): void;

import { assertNatural } from "kdim";

assertNatural(300); // ok
assertNatural(1, 2, 10); // ok
assertNatural(-1); // Error: Arguments must be natural numbers

NOTE: throws a RangeError if any number has a fractional part, or is not zero or greater.

assertCounting

Assert that number(s) have no fractional part and are greater than zero.

Function Signature 

function assertCounting(...numbers: number[]): void;

import { assertCounting } from "kdim";

assertCounting(300); // ok
assertCounting(1, 2, 10); // ok
assertCounting(0); // Error: Arguments must be counting numbers

NOTE: throws a RangeError if any number has a fractional part, or is not greater than zero.

assertValidRange

Assert that the arguments constitute a valid range, in which min < max, and, if present, min <= value && value <= max.

Function Signature 

function assertValidRange(min: number, max: number, value?: number): void;

assertValidRange(0, 5);
assertValidRange(0, 5, 1);
assertValidRange(5, 0); // Error: Minimum must be less than maximum
assertValidRange(0, 5, 9); // Error: Value must be between minimum and maximum

Utility Types

Tuple

A typed tuple of generic length.

import { Tuple } from "kdim";

const threeBoolTuple: Tuple<boolean, 3> = [true, false, false];
const wrongBoolTuple: Tuple<boolean, 3> = [false, true]; // Error: Source has 2 element(s) but target requires 3

This also composes to allow for strongly-typed multidimensional arrays, such as a chess board:

import { Tuple } from "kdim";

type Piece = "pawn" | "rook" | "knight" | "bishop" | "queen" | "king";
type Board = Tuple<Tuple<Piece | null, 8>, 8>;

const board: Board = [
  ["rook", "knight", "bishop", "queen", "king", "bishop", "knight", "rook"],
  ["pawn", "pawn", "pawn", "pawn", "pawn", "pawn", "pawn", "pawn"],
  [null, null, null, null, null, null, null, null],
  [null, null, null, null, null, null, null, null],
  [null, null, null, null, null, null, null, null],
  [null, null, null, null, null, null, null, null],
  ["pawn", "pawn", "pawn", "pawn", "pawn", "pawn", "pawn", "pawn"],
  ["rook", "knight", "bishop", "queen", "king", "bishop", "knight", "rook"],
];

Vec

A convenience type equivalent to Tuple<number, K extends number>, useful for mathematical computation and data structures.

import { Vec } from "kdim";

const position: Vec<3> = [1.0, 69, 420];
const speed: Vec<3> = [-1, 3, 11];
const accel: Vec<3> = [0, 0, "no"]; // Error: type 'string' is not assignable to type 'number'

License

MIT © Tobias Fried

About

Exploring mathematical types and data structures in TypeScript

rektdeckard.github.io/kdim

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