svg-path-properties/src/bezier-functions.ts

Calculate the length for an SVG path, to use it with node or a Canvas element

/* eslint-disable security/detect-object-injection */
import { tValues, cValues, binomialCoefficients } from './bezier-values.ts'
import { Point } from './types.ts'
export const cubicPoint = (xs: number[], ys: number[], t: number): Point => {
  const x =
    (1 - t) * (1 - t) * (1 - t) * xs[0] +
    3 * (1 - t) * (1 - t) * t * xs[1] +
    3 * (1 - t) * t * t * xs[2] +
    t * t * t * xs[3]
  const y =
    (1 - t) * (1 - t) * (1 - t) * ys[0] +
    3 * (1 - t) * (1 - t) * t * ys[1] +
    3 * (1 - t) * t * t * ys[2] +
    t * t * t * ys[3]

  return { x, y }
}

export const cubicDerivative = (xs: number[], ys: number[], t: number) => {
  const derivative = quadraticPoint(
    [3 * (xs[1] - xs[0]), 3 * (xs[2] - xs[1]), 3 * (xs[3] - xs[2])],
    [3 * (ys[1] - ys[0]), 3 * (ys[2] - ys[1]), 3 * (ys[3] - ys[2])],
    t
  )
  return derivative
}

export const getCubicArcLength = (xs: number[], ys: number[], t: number) => {
  let sum: number
  let correctedT: number

  /* if (xs.length >= tValues.length) {
        throw new Error('too high n bezier');
      } */

  const n = 20

  const z = t / 2
  sum = 0
  for (let i = 0; i < n; i++) {
    correctedT = z * tValues[n][i] + z
    sum += cValues[n][i] * BFunc(xs, ys, correctedT)
  }
  return z * sum
}

export const quadraticPoint = (xs: number[], ys: number[], t: number): Point => {
  const x = (1 - t) * (1 - t) * xs[0] + 2 * (1 - t) * t * xs[1] + t * t * xs[2]
  const y = (1 - t) * (1 - t) * ys[0] + 2 * (1 - t) * t * ys[1] + t * t * ys[2]
  return { x, y }
}

export const getQuadraticArcLength = (xs: number[], ys: number[], t: number) => {
  if (t === undefined) {
    t = 1
  }
  const ax = xs[0] - 2 * xs[1] + xs[2]
  const ay = ys[0] - 2 * ys[1] + ys[2]
  const bx = 2 * xs[1] - 2 * xs[0]
  const by = 2 * ys[1] - 2 * ys[0]

  const A = 4 * (ax * ax + ay * ay)
  const B = 4 * (ax * bx + ay * by)
  const C = bx * bx + by * by

  if (A === 0) {
    return t * Math.sqrt(Math.pow(xs[2] - xs[0], 2) + Math.pow(ys[2] - ys[0], 2))
  }
  const b = B / (2 * A)
  const c = C / A
  const u = t + b
  const k = c - b * b

  const uuk = u * u + k > 0 ? Math.sqrt(u * u + k) : 0
  const bbk = b * b + k > 0 ? Math.sqrt(b * b + k) : 0
  const term =
    b + Math.sqrt(b * b + k) !== 0 && (u + uuk) / (b + bbk) !== 0
      ? k * Math.log(Math.abs((u + uuk) / (b + bbk)))
      : 0

  return (Math.sqrt(A) / 2) * (u * uuk - b * bbk + term)
}

export const quadraticDerivative = (xs: number[], ys: number[], t: number) => {
  return {
    x: (1 - t) * 2 * (xs[1] - xs[0]) + t * 2 * (xs[2] - xs[1]),
    y: (1 - t) * 2 * (ys[1] - ys[0]) + t * 2 * (ys[2] - ys[1])
  }
}

function BFunc (xs: number[], ys: number[], t: number) {
  const xbase = getDerivative(1, t, xs)
  const ybase = getDerivative(1, t, ys)
  const combined = xbase * xbase + ybase * ybase
  return Math.sqrt(combined)
}

/**
 * Compute the curve derivative (hodograph) at t.
 */
const getDerivative = (derivative: number, t: number, vs: number[]): number => {
  // the derivative of any 't'-less function is zero.
  const n = vs.length - 1
  let _vs
  let value

  if (n === 0) {
    return 0
  }

  // direct values? compute!
  if (derivative === 0) {
    value = 0
    for (let k = 0; k <= n; k++) {
      value += binomialCoefficients[n][k] * Math.pow(1 - t, n - k) * Math.pow(t, k) * vs[k]
    }
    return value
  } else {
    // Still some derivative? go down one order, then try
    // for the lower order curve's.
    _vs = new Array(n)
    for (let k = 0; k < n; k++) {
      _vs[k] = n * (vs[k + 1] - vs[k])
    }
    return getDerivative(derivative - 1, t, _vs)
  }
}

export const t2length = (
  length: number,
  totalLength: number,
  func: (t: number) => number
): number => {
  let error = 1
  let t = length / totalLength
  let step = (length - func(t)) / totalLength

  let numIterations = 0
  while (error > 0.001) {
    const increasedTLength = func(t + step)
    const increasedTError = Math.abs(length - increasedTLength) / totalLength
    if (increasedTError < error) {
      error = increasedTError
      t += step
    } else {
      const decreasedTLength = func(t - step)
      const decreasedTError = Math.abs(length - decreasedTLength) / totalLength
      if (decreasedTError < error) {
        error = decreasedTError
        t -= step
      } else {
        step /= 2
      }
    }

    numIterations++
    if (numIterations > 500) {
      break
    }
  }

  return t
}