# modeling/src/curves/bezier/arcLengthToT.js

``````const lengths = require('./lengths')

/**
* Convert a given arc length along a bezier curve to a t value.
* Useful for generating equally spaced points along a bezier curve.
*
* @example
* const points = [];
* const segments = 9; // this will generate 10 equally spaced points
* const increment = bezier.length(100, bezierCurve) / segments;
* for(let i = 0; i <= segments; i++) {
*   const t = bezier.arcLengthToT({distance: i * increment}, bezierCurve);
*   const point = bezier.valueAt(t, bezierCurve);
*   points.push(point);
* }
* return points;
*
* @param {Object} [options] options for construction
* @param {Number} [options.distance=0] the distance along the bezier curve for which we want to find the corresponding t value.
* @param {Number} [options.segments=100] the number of segments to use when approximating the curve length.
* @param {Object} bezier a bezier curve.
* @returns a number in the [0, 1] interval or NaN if the arcLength is negative or greater than the total length of the curve.
* @alias module:modeling/curves/bezier.arcLengthToT
*/
const arcLengthToT = (options, bezier) => {
const defaults = {
distance: 0,
segments: 100
}
const { distance, segments } = Object.assign({}, defaults, options)

const arcLengths = lengths(segments, bezier)
// binary search for the index with largest value smaller than target arcLength
let startIndex = 0
let endIndex = segments
while (startIndex <= endIndex) {
const middleIndex = Math.floor(startIndex + (endIndex - startIndex) / 2)
const diff = arcLengths[middleIndex] - distance
if (diff < 0) {
startIndex = middleIndex + 1
} else if (diff > 0) {
endIndex = middleIndex - 1
} else {
endIndex = middleIndex
break
}
}
// if we have an exact match, return it
const targetIndex = endIndex
if (arcLengths[targetIndex] === distance) {
return targetIndex / segments
}
// we could get finer grain at lengths, or use simple interpolation between two points
const lengthBefore = arcLengths[targetIndex]
const lengthAfter = arcLengths[targetIndex + 1]
const segmentLength = lengthAfter - lengthBefore
// determine where we are between the 'before' and 'after' points
const segmentFraction = (distance - lengthBefore) / segmentLength
// add that fractional amount and return
return (targetIndex + segmentFraction) / segments
}

module.exports = arcLengthToT
``````