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bezier-bezier-intersection

function bezierBezierIntersection(ps1: number[][], ps2: number[][]): X[]

Defined in intersection/bezier-bezier-intersection/bezier-bezier-intersection.ts:59

Returns an array of intersections between two bezier curves up to cubic order (i.e. points, linear, quadratic or cubic bezier curves (i.e. order 0,1,2 or 3 curves). The algorithm employed uses advanced techniques such as floating point error bounding, adaptive multi-precision floating point arithmetic, pre-filtering of easy cases, certified root finding and algebraic implicitization of the curves in order to find guaranteed accurate results.

  • this algorithm is mathematically guaranteed accurate to within 4 * Number.EPSILON in the returned t parameter values of the bezier curves (bar underflow/overflow)

  • the returned intersections are ordered by t parameter value of the first bezier curve

  • if the two curves have an infinite number of intersections then the intersection of the endpoints of each curve with the other is returned instead (and the intersection kind property will equal 5)

  • each intersection in the returned array of intersections is an object with the following properties (see the type [[X]]`):

    • p: point of intersection (calculated from the guaranteed root interval)
    • t1: first bezier curve's parameter t value (calculated from the guaranteed root interval)
    • t2: second bezier curve's parameter t value (calculated from the guaranteed root interval)
    • kind: kind of intersection (see [[X]] for details)
    • ri1: first bezier curve's root interval guaranteed to contain the correct t value in the form { tS, tE, multiplicity }, where tS and tE are the start and end of the interval
    • ri2: second bezier curve's root interval guaranteed to contain the correct t value in the form { tS, tE, multiplicity }, where tS and tE are the start and end of the interval
    • box: small box that is guaranteed to contain the intersection (calculated from the guaranteed root interval)

Some examples are shown below:

This example illustrates Bezout's Theorem: a cubic-cubic intersection implies 3x3 = 9 intersections at most.

Note!
Green circles are draggable!
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Overlapping algebraically identical bezier curves have an infinite number of intersecions; this is indicated by the kind of intersection (kind === 5) and only the intersections at the endpoints of each curve are retrurned

Note!
Green circles are draggable!
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point / cubic bezier curve - point is exactly on curve

Note!
Green circles are draggable!
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quadratic / quadratic bezier curves

Note!
Green circles are draggable!
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Parameters:​

NameTypeDescription
ps1number[][]an order 0,1,2 or 3 bezier curve given as an ordered array of its control point coordinates, e.g. [[0,0], [1,1], [2,1], [2,0]]
ps2number[][]an order 0,1,2 or 3 bezier curve given as an ordered array of its control point coordinates, e.g. [[0,0], [1,1], [2,1], [2,0]]