eval-de-casteljau-error

function evalDeCasteljauError(ps: number[][], t: number[]): number[]

Defined in local-properties-at-t/evaluate/eval-de-casteljau-error.ts:70

Returns a representation of the error (from which an absolute error bound can be calculated) when evaluating the given bezier curve at the parameter t using De Casteljau's algorithm.

The returned error representation needs to be multiplied with Stewart error counters¹ and an appropriate error function, γ, depending on the precision used (e.g. double or double-double). This is explained in more detail below. See also Higham 2002 p. 68 near the bottom.

(1) G. W. Stewart. Introduction to Matrix Computations. Academic Press, New York, 1973. xiii+441 pp. ISBN 0-12-670350-7

The absolute erros below can be calculated as follows (where <E> are the error counters as indicated in the comments of the return value below):

• double precision: <E> * (γ(1)) * result_
• double-double precision: <E> * (2*γγ(3)) * result_

where [[γ]] and [[γγ]] are the usual error functions with γ(1) === 1.1102230246251568e-16 and γγ(3) === 3.697785493223493e-32. The T in the error counter formula is the input error given as an error counter on t. For example, if the exact t (let's call it te) is bounded by (|t| - 5u) < |te| < (|t| + 5u) where u === Number.EPSILON/2 then T should be given as 5. If t is exact then T is zero.

// for cubic bezier curves
return [ 
    x_,  // <E> === 3T + 9
    y_   // <E> === 3T + 9
];
// for quadratic bezier curves
return [ 
    x_,  // <E> === 2T + 6
    y_   // <E> === 2T + 6
];
// for linear bezier curves (i.e. lines)
return [ 
    x_,  // <E> === T + 3
    y_   // <E> === T + 3
];

Parameters:

Name	Type	Description
ps	number[][]	an order 0,1,2 or 3 bezier curve given by an ordered array of its control points, e.g. [[0,0],[1,1],[2,1],[2,0]]
t	number[]	the parameter value where the bezier should be evaluated (given in double-double precision)