to-power-basis-error-counters
function toPowerBasisErrorCounters(ps: number[][]): number[][]
Defined in to-power-basis/to-power-basis/to-power-basis-error-counters.ts:83
Returns a representation of the error (from which an absolute error bound
can be calculated) when calculating the power basis representation of a
bezier curve of order <= 3 (using e.g. toPowerBasis
or toPowerBasisDd
).
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> * (γγ(3)) * result_
where [[γ]] and [[γγ]] are the usual error functions with γ(1) === 1.1102230246251568e-16
and γγ(3) === 3.697785493223493e-32
.
// for cubic bezier curves
return [
[
x3, // <E> === 3
x2, // <E> === 3
x1, // <E> === 2
0,
],
[
y3, // <E> === 3
y2, // <E> === 3
y1, // <E> === 2
0,
]
]
// for quadratic bezier curves
return [
[
x2, // <E> === 2
x1, // <E> === 1
0,
],
[
y2, // <E> === 2
y1, // <E> === 1
0,
]
];
// for linear bezier curves (i.e. lines)
return [
[
x1_, // <E> === 1
x0_ // <E> === 0
],
[
y1_, // <E> === 1
y0_ // <E> === 0
]
];
Note!
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]] |