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From "Martin Desruisseaux (Jira)" <j...@apache.org>
Subject [jira] [Closed] (SIS-455) Compute length of cubic Bézier curve
Date Sun, 20 Oct 2019 19:43:00 GMT
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[ https://issues.apache.org/jira/browse/SIS-455?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel
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Martin Desruisseaux closed SIS-455.
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> Compute length of cubic Bézier curve
> ------------------------------------
>
>                 Key: SIS-455
>                 URL: https://issues.apache.org/jira/browse/SIS-455
>             Project: Spatial Information Systems
>          Issue Type: Improvement
>          Components: Geometry, Referencing
>            Reporter: Martin Desruisseaux
>            Assignee: Martin Desruisseaux
>            Priority: Major
>             Fix For: 1.0
>
>
> We need a way to estimate the length of a cubic Bézier curve from its starting point
at _t_=0 to an arbitrary _t_ value where _t_ ∈ [0…1]. Conversely, we need to estimate
the _t_ parameter for a given length since the starting point. There is no exact solution
for this problem, but we may estimate the length using Legendre-Gauss approach documented
in [A Primer on Bézier Curves|https://pomax.github.io/bezierinfo/#arclength] page. The accuracy
is determined by the number [Gaussian Quadrature Weights and Abscissae|https://pomax.github.io/bezierinfo/legendre-gauss.html]
used. For example with 3 terms:
> {noformat}
> w₁ = 0.8888888888888888; a₁ = 0;
> w₂ = 0.5555555555555556; a₂ = -0.7745966692414834;
> w₃ = 0.5555555555555556; a₃ = +0.7745966692414834;
> length(t) ≈ t/2 * (w₁*f(a₁*t/2 - t/2) + w₂*f(a₂*t/2 - t/2) + w₃*f(a₃*t/2
- t/2))
> {noformat}
> with _f(t)_ defined as {{hypot(x′(t), y′(t))}} and with _x′(t)_ and _y′(t)_ the
first derivatives of Bézier equations for _x(t)_ and _y(t)_.
> Once we have the length for a given _t_ value, we can try to find the converse by using
an iterative approach as described in the [Moving Along a Curve with Specified Speed|https://www.geometrictools.com/Documentation/MovingAlongCurveSpecifiedSpeed.pdf]
paper from geometric tools.
> Once we are able to estimate the _t_ parameters for a given length, we should delete
the {{Bezier.isValid(x, y)}} method (and consequently remove its use and the {{TransformException}}
in the {{curve}} method). Instead, given the geodesic distance from Bézier start point to
¼ of the distance from start point to end point, estimate the _t_ parameter at that position.
It should be a value close but not identical to _t_≈¼. We can then compute the (_x_, _y_)
coordinates of the point on that curve at that _t_ parameter value and compare with the expected
coordinates. It should (hopefully) be a point closer to expected than the point computed at
exactly _t_=¼, thus removing the need for the {{Bezier.isValid(x,y)}} hack.
> Alternatively, all the above is a complicated way to say that we want the shortest distance
between a point on the geodesic path and a point on the curve which is known to be at position
close to (but not exactly at) _t_≈¼ and _t_≈¾.

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