[ https://issues.apache.org/jira/browse/SIS-455?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel ]
Martin Desruisseaux updated SIS-455:
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Issue Type: Improvement (was: New Feature)
> 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
> Affects Versions: 1.0
> Reporter: Martin Desruisseaux
> Priority: Major
>
> 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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