[ https://issues.apache.org/jira/browse/SIS-455?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel ] Martin Desruisseaux updated SIS-455: ------------------------------------ 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. -- This message was sent by Atlassian JIRA (v7.6.3#76005)