Martin Desruisseaux created SIS455:

Summary: Compute length of cubic Bézier curve
Key: SIS455
URL: https://issues.apache.org/jira/browse/SIS455
Project: Spatial Information Systems
Issue Type: New Feature
Components: Geometry, Referencing
Affects Versions: 1.0
Reporter: Martin Desruisseaux
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 LegendreGauss approach documented in [A Primer on Bézier
Curveshttps://pomax.github.io/bezierinfo/#arclength] page. The accuracy is determined by
the number [Gaussian Quadrature Weights and Abscissaehttps://pomax.github.io/bezierinfo/legendregauss.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 Speedhttps://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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