Martin Desruisseaux created SIS-455:
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Summary: Compute length of cubic Bézier curve
Key: SIS-455
URL: https://issues.apache.org/jira/browse/SIS-455
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 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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