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From "Martin Desruisseaux (JIRA)" <>
Subject [jira] [Created] (SIS-455) Compute length of cubic Bézier curve
Date Fri, 24 May 2019 08:43:00 GMT
Martin Desruisseaux created SIS-455:

             Summary: Compute length of cubic Bézier curve
                 Key: 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|] page. The accuracy is determined by
the number [Gaussian Quadrature Weights and Abscissae|]
used. For example with 3 terms:

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))

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|]
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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