One solution is to simply raise the number of control points used to form the curve, though as the number of control points rises overfitting will become more noticeable. Global control benefits my solution over a short range, maintaining a natural arc, though over longer more intricate paths global control across many more control points may unfavourably alter the path defined by others elsewhere in the curve.
A better method is to join a series of cubic curves together to form a piecewise cubic curve.
While each control point has global control within its cubic curve, it will have no influence over the curve during any other cubic section. This will lessen overfitting while offering suitable control to create complicated pathways.
In a piecewise cubic curve, the final control point of each cubic section is shared with the subsequent section which uses the same control point as its first. This shared control point is called the “knot”. This is shown in figure 1.
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| Figure 1: Curve segments join at "knots" (Sellarès, not dated) |
Depending on the position of control points of adjacent cubic sections, preserving curve continuity across sections can be relatively straightforward. An example of such a configuration is shown in figure 2.
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| Figure 2: Join of curve sections with intuitively maintains continuity (Zhou, Song and Tian, 2011, p. 2448) |
However, if control points formation creates an acute angle between cubic sections it can prevent higher orders of C continuity as “don’t enforce derivative continuity at join points” (Sellarès, not dated).
In these instances, Zhou, Song and Tian (2011, p. 2447) suggest creating supplemental control points (shown in figure 3) at mid-points on adjacent sides of both Bézier polygons.
These supplemental control points are used with the original control point to create a quadratic Bézier curve to join the two cubic sections (where the supplemental control points have become the end/start points of the respective cubic sections).
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| figure 3: Addition of supplemental control points to form linking quadratic Bézier curve (Zhou, Song and Tian, 2011, p. 2448) |
This allows higher orders of continuity to be maintained throughout the piecewise curve.
References
Sellarès, T. (n.d.). Curves. Available at: http://ima.udg.edu/~sellares/MEG/CurvesMEG09.pdf [Accessed 14 Jan. 2019].
Zhou, F., Song, B. and Tian, G. (2011). Bézier Curve Based Smooth Path Planning for Mobile Robot. Journal of Information & Computational Science 8: 12 (2011) 2441–2450, [online] pp.2444 - 2448. Available at: https://www.researchgate.net/publication/285739464_Bezier_curve_based_smooth_path_planning_for_mobile_robot [Accessed 12 Jan. 2019].
