Editing Composite Bézier curve (section)

=== Smoothly joining cubic Béziers ===
Given two cubic Bézier curves with control points <math>[\mathbf  P_0,\mathbf  P_1,\mathbf  P_2,\mathbf  P_3]</math> and <math>[\mathbf  P_3,\mathbf  P_4,\mathbf  P_5,\mathbf  P_6]</math> respectively, the constraints for ensuring continuity at <math>\mathbf  P_3</math> can be defined as follows:

* <math>C^0/G^0</math> (positional continuity) requires that they meet at the same point, which all Bézier splines do by definition. In this example, the shared point is <math>\mathbf  P_3</math>

* <math>C^1</math> (velocity continuity) requires the neighboring control points around the join to be mirrors of each other. In other words, they must follow the constraint of <math>\mathbf  P_4=2\mathbf  P_3-\mathbf  P_2</math>

* <math>G^1</math> (tangent continuity) requires the neighboring control points to be [[collinearity|collinear]] with the join. This is less strict than <math>C^1</math> continuity, leaving an extra degree of freedom which can be parameterized using a scalar <math>\beta_1</math>. The constraint can then be expressed by <math>\mathbf  P_4=\mathbf  P_3+(\mathbf  P_3-\mathbf  P_2)\beta_1</math>

While the following continuity constraints are possible, they are rarely used with cubic Bézier splines, as other splines like the [[B-spline]] or the [[Beta-spline|β-spline]]<ref>
{{Cite journal|title=Properties of β-splines|last=Goodman|first=T.N.T|journal=Journal of Approximation Theory |language=en|date=1983-12-09|volume=44 |issue=2 |pages=132–153 |doi=10.1016/0021-9045(85)90076-0 |doi-access=free}}</ref> will naturally handle higher constraints without loss of local control.

* <math>C^2</math> (acceleration continuity) is constrained by <math>\mathbf  P_5 =\mathbf  P_1+4(\mathbf  P_3-\mathbf  P_2)</math>. However, applying this constraint across an entire cubic Bézier spline will cause a cascading loss of local control over the tangent points. The curve will still pass through every third point in the spline, but control over its shape will be lost. In order to achieve <math>C^2</math> continuity using cubic curves, it's recommended to use a cubic uniform B-spline instead, as it ensures <math>C^2</math> continuity without loss of local control, at the expense of no longer being guaranteed to pass through specific points

* <math>G^2</math> (curvature continuity) is constrained by <math>\mathbf  P_5=\mathbf  P_3+(\mathbf  P_3-\mathbf  P_2)(2\beta_1+\beta_1^2+\beta_2/2)+(\mathbf  P_1-\mathbf  P_2)\beta_1^2</math>, leaving two degrees of freedom compared to <math>C^2</math>, in the form of two scalars <math>\beta_1</math> and <math>\beta_2</math>. Higher degrees of geometric continuity is possible, though they get increasingly complex<ref>
{{Cite web|url=https://www2.eecs.berkeley.edu/Pubs/TechRpts/1986/6081.html|title=Geometric Continuity: A Parametrization Independent Measure of Continuity for Computer Aided Geometric Design|last=DeRose|first=Anthony D.|language=en|date=1985-08-01}}
</ref>

* <math>C^3</math> (jolt continuity) is constrained by <math>\mathbf  P_6=\mathbf  P_3+(\mathbf  P_3-\mathbf  P_0)+6(\mathbf  P_1-\mathbf  P_2+\mathbf  P_3-\mathbf  P_2)</math>. Applying this constraint to the cubic Bézier spline will cause a complete loss of local control, as the entire spline is now fully constrained and defined by the first curve's control points. In fact, it is arguably no longer a spline, as its shape is now equivalent to extrapolating the first curve indefinitely, making it not only <math>C^3</math> continuous, but <math>C^\infin</math>, as joins between separate curves no longer exist