Letting
B0 = A3 and
B3 = C0 ensures
G0 continuity (location). Choosing
B1 to be collinear with
A2A3 will ensure
G1 continuity (direction) at
B0, and similarly choosing
B2 to be collinear with
C0C1 ensures
G1 continuity at
B3.
Now there remain two degrees of freedom for the placement of
B1 and
B2. I guess that these can be used to achieve
G2 continuity (curvature) as well.
Now a little bit of math. In the following document
http://www.tsplines.com/resources/class_notes/Bezier_curves.pdf[
^], you find an interesting formula on page 22, telling you that the curvature equals
h/a^2 (dropping the constant
n-1/n), where
a is the length of the first leg of the control polygon, and
h is the perpendicular distance from the second control point to the first leg of the control polygon.
We state that
B1 = B0 + a U and
B2 = B3 + a' V, where
U and
V are the unit vectors in the directions of
B0B1 and
B3B2 respectively.
h and
h' are obtained by projecting
B1B2 onto
U and
V:
h = B1B2 /\ U = (B0B3 + a' V) /\ U = p + q a',
h' = B1B2 /\ V = (B0B3 + a U) /\ V = p' - q a. (
/\ is the vector product operator, a 2x2 determinant in 2D).
This can be put together with the known curvatures at endpoints computed from the other two arcs:
h = K a^2 and
h' = K' a'^2.
Elimination of three unknowns gives us a quartic equation in
a, hence 0 to 4 solutions.
p + q a' = K a^2<br />
p' - q a = K' a'^2<br />
<br />
=><br />
<br />
(p' - q a) q^2 = K' (K a^2 - p)^2
:-(
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