the relationship between C (parametric) and G (geometric) continuity between curves
• C0/G0 are equivalent
• G2 continuity is also G1 and G0 continuous
• C2 continuity is also G2, C1, G1 and G0 continuous
• C0/G0 are equivalent
• G2 continuity is also G1 and G0 continuous
• C2 continuity is also G2, C1, G1 and G0 continuous
it's a little difficult to wrap your head around, but this is how they relate to the derivatives (top)
notably, G-continuity has no restriction on parametric continuity.
G2 *can* be C1 too, but it only has to be the lesser strict G1
G3 *can* be C2 & C1, but has to be G2 & G1
notably, G-continuity has no restriction on parametric continuity.
G2 *can* be C1 too, but it only has to be the lesser strict G1
G3 *can* be C2 & C1, but has to be G2 & G1
a general definition of Gn continuity:
the join between two curves A(t) and B(t), is Gn continuous, if a function g(t) exists, so that A(t) and B(g(t)) is Cn continuous
the join between two curves A(t) and B(t), is Gn continuous, if a function g(t) exists, so that A(t) and B(g(t)) is Cn continuous
for G1 continuity, the first derivative has matching directions, but not necessarily magnitude
for example, here's a G1 continuous bézier spline. Note how the first derivative is discontinuous (in other words, not C1), but the directions/tangents match
for example, here's a G1 continuous bézier spline. Note how the first derivative is discontinuous (in other words, not C1), but the directions/tangents match
this relationship, however, doesn't generalize to higher G-continuities - it's not as simple as "matching directions but not magnitude" as it is for G1 in particular
this is a G3 continuous join, with the first two derivatives on the right
this is a G3 continuous join, with the first two derivatives on the right
• • •
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