Parametric & Geometric Continuity of Curves in Computer Graphics
Prerequisite: B-spline curve, Bezier curve
The continuity condition represents that how smoothly a curve transit from one curve segment to another segment. Consider you are given a curve as shown below:
There are three kinds of Parametric continuities that exist:
(a) Zero-order parametric continuity(_C0 ) : A curve is said to be zero-order parametric continuous if both segments of the curve intersect at one endpoint.
P(t2) = Q(t1)
(b) First-order parametric continuity(C1) : A curve is said to be first-order parametric continuous if it is Co Continuous and the first-order derivative of segment P at t=t2 is equal to the first-order derivative of segment Q at t=t1. Such kinds of curves have the same tangent line at the intersection point.
P'(t2) = Q'(t1)
(c) Second-order parametric continuity(C2) : A curve is said to be second-order parametric continuous if it is Co and C1 Continuous and the second-order derivative of the segment P at t=t1 is equal to the second-order derivative of segment Q at t=t2.
P''(t2) = Q''(t1)
Geometric Continuity : It is an alternate method for joining two curve segments, where it requires the parametric derivation of both segments which are proportional to each other rather than equal to each other.
(a) Zero-order geometric continuity(Go) : It is similar to the zero-order parametric curve continuity condition.
P(t2) = Q(t1)
(b) First-order geometric continuity(G1) : The junction point between two points is said to be G1 continuous if the coordinate of both curve segments is G0 continuous and following the below condition:
P'(t2) = k * Q'(t1) for all x, y, z and k > 0.
(c) Second-order geometric continuity(G2) : The junction point between two points is said to be G2 continuous if the coordinate of both curve segments is G1 continuous and following the below condition:
P''(t2) = k * Q''(t1) for all x, y, z and k > 0.
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