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 C^o 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 C^o 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(G^o) : 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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