# 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(_C^{0 }) : **A curve is said to be zero-order parametric continuous if both segments of the curve intersect at one endpoint.

P(t_{2) = Q(t1)}

**(b) First-order parametric continuity(C^{1}) : **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=t

_{2 }is equal to the first-order derivative of segment Q at t=t

_{1}. Such kinds of curves have the same tangent line at the intersection point.

P'(t_{2) = Q'(t1)}

**(c) Second-order parametric continuity(C^{2}) : **A curve is said to be second-order parametric continuous if it is C

^{o}and C

^{1}Continuous and the second-order derivative of the segment P at t=t

_{1}is equal to the second-order derivative of segment Q at t=t

_{2}.

P''(t_{2) = 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(t_{2) = Q(t1)}

** (b) First-order geometric continuity(G^{1}) : **The junction point between two points is said to be G

^{1}continuous if the coordinate of both curve segments is G

^{0}continuous and following the below condition:

P'(t_{2) = k * Q'(t1) for all x, y, z. }

**(c) Second-order geometric continuity(G^{2}) : **The junction point between two points is said to be G

^{2}continuous if the coordinate of both curve segments is G

^{1}continuous and following the below condition:

P''(t_{2) = k * Q''(t1) for all x, y, z.}