Parametric & Geometric Continuity of Curves in Computer Graphics

In this video, we will be covering what is Parametric & Geometric Continuity of Curves in Computer Graphics.

Parametric continuity is the smooth transition of one curve to another curve while representing continuity condition. Parametric continuities are of 3 types, which are as follows:
1. Zero-Order Parametric Continuity (C⁰)
2. First-Order Parametric Continuity (C¹)
3. Second-Order Parametric Continuity (C²)

1. Zero-Order Parametric Continuity (C⁰): The curve that shares the same point where they are joined (Point 0) is called Zero-order parametric else if both segments of the curve intersect at one endpoint they are known as zero-order parametric continuous.

P(t2) =  Q(t1)

2. First-Order Parametric Continuity (C¹): This is an addition to C⁰ continuity, where the first derivative or tangents at the point of intersection of both curves are the same in magnitude and direction which means, the first order derivative of segment P at t=t2 is equal to first order derivative of segment Q at t = t1. 

P’ (t2) = Q’ (t1)

3. Second-Order Parametric Continuity (C²): This is an addition to C⁰ and C¹ continuity, where the second derivatives of both curves are the same at the point of intersection, which means the second order derivatives of segment P at t=t1 are equal to second order derivative of segment Q at t = t2.

P‘’ (t2) = Q‘’ (t1)

Geometric Continuity: This is an alternate method for joining two curve segments. It uses only the shape of the curve. It is usually defined in terms of parametrizations, but the choice of parametrization does not affect the outcome. So a junction between two curves is said to be G⁰-continuous if the (x, y, z) values of the two curves agree on the axis. 

There are 3 kinds of Geometric Continuities:

1. Zero-Order  Geometric Continuity (G⁰): It is the same as Zero-order parametric curve continuity condition.

                  P(t2) =  Q(t1)

2. 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 follows the below condition:

            P'(t2) =  k * Q'(t1)  for all x, y, z and k > 0.

3. 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 follows the below condition:

             P‘’ (t2) =  k * Q‘’ (t1)  for all x, y, z and k > 0.
 
Parametric & Geometric Continuity of Curves in Computer Graphics: https://www.geeksforgeeks.org/parametric-geometric-continuity-of-curves-in-computer-graphics/