In CATIA, A, B, and C surfaces are reference planes that are used as the basis for defining geometry in 3D space. These surfaces are typically used in conjunction with sketches and other 3D features to create complex parts and assemblies.

​​​​​​​Here's a brief overview of each surface:

Class A surfaces are the highest quality surfaces, used in applications where visual appearance and aerodynamics are critical, such as automotive exteriors, consumer products, and aerospace. They are continuous, curvature continuous, and have no defects or visible seams;

Class B surfaces are of lower quality than Class A surfaces, but still of high quality. They are used in applications where visual appearance is important, but not critical, such as internal components of automobiles or consumer products. They may have small defects or visible seams but still maintain a smooth appearance;

Class C surfaces are the lowest quality surfaces, used in applications where appearance is not critical, such as structural components or parts that will not be visible. They may have visible defects, seams, or discontinuities.

On the mathematical side, they are designed based on C0/C1/C2/C3 and G0/G1/G2/G3 tangency and curvature continuity, usually based on NURBs curves and NURBs surfaces in CATIA.

For designing, with the interface, we can use control points and geometrical elements to control the expected shapes.

C0/C1/C2/C3 continuity refers to the smoothness and seamless connection between adjacent surfaces or curves. The higher the continuity, the smoother the transition;

G0/G1/G2/G3 continuity refers to geometric continuity. G0 continuity means tangency, G1 means curvature continuity, G2 means continuous curvature, and G3 means continuous curvature and torsion;

The difference between G0/G1/G2/G3 continuity and C0/C1/C2/C3 continuity is that G continuity measures the smoothness of the curve or surface geometry, while C continuity measures the smoothness of the curve or surface and its derivatives (slope and curvature). G continuity only considers position, while C continuity considers both position and shape.

Overall, the A, B, and C surfaces are critical components of CATIA's modeling environment. By providing a reliable and consistent reference system, these surfaces help designers and engineers create accurate and precise 3D models.