Introduction to Surface Continuities and its Types

You might have come across the term, surface continuity quite frequently while working with different concepts in surface modelling of plastic modelling. The term ‘continuity’ is valid only when there is a connection or contact between two surfaces or surface layers. Technically speaking, this is the terminology used in modelling just to define the flow of curves/surfaces. There are different types of surface continuity based on the ways in which two surfaces meet. 

Surface continuity is crucial in Class-A surface modelling. Surface modelling is the art of building complex geometrical features, one surface at a time. Usually, surface modelling is used to create continuous organic shapes, therefore the junctions between surfaces are quite important.

Generally, we describe the 'smoothness' of a junction, where one surface meets another, using the term 'G' (G stands for 'Geometry'). We can add a suffix to our G to describe how many similar characteristics each surface has. The more alike the surfaces (the more conditions that match across the pair of inputs), the more smoothly the adjoining surfaces flow. If two surfaces are not touching, they have no continuity which can be represented as the surfaces having ‘No-G’.

 

What are the Types of Continuity Curves?

There are four types of surface continuities based on the variations in which surfaces establish contact with each other. These include:

1. G0 – Position (touching)
2. G1 – Tangent (angle)
3. G2 – Curvature (radius)
4. G3 – Acceleration/Torsion (rate of change of curvature)
   

 

G0 - Position Continuity :

G0 or Point Continuity is simply when one surface or curve touches another and they share the same boundary. When two curves are having a common endpoint, with no further conditions, and are positionally continuous, a GO surface continuity is created. This is the less smooth type of joint, as the only condition to satisfy is the common endpoint, regardless of the flow of the two curves/surfaces around the joint. In the examples below, you can see what it looks like on both curves and surfaces.

 

  

 

G1 – Tangent Continuity :

G1 or Tangent continuity or Angular continuity implies that two faces/surfaces meet along a common edge and that the tangent plane, at each point along the edge, is equal for both faces/surfaces. They share a common angle; the best example of this is a fillet (Engineering Fillet), a blend with Tangent Continuity, or in some cases a Conic. 

You can't have G1 continuity unless you at least have G0 continuity. The further we move ahead in continuities, keep in mind that the preceding number(s) must exist in order for it to be true. Sometimes, it is also known as the second-order of continuity. In the examples below, you can see what it looks like on both curves and surfaces.

 

  

 

 G2 – Curvature Continuity

G2 continuity adds another order (condition) that must match across our pair of curves. In this type of surface continuity, the curves must touch, be tangent with each other, and have the same curvature, at the point where they touch. Technically, this means that our input geometry must now be curved as well. Straight lines have no curvature (or infinite curvature, depending on your point of view). Therefore, a pair of inputs containing a straight line can't meet G2 conditions.

This is the minimum math requirement for Class A Surface. Another way to describe this is in a situation where a reflection is cast upon the surfaces and you would not be able to tell where one patch ends and the other begins. In the examples below, you can see what a G2 continuity looks like on both curves and surfaces.

 

  

 

G3 – Acceleration of Curvature / Torsion Continuity

G3 curvature continuity includes the concept of 'Acceleration' of curvature. That is the amount of change in the curvature, over the length of the curve. G3 is looking for balance on the rate of curvature – in other words, the max value of the curvature hits its peak about the middle of the transition area.

Let's take a look at our 2D curvature combs shown above. To achieve G3 continuity, the curvature combs should be heading in the same direction (Tangency), be the same length (Curvature), and, more importantly, be tangent (Acceleration, or rate of change). 

 

What is beyond G3? 

Mathematically you can keep on adding more 'orders' across curves to create G4, G5 - Gn continuity, but we are realists, so we are going to stop here. Even if you could create 'smoother' surfaces it would be pretty difficult to measure it.

 

CAD Software and Continuity

CAD platforms are capable of demonstrating continuity types in the form of mathematical equations to define the continuity in the backend. To boost your imagination on these explanations, you can take a look at the below images with different types of continuity.