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Continuity G0 G1 G2 G3

The continuity settings G0, G1, G2, G3 specify the smoothness between adjoining curves or surface patches.

In the curve example below, the Align tool is being used to modify the red curve to match the blue curve with G0, G1, G2, or G3 continuity.

  • Curvature combs - Illustrate the smoothness of the connection between the curves for each continuity level.
  • CV Placement - Shows the number of CVs that need to be positioned and constrained for each level of continuity.
  • Criteria - The technical definitions and requirements for each continuity level:
    • G0 Position - The curves share the same point at the junction.
    • G1 Tangent - As above plus the end tangent directions are matched at the junction.
    • G2 Curvature - As above plus the radius/curvature values are matched at the junction.
    • G3 Curvature - As above plus the rate of change of the radius/curvature values are matched.

Tools

Many surface and curve creation tools have options to set the required continuity with input geometry, such as the Curve Fillet and Surface Fillet tools.

To create a continuity relationship on existing geometry:

To measure continuity relationships numerically:

To measure continuity relationships visually:

To set the continuity tolerances:

Continuity and construction tolerances

When geometry is being created, any requested continuity relationship is calculated to within tolerances set in the Construction Options.

  • G0 Positional continuity - This is important when building closed volumes for manufacture and data export.

  • G1 Tangent continuity - The tangent angle is matched between adjoining geometry.

  • G2 Curvature continuity - The radius values of the geometry are matched at the join.

    The curvature deviation at a boundary is (ignoring negative values): dev = (R1 - R2) / (R1 + R2).

    This is used for calculations that the software makes and cannot be usefully interpreted by the user. Visual evaluation methods are typically used instead for design decisions.

Surface continuity: Colinear and non-colinear alignment

When aligning to a surface, there are two situations that can arise: colinear and non-colinear.

Colinear alignment

Colinear alignment is possible between two surfaces when:

  • They meet at a natural boundary (not a trim edge or curve-on-surface).
  • And, have the same degree and spans in the direction of the connected edge.

Continuity can then be achieved without increasing complexity by aligning Cvs, using one of the following:

  • Align tool with Edge Alignment type.
  • Co-linear options in Rail or Square tools.
  • Edge Align options in Freeform Blend.
  • Manually placing CVs using Transform CV tool options.

Non-Colinear alignment

Non-colinear situations include:

  • Aligning to the interior of a surface.
  • Aligning to a trimmed edge.

Achieving continuity typically produces more complex surfaces with more spans and/or uneven spacing.

Deep dive: G3 and G4 continuity

In practice, the higher levels of continuity are often achieved "by eye" using modeling skills and evaluation tools. There are however, some tools that build to higher continuity levels.

  • G3 is available in Alias Autostudio and Alias Surface in most curve and surface tools that offer continuity (excluding Rail and Square). G3 matches the rate of change of curvature at the boundary.

  • G4 is only available in the Blend Curve tools. G4 matches the same rate of change of curvature at the boundary as G3, but additionally the acceleration of the curvature must be equal in three dimensions.

To understand the criteria for G3 and G4, it is useful to look at the red outline curves on the curvature combs and the continuity between them. This will always be two levels lower than the continuity achieved on the actual curves.

Deep dive: G1 and C1 - Geometric and analytic Continuity

Alias uses Geometrical continuity (G0, G1, G2) rather than Analytical continuity (C0, C1, C2) to allow more modelling flexibility while maintaining good results.

  • Geometric continuity G1 - The tangent directions are matched, but not the tangent length.
  • Analytic continuity C1 - Both the tangent directions and the tangent length are matched.

C1 Tangent continuity is measured by some CAD systems but is not created automatically by Alias modeling tools.

g1 c1 image

Deep dive: Dev=1.0 (Flat) continuity

When measuring G2 curvature continuity at the boundary of two surfaces, a result of dev=1.0 is sometimes shown. This was displayed as "flat" in earlier versions of Alias and only occurs when one of the surfaces is flat in one direction.

The result of dev=1.0 is an exception rather than an error and may not indicate a visual problem. Apply visual evaluation tools to verify the results. In the example below, applying visual evaluation tools shows that the surfaces are smooth. If however there is design requirement to avoid any flat areas, then the dev=1.0 provides useful feedback.

Understanding 'Flat' curvature calculations

The calculation for G2 Curvature deviation at a boundary is (ignoring negative values): dev = (R1 - R2) / (R1 + R2).

An internal Infinity Tolerance is set to 100000 cm by default. You can modify this value by changing the environment variable : ALIAS_G2_INFINITY_TOL.

In the following example:

  1. R1 and R2 are both infinity (flat). The calculation delivers a deviation of zero.
  2. R1 is not infinity, but it is larger than the infinity tolerance so is treated as infinity in the calculation, hence deviation is zero.
  3. R1 is a value less than the infinity tolerance. The deviation calculation then becomes: value-infinity/value+infinity, which is infinity/infinity, which is 1.0.

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