Understanding curve and surface continuity

What is curve and surface continuity?

Continuity describes how two curves or surfaces connect at a shared point or boundary. It helps you understand whether geometry simply meets, transitions without a sharp change in direction, or blends with a gradual change in curvature.

In Shapr3D, continuity matters when preparing sketches and curves and when creating or refining geometry with tools such as Fill, Loft, Sweep, and Fillet. It also helps when creating curves that will become inputs to other modeling operations.

There are three commonly used levels of geometric continuity:

• G0 (Positional) – Geometry meets at a shared position.
• G1 (Tangent) – Geometry meets at a shared position with matching tangent direction.
• G2 (Curvature) – Geometry meets at a shared position with matching tangent direction and curvature.

Each higher level includes the requirements of the levels below it. For example, a G2 connection also satisfies G1 and G0 continuity.

Understanding G0, G1, and G2 continuity

G0 (Positional) continuity

G0 continuity connects geometry at a shared position without requiring matching directions.

• Curves: Meet at the same position, but their tangent directions do not need to match.
• Surfaces: Share a boundary but may form a visible corner or sharp edge.
• Example: Two sketch lines meet at a corner. In 3D, two faces share an edge while forming a sharp angle.
• When to use: For intentional corners, hard edges, or connections where a smooth transition is not required.

G1 (Tangent) continuity

G1 continuity creates a smooth transition by matching the direction of connected geometry.

• Curves: Meet at the same position with matching tangent directions, removing abrupt changes in direction.
• Surfaces: Share a boundary with matching tangent planes, creating a transition without a sharp crease. Curvature may still change abruptly.
• Example: A straight line connects tangentially to an arc. In 3D, a rounded transition meets a neighboring face without a visible hard edge.
• When to use: For smooth-looking transitions where matching curvature is not necessary.

G2 (Curvature) continuity

G2 continuity creates a more gradual transition by matching tangent directions and curvature at the connection. Curvature describes how quickly a curve changes direction. A straight line has zero curvature, while a circle has constant curvature.

• Curves: Meet at the same position with matching tangent directions and curvature.
• Surfaces: Share a boundary with matching tangent planes and curvature, creating a more gradual transition and often a smoother flow of reflections.
• Example: Two spline segments meet with matching tangent directions and curvature. In 3D, a curvature-continuous blend on a product housing creates a more gradual transition than a tangent-only blend.
• When to use: For prominent exterior surfaces where smooth transitions, highlights, and reflections are especially important.

Note: G2 is not always the best choice. G0 or G1 may be more appropriate for intentional edges, mechanical features, or simpler transitions.

Curve continuity

Curve continuity describes the connection between line segments, arcs, splines, or other curves. It applies to curves in a 2D sketch and to curves positioned in 3D space.

Continuity between sketch curves

When connecting sketch lines, arcs, or splines, use sketch constraints to control how the curves meet.

• Coincident: Makes supported sketch entities meet at a shared position, establishing the positional connection needed for G0.
• Tangent: Aligns the tangent directions of supported sketch entities, helping establish G1 continuity.

Note: Sketch constraints help control curve relationships, but the resulting continuity depends on the geometry and constraints involved.

Continuity within and between splines

A spline can change direction and curvature along its length. Even if a spline appears smooth, it may not form a curvature-continuous connection with another spline, arc, or line.

When two spline segments meet, consider their continuity at the shared endpoint:

• G1 (Tangent): The tangent directions must match at the connection.
• G2 (Curvature): Both the tangent directions and curvature must match at the connection.

Control points and handles influence the spline's shape, but visual alignment alone does not guarantee the required continuity.

Example: A spline can connect tangentially to a straight line and achieve G1 continuity. To achieve G2, the spline's curvature at the connection must also match the line's zero curvature.

Continuity in 3D curve networks

Curves used to define 3D geometry may lie on different planes or follow different paths. Two curves that appear connected in one view may actually be separated in 3D space.

When preparing a curve network, consider the following:

• Connections: Check that the intended endpoints and intersections meet in 3D space, not just from a particular viewing angle.
• Direction: Check for sharp changes in direction where curves meet, especially when a smooth transition is required.
• Curvature: Even when curves connect tangentially, differences in curvature can affect the resulting geometry.

Example: When using Sweep, the continuity of the path affects how the geometry changes direction. Sharp corners may create abrupt transitions, while tangent-connected curves can produce smoother results. Changes in curvature can still affect the resulting shape, even with tangent continuity.

Surface continuity

Surface continuity describes how neighboring surfaces meet along a shared boundary. It is related to curve continuity, but it is not the same: surfaces have tangent planes and curvature in multiple directions, not just one tangent direction along a curve.

• G0 surfaces share a boundary, but can form a sharp crease.
• G1 surfaces share a boundary and have matching tangent planes along it, removing the sharp crease.
• G2 surfaces share a boundary with matching tangent behavior and compatible curvature across it, producing a more gradual blend.

Note: Two surfaces can appear smooth at a distance but still show a change in reflections at their connection. Lighting, reflections, and inspection views can make differences between G1 and G2 easier to see.

[Illustration: Three adjacent-surface pairs with G0, G1, and G2 connections. Add reflection or zebra-style stripes crossing the shared boundary to reveal differences.]

How to inspect surface transitions

• Look for sharp edges: A visible crease at a shared boundary indicates that the surfaces may only meet positionally.
• Observe highlights and reflections: An abrupt change in the flow of highlights can reveal a change in curvature, even when there is no sharp edge.
• Inspect from several angles: A transition may appear smooth from one viewpoint but show irregularities from another.

Note: Visual inspection is useful for assessing the result, but it is not a precise measurement of geometric continuity.

How input geometry affects continuity

The continuity of the resulting geometry depends not only on the continuity setting you request, but also on the geometry used to construct it. Profiles, guide curves, surrounding faces, and their connections can constrain the transitions that are possible.

Input curve connections

The continuity of input curves can affect the smoothness of the resulting surface. When preparing curves for surface creation, consider the following:

• Sharp corners (G0): Curves that meet at a corner may introduce abrupt changes in direction or prevent the surface from achieving higher continuity at that location.
• Tangent connections (G1): Curves with matching tangent directions may still have different curvature, which can affect the smoothness of the resulting surface.
• Compatibility with boundaries: Guide curves that meet a surface boundary at an incompatible angle may prevent the intended smooth transition.

Practical example: If two guide-curve segments meet at a sharp corner, the resulting surface may develop a ridge or fail to produce the expected smooth blend. Refining the curve connection before creating the surface can improve the result.

Surrounding surface geometry

When creating a surface that blends into existing faces, the surrounding geometry affects which continuity levels are achievable.

Consider the following:

• G0 (Positional): Matching the position of surrounding boundaries is the least restrictive requirement.
• G1 (Tangent): Matching tangent planes introduces additional constraints on the surface.
• G2 (Curvature): Matching curvature introduces further constraints and may be more difficult to achieve.
• Geometric limitations: Sharp corners, rapidly changing curvature, incompatible boundary conditions, or insufficient space can make higher continuity difficult or impossible.

Practical example: A surface patch may connect to several surrounding faces at G0, but requesting G2 along every boundary can introduce conflicting requirements. Adjusting individual boundaries, refining input curves, or simplifying the curve network may help achieve a smoother transition.

Why smooth inputs do not guarantee smooth surfaces

Smooth-looking input curves do not necessarily produce a smooth surface. The result depends on the compatibility of the entire curve network and its surrounding geometry.

• Individual curves: May appear smooth but have incompatible connections.
• Combined inputs: May impose conflicting requirements on the resulting surface.
• Resulting surface: May still appear smooth even when the inputs do not have matching continuity everywhere.

Note: Evaluate the complete curve network and resulting surface rather than relying on individual curves or a single continuity setting.

When to use each continuity level

Best practices for smooth transitions

• Choose the continuity level for the design: Use G0 for intentional edges, G1 for tangent blends, and G2 when a more gradual curvature transition is needed.
• Check that inputs connect in 3D: Confirm that curve endpoints and intersections meet rather than only appearing to touch in a particular view.
• Inspect curve joins: Check for unexpected corners, tangent mismatches, or abrupt curvature changes before using curves to construct a surface.
• Keep input networks manageable: Start with the curves and profiles needed to define the shape. Add more control only where it improves the result.
• Increase continuity selectively: If a higher continuity setting fails or distorts the result, try applying it to fewer boundaries or refining the surrounding geometry.
• Inspect the final surface: Look at the transition from multiple angles and use highlights or reflections to identify unexpected changes in shape.

Continuity in Shapr3D tools

Continuity principles apply across several modeling workflows, although the available controls and input requirements differ by tool.

• Fill – Uses boundary profiles, guide curves, or connected curve networks to create geometry. Available continuity settings depend on the selected inputs and surrounding faces.
• Loft – Creates geometry between cross-sectional profiles. The arrangement and continuity of profiles and guide curves influence the resulting transition.
• Sweep – Creates geometry by sweeping a profile along a path. The continuity of the path influences how smoothly the resulting geometry changes direction. Sharp corners or abrupt changes in curvature along the path can affect the resulting surface.
• Fillet – Rounds edges between faces. The selected blend and surrounding surfaces affect the transition into adjacent geometry.