Geometry and Topology

NOTE

Target Audience: Advanced Implementers & Geometry Kernel Developers
Prerequisites: Please read the Glossary and Data Model Map first.

This page provides a deep dive into the mathematical and structural representation of geometry in STEP, covering NURBS, complex topology, and geometric continuity.

1. Foundations of NURBS

NURBS (Non-Uniform Rational B-Splines) are the mathematical backbone of STEP B-rep.

Relationships: Bézier, B-Spline, and NURBS

Think of these as a hierarchy of increasing control and mathematical complexity:

Curve Type	Hierarchy	Key Characteristics
Bézier Curve	Base	The simplest form. A single polynomial segment. Moving one control point affects the entire curve.
B-Spline	Extension	A sequence of Bézier curves joined together. Moving a control point only affects a local segment (Local Control).
NURBS	Final Form	Adds Weights (Rational) and Non-Uniform knots. Can perfectly represent circles and ellipses, which regular B-splines cannot.

The "NURBS" Name Breakdown:

Non-Uniform: Knots don't have to be equally spaced (allows for varying "speed" along the parameter $u$ ).
Rational: Uses Weights. This is required to define exact conic sections (circles, ellipses, hyperbolas).
B-Spline: Basis-Spline. A method of joining multiple polynomial segments with a specific level of continuity.

Note on T-Splines: Standard STEP (ISO 10303) does not natively support T-Splines as a distinct mathematical entity. When CAD systems (like Fusion 360 or Rhino) export T-Spline models to STEP, they are automatically converted into a collection of standard NURBS patches. This ensures compatibility with all STEP-compliant software but may result in a higher number of faces.

Key Components

Term	Meaning	Impact on Shape
Degree (次数)	The degree of the polynomial (e.g., 3 for cubic splines).	Higher degrees allow for smoother curves but increase calculation cost.
Knot Vector (ノットベクトル)	A sequence of parameter values that determine where the spline segments meet.	Multiple knots at the same position (multiplicity) create "sharp" corners or discontinuities.
Control Points	Points that "pull" the curve toward them.	Moving a control point locally affects the shape.
Weights (重み)	Used in Rational B-splines.	Allows exact representation of conic sections (circles, ellipses, hyperbolas).

Mathematical Definition

NURBS can represent both curves and surfaces.

1. NURBS Curve $C (u)$

A NURBS curve is defined as:

C (u) = \frac{\sum_{i = 0}^{n} N_{i, p} (u) w_{i} P_{i}}{\sum_{i = 0}^{n} N_{i, p} (u) w_{i}}

(In plain text: Sum of (Basis * Weight * ControlPoint) / Sum of (Basis * Weight))

2. NURBS Surface $S (u, v)$

A NURBS surface is defined as the bivariate extension:

S (u, v) = \frac{\sum_{i = 0}^{n} \sum_{j = 0}^{m} N_{i, p} (u) N_{j, q} (v) w_{i, j} P_{i, j}}{\sum_{i = 0}^{n} \sum_{j = 0}^{m} N_{i, p} (u) N_{j, q} (v) w_{i, j}}

(In plain text: Sum of (Basis_u * Basis_v * Weight * ControlPoint) / Sum of (Basis_u * Basis_v * Weight))

Interactive NURBS Playground

Use the viewer below to experiment with NURBS parameters. Switch between Curve and Surface modes to see how Weights (Rational) and Knots (Non-Uniform) influence the geometry.

Where:

$P$ : Control Points (grid for surfaces).
$w$ : Weights.
$N$ : B-spline Basis Functions of degree $p$ (and $q$ for surfaces), defined by the Cox-de Boor recursion formula.
$u, v$ : Parameters.

Basis Function Recursion (Cox-de Boor)

The basis functions $N_{i, p} (u)$ are defined recursively:

Degree 0: $N_{i, 0} (u) = 1$ if $u_{i} \leq u < u_{i + 1}$ , else $0$ .
Degree $p$ : $N_{i, p} (u) = \frac{u - u_{i}}{u_{i + p} - u_{i}} N_{i, p - 1} (u) + \frac{u_{i + p + 1} - u}{u_{i + p + 1} - u_{i + 1}} N_{i + 1, p - 1} (u)$

Matrix Representation (Implementation Perspective)

In computer graphics and CAD kernels, B-splines can be evaluated using matrix forms. For a single segment:

C (u) = U M P

Where:

$U$ : The parameter vector $[u^{p}, u^{p - 1}, \dots, u, 1]$ .
$M$ : The basis matrix (determined by the degree and knot vector).
$P$ : The vector of control points.

Note: For NURBS, the same operation is performed on weighted control points in homogeneous coordinates $(w_{i} x_{i}, w_{i} y_{i}, w_{i} z_{i}, w_{i})$ , then projected back.

How is it uniquely determined?

A NURBS curve is uniquely defined by four inputs:

Degree ( $p$ ): Determines the polynomial complexity.
Knot Vector ( $U$ ): A non-decreasing sequence of numbers (e.g., ${0, 0, 0, 1, 2, 3, 3, 3}$ ). It dictates how the control points influence the curve along the parameter $u$ .
Control Points ( $P_{i}$ ): The skeleton of the curve.
Weights ( $w_{i}$ ): The "strength" of each control point. If all weights are 1.0, the curve is a non-rational B-spline.

Uniqueness Note: In STEP, these are explicitly stored in entities like B_SPLINE_CURVE_WITH_KNOTS. A NURBS curve is only "determined" within the range of its knot vector (typically $[u_{m i n}, u_{m a x}]$ ).

Parameter Space: 2D vs. 3D

STEP uses two ways to define curves on surfaces:

Space Curve (3D): Defined directly in 3D XYZ space (e.g., B_SPLINE_CURVE_WITH_KNOTS).
PCurve (2D): Defined in the 2D UV parameter space of a surface (e.g., PCURVE).

Why both?
Robust geometry kernels use PCURVE to ensure that the boundary of a face (the trim line) lies exactly on the surface, preventing "gaps" or "leaks" in the model during import/export.

2. Topology and B-rep Types

STEP strictly separates Topology (connectivity) from Geometry (mathematical shape).

Geometry vs. Topology Separation

Topology: Defines the "skeleton" of the model (who is next to whom).
Geometry: Defines the "meat" of the model (the exact coordinates and formulas).

Closed vs. Open Shells

MANIFOLD_SOLID_BREP: Represents a "water-tight" solid. It must have a volume.
SHELL_BASED_SURFACE_MODEL: Represents surfaces with no volume (e.g., a thin sheet of metal).

Voids and Hollow Solids

For solids with internal cavities (voids), STEP uses ORIENTED_CLOSED_SHELL.

Outer Bound: The exterior shell (oriented "out").
Voids: One or more internal closed shells (oriented "in").

3. Geometric Entities & Primitives

Primitives (CSG)

While STEP is primarily B-rep, it supports "Primitives" which can be used in CSG (Constructive Solid Geometry) operations or as B-rep underlying geometry:

BLOCK: A rectangular box.
CYLINDER: A cylinder.
SPHERE: A sphere.
TORUS: A doughnut shape.

Complex Surfaces & Fillets

CAD kernels prioritize using the simplest mathematical definition (Elementary Surfaces) to keep files lightweight and precise before falling back to NURBS.

Elementary Surfaces for Fillets:
- CYLINDRICAL_SURFACE: Often used for constant-radius fillets along straight edges.
- TOROIDAL_SURFACE / SPHERICAL_SURFACE: Used for fillets at corners or along circular edges.
NURBS (B_SPLINE_SURFACE): Used for Variable Radius Fillets or complex "blends" where multiple surfaces meet.
Note: Many CAD systems offer an "Export all surfaces as NURBS" option. While this increases file size, it is sometimes used to improve compatibility between systems that might struggle to interpret complex elementary surface intersections.
Offset Surface: A surface defined at a constant distance from a base surface (OFFSET_SURFACE).

4. Continuity and Quality

Geometric Continuity (G0, G1, G2)

When two surfaces or curves meet, their "smoothness" is defined by continuity.

Type	Name	Visual Result
G0	Positional	The elements touch, but there is a visible "crease".
G1	Tangential	The transition is smooth (no crease), but the reflection might "jump".
G2	Curvature	The transition is perfectly smooth; reflections flow across the boundary.

STEP Implementation: Continuity is often explicitly declared in NURBS entities (e.g., continuity: geometric_continuity_2).

5. Physical Properties (Mass Properties)

STEP can store calculated physical data so that the receiving system can verify the integrity of the geometry.

Volume: Stored in GEOMETRIC_ITEM_SPECIFIC_USAGE linked to a VOLUME_UNIT.
Mass / Center of Gravity: Often stored under PROPERTY_DEFINITION entities.
Validation Properties: Special entities like VALUATION_PROPERTY are used to store "check values" for area and volume. If the imported model's volume differs from this value, the import is considered failed or "lossy".

💡 Implementation Tips

Check for Water-tightness: If you are importing a MANIFOLD_SOLID_BREP, verify that every EDGE_CURVE is shared by exactly two ADVANCED_FACEs (manifold condition).
Handle Degree 1 Splines: Often, CAD systems export polylines as degree 1 NURBS.
Torus Inversion: Pay attention to the axis and radius attributes of TOROIDAL_SURFACE. If the minor radius is larger than the major radius, it becomes a self-intersecting "spindle torus".

📚 Next Steps

Common Pitfalls - Learn about tolerance issues.
Validation and CAx-IF - How to use validation properties.

🔗 Further Reading & References

The NURBS Book (Les Piegl and Wayne Tiller): The "Bible" of NURBS. Essential for anyone implementing geometry kernels.
NURBS-Python (geomdl) Documentation: Excellent practical examples and visual explanations of NURBS math.
CAx-IF Recommended Practices: Official guidelines on how to implement STEP geometry to ensure interoperability between CAD systems.
ISO 10303-42: The official part of the STEP standard that defines "Geometric and topological representation."

Back to README

Geometry and Topology ​

1. Foundations of NURBS ​

Relationships: Bézier, B-Spline, and NURBS ​

Key Components ​

Mathematical Definition ​

1. NURBS Curve C(u) ​

2. NURBS Surface S(u,v) ​

Interactive NURBS Playground ​

Basis Function Recursion (Cox-de Boor) ​

Matrix Representation (Implementation Perspective) ​

How is it uniquely determined? ​

Parameter Space: 2D vs. 3D ​

2. Topology and B-rep Types ​

Geometry vs. Topology Separation ​

Closed vs. Open Shells ​

Voids and Hollow Solids ​

3. Geometric Entities & Primitives ​

Primitives (CSG) ​

Complex Surfaces & Fillets ​

4. Continuity and Quality ​

Geometric Continuity (G0, G1, G2) ​

5. Physical Properties (Mass Properties) ​

💡 Implementation Tips ​

📚 Next Steps ​

🔗 Further Reading & References ​