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B-Spline

In one sentence: Draw smooth, free-form curves using B-spline mathematics — either by placing control points that pull the curve like magnets, or by specifying interpolation points that the curve passes through exactly.

Overview

Workbench: Sketcher · Menu: Sketch → Sketcher geometries → B-Spline
Toolbar: Sketcher Geometries (B-Spline dropdown) · Shortcut: G, B, B (default: B-Spline by Control Points)

Four tools create B-spline curves, organised along two axes:

Open (ends free) Periodic (closed loop)
By control points B-Spline Periodic B-Spline
By interpolation points B-Spline by Interpolation Periodic B-Spline by Interpolation
  • Control points pull the curve toward them but the curve does not pass through them (except at the endpoints for a non-uniform spline).
  • Interpolation points are points the curve passes through exactly.
  • Open splines have a distinct start and end; Periodic splines close smoothly back on themselves.

Intuition

Control points vs interpolation

Think of a control-point B-Spline like a flexible ruler bent by a series of magnets: each magnet pulls the ruler toward it but the ruler never quite touches the magnet (except at the ends). Moving a magnet changes the curve globally — all nearby points shift.

An interpolation B-Spline is more like threading a wire through a series of eyelets: the wire is forced through every eyelet exactly, but it curves smoothly between them. If you know where the curve must pass, use interpolation; if you want to sculpt the shape by feel, use control points.

Open vs periodic

An open B-Spline has a start and end point: it can connect to other geometry (lines, arcs) at those endpoints. A periodic B-Spline forms a seamless closed loop with no start or end — ideal for completely enclosed profiles like organic outlines or cam lobe shapes.

When to use it

  • You need a smooth, organic curve that cannot be described by a circle or ellipse (e.g. an airfoil section, a cam profile, a decorative scroll).
  • You are fitting a curve through a set of known data points (interpolation).
  • You need a smooth closed outline for a body whose profile is not a standard shape (periodic B-Spline).

When NOT to use it

  • Simple arcs and lines suffice — B-Splines are harder to constrain and slower for the solver. Prefer arcs for round shapes and lines for straight ones.
  • You need an ellipse — use Create Circle / Ellipse; it is exact and much easier to constrain.
  • The curve must meet another edge tangentially — possible with B-Splines but fiddly; for smooth joins between an arc and a line, use Fillet or Tangent on simpler geometry first.

Step-by-step walkthrough

B-Spline by Control Points (open)

  1. Press G, B, B or select B-Spline from the dropdown.
  2. Click to place control points one by one. The curve updates in real time, pulled toward each control point.
  3. Right-click or press M to finish placing points and exit the tool. The spline is open (start and end are distinct).
  4. Constrain the spline:
    • Fix endpoint positions with Coincident or Lock Position.
    • Constrain the control-point positions (the weight circles) to fix the shape.
    • Use Tangent at endpoints to ensure smooth joins with adjacent geometry.

Periodic B-Spline (closed loop) by Control Points

  1. Select Periodic B-Spline from the dropdown.
  2. Click control points. The curve closes automatically after three or more points.
  3. Right-click to finish. There is no start or end — the curve wraps.
  4. Constrain control point positions to fix the shape.

B-Spline by Interpolation

  1. Select B-Spline by Interpolation from the dropdown.
  2. Click interpolation points — the curve passes through each one exactly.
  3. Right-click to finish.
  4. Constrain the interpolation point positions.

Periodic B-Spline by Interpolation

Same as Interpolation, but closes the curve smoothly back through the first interpolation point.

Parameters

B-Splines in Sketcher have rich internal geometry:

Element Description
Control polygon The dashed polygon connecting control points (visible in edit mode)
Control points The blue squares that pull the curve; each has 2 DoF (X, Y)
Knot points The red squares on the curve body marking knot locations
Weight circles Circles at control points showing relative weight (drag to adjust)
Degree The polynomial degree of each segment (default 3 = cubic)

The most important parameters to constrain:

Parameter Type Applied via
Endpoint position (open spline) Coordinate Coincident or Lock Position
Control point positions Coordinate Lock Position on each control point
Endpoint tangent direction Angle Tangent between endpoint and adjacent edge
Interpolation point positions Coordinate Coincident or Lock Position

B-Spline creation in depth

Each of the four creation modes has qualitatively different behaviour when you change the shape or add constraints.

B-Spline by Control Points (open) (default)

The curve is pulled toward the control points but does not pass through interior ones. Moving a control point smoothly deforms the whole neighbourhood. Endpoints are interpolated for a standard clamped B-Spline.

Use this mode when:

  • You want sculpting-style shape control: place magnets and move them until the shape looks right.
  • You do not care exactly where the curve passes at interior points.

Watch out for: Each control point adds 2 DoF. A B-Spline with N control points needs 2N constraints to be fully constrained (in addition to the endpoint constraints). Large numbers of unconstrained control points are the most common cause of B-Spline-related solver slowdowns.

Periodic B-Spline by Control Points

A closed loop with no endpoints. The curve wraps smoothly through all control points. Control points near each other can create a sharp cusp if too close.

Use this mode when:

  • You need a fully enclosed smooth profile (no start/end seam).
  • You are designing a cam lobe, organic enclosure, or decorative profile.

Watch out for: It is impossible to apply an endpoint Tangent constraint (there are no endpoints). To control the curve direction at a specific location, constrain the positions of the two surrounding control points.

B-Spline by Interpolation (open)

The curve passes exactly through every interpolation point. Moving a point moves the curve at that exact location. Good for fitting a curve to known data points.

Use this mode when:

  • You know where the curve must pass at specific locations.
  • You are reverse-engineering a shape from measurements.

Watch out for: Interpolation splines can have higher oscillation between points (Runge's phenomenon) if the points are unevenly spaced. Use fewer, more evenly spaced points to get a smoother result.

Periodic B-Spline by Interpolation

Closed loop through exact interpolation points. The curve joins the last point back to the first with full C2 continuity.

Use this mode when:

  • You need a closed profile and know the exact points it must pass through.

Watch out for: The closure point (where the curve wraps back) may have a kink if the start and end tangents are not consistent. Ensure the first and last interpolation points are not too close together.

Advanced usage

Increasing / decreasing degree

Right-click a B-Spline and choose Increase B-Spline Degree to add polynomial degree (smoother, more control) or Decrease B-Spline Degree (fewer parameters, less smooth). Degree 3 (cubic) is standard for most organic profiles.

Converting a B-Spline to construction

Select the B-Spline and press G, W or toggle the Construction button to make it a reference element. This is useful when the spline is a guide rather than a profile edge.

B-Spline weights

Each control point has an associated weight that biases how strongly the curve is attracted to it. Higher weight pulls the curve closer. Drag the weight circles to adjust. Weights can be set equal with Equal for a uniform NURBS curve.

Common mistakes and pitfalls

B-Spline has many remaining DoF despite seeming finished

Cause: Every control point contributes 2 DoF. A 5-point spline has 10 DoF from the control points alone, minus whatever has been applied. The solver reports the true count.
Fix: Constrain every control point with Lock Position or relative-position constraints. Use Coincident to anchor endpoints to known vertices.

Solver becomes slow after constraining a B-Spline

Cause: B-Splines with many control points create large constraint systems. Adding dimensional constraints to control points can trigger complex solver iterations.
Fix: Use fewer control points where possible. Consider a lower-degree spline or break the curve into shorter segments.

Tangent constraint at endpoint shows as conflicting

Cause: The endpoint is already fully constrained by a Coincident and a Lock Position, and the tangent constraint cannot be satisfied simultaneously.
Fix: Remove the Lock Position and let the Tangent + Coincident define the endpoint's behaviour. Or remove the Tangent and manually align the control polygon direction.

Python API

Minimal example — cubic B-Spline

import FreeCAD as App
import Sketcher
import Part

doc = App.activeDocument()
sketch = doc.getObject("Sketch")

# Add a simple cubic B-Spline through four control points
poles = [
    App.Vector(0, 0, 0),
    App.Vector(10, 20, 0),
    App.Vector(30, 20, 0),
    App.Vector(40, 0, 0),
]
weights = [1.0, 1.0, 1.0, 1.0]
knots = [0.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 1.0]  # clamped cubic
mults = [4, 4]
degree = 3
periodic = False

bspline = Part.BSplineCurve()
bspline.buildFromPolesMultsKnots(poles, mults, knots, periodic, degree, weights)

idx = sketch.addGeometry(bspline, False)
print(f"B-Spline added at index {idx}")
doc.recompute()

Key geometry properties

Part.BSplineCurve from sketch.Geometry[idx]:

Property Type Description
NbPoles int Number of control points
NbKnots int Number of knot values
Degree int Polynomial degree
isPeriodic() bool True if the spline is closed
getPoles() list List of control point App::Vector positions

See also