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Approximate Polynomial Surface

In one sentence: Fit a degree-2 Bézier surface patch to a mesh region, producing a single quadratic surface that represents planes, cylinders, spheres, cones, and saddle shapes.

Overview

Workbench: Reverse Engineering
Menu: Reverse Engineering → Approximation → Polynomial Surface
Shortcut: none

Fits a degree-2 polynomial z = f(x, y) to the mesh points and converts the polynomial coefficients to a 3×3 Bézier control-point grid. Creates a Part::Spline (Geom_BezierSurface) that represents the quadratic approximation.

Intuition

A degree-2 polynomial surface is the simplest possible curved surface — it can represent all the quadric primitives: planes (all zero), cylinders (one curved direction), spheres and paraboloids (both directions curved equally), and saddle shapes (curvatures in opposite directions). One mathematical formula covers all of them.

Use this when a simple, single-patch surface is sufficient — it is faster to compute than a full B-spline fit and gives a clean result for regions that are close to a quadric shape.

When to use it

  • The mesh region is approximately planar, cylindrical, spherical, or saddle-shaped, and you want a smooth, single-patch surface.
  • You need a parametric surface that can be easily edited (the 3×3 control grid is simple to manipulate).
  • You want a quick approximate surface before deciding whether a full B-spline fit is needed.

When NOT to use it

  • Complex curvature — a degree-2 patch cannot represent sharp bends, high-curvature transitions, or regions that require more than one patch. Use Approximate B-Spline Surface instead.
  • Large regions — a single quadratic patch cannot accommodate significant shape variation across a large area without high residuals.

Step-by-step

  1. Isolate a mesh segment of approximately quadric shape using segmentation tools.
  2. Select the Mesh::Feature.
  3. Choose Reverse Engineering → Approximation → Polynomial Surface.
  4. FreeCAD creates a Part::Spline containing the 3×3 Bézier surface.

Parameters

No task panel — the fit is automatic.

Common mistakes and pitfalls

High residual — region is not quadric

Cause: The mesh has more complex curvature than a degree-2 polynomial can represent.
Fix: Use Approximate B-Spline Surface with more control points for better accuracy.

Python API

import FreeCADGui as Gui
Gui.doCommand("Gui.runCommand('Reen_ApproxPolynomial', 0)")

See also