Study notes for Ian Stroud, Boundary Representation Modelling Techniques (Springer, 2006).

Free-form geometry is where analytic comfort ends. This chapter provides the essential toolkit: Bézier, B-splines, rational forms, and NURBS—plus practical notes on how to build surfaces from curves and patches.

1) Basics: curve/surface families

  • Bézier: intuitive control-point formulation; good for local modelling but global influence for high degree.
  • B-spline: knot vectors and piecewise polynomials; local control and stability.
  • Rational forms: represent conics exactly (circles/ellipses) and enable projective behaviour.
  • NURBS: the workhorse representation for industrial CAD.

2) Interpolation and lofting

Real workflows require:

  • interpolating curves through points (with continuity constraints)
  • lofting surfaces from multiple section curves
  • handling mixed constraints (positional, tangency, curvature)

The chapter also discusses the “engineering reality”:

  • parameterization choices matter
  • continuity conditions are easy to state but hard to satisfy robustly
  • input data is noisy

3) Handling problems and sculpting methods

Free-form operations can fail due to:

  • self-intersections
  • oscillations from poor parameterization
  • patch mismatch and continuity breaks

“Sculpting” style methods and smoothing are introduced as pragmatic tools:

  • adjust control points
  • apply fairness energies
  • trade accuracy for smoothness

Chapter outline (from the book)

Major sections

  • 13.1 Basics
  • 13.2 Interpolation
  • 13.3 Lofting

Selected subsections

  • 13.1.1 B ezier geometry
  • 13.1.2 B-spline geometry
  • 13.1.3 Rational forms
  • 13.1.4 NURBS geometry

Implementation checklist

  • Standardize on NURBS as your canonical free-form curve/surface representation.
  • Provide utilities:
    • evaluation (point/derivatives)
    • closest point / projection
    • intersection (curve/curve, curve/surface, surface/surface)
    • reparameterization and knot insertion
  • Add “quality tools”:
    • curvature plots
    • deviation metrics
    • fairness/smoothing controls

Free-form geometry: the hidden costs

  • evaluation is easy; robust intersection is not
  • trimming is where most bugs live
  • continuity (G1/G2) is hard to preserve under edits

Invest in good numerical tooling: robust solvers, tolerance-aware predicates, and stable parameterizations.

Practical exercises

  • Fit a NURBS curve to sample points and measure deviation.
  • Implement knot insertion and show it preserves the curve shape.