An Interactive Introduction to Splines

New 2013 version HTML5 based

Spline Curves (with interactive 2D Java applets)
• Bezier spline curves
  DeCasteljau algorithm. Linear, quadratic and cubic Bezier splines. Bezier spline subdivision. Bernstein polynomials. Recurrence relations. How to plot Bezier spline and basis functions. Proof of the deCasteljau algorithm.
• More Bezier splines Math
  Affine invariance. Convex hull. Linear precision. Differentiation of the Bezier curve. Deriving the deCasteljau algorithm. Two remarks.
• Interpolating Lagrange curve
  Interpolating curves. Lagrange polynomials. Knots. Aitken algorithm. Lagrange curve oscillations.
• Interpolating Cardinal and Catmull-Rom splines
  Continuity. Building a complex spline. Getting rid of tangents. Cardinal curve subdivision.
• B-splines
  • Building cubic B-spline
    C² continuity. de Boor points. Cox - de Boor algorithm. Local coordinates.
  • B-spline basis functions
    Recurrence relations. Knot vectors. Uniform B-splines. Closed curves.
  • Nonuniform B-splines
    Handling endpoints.
  • Interpolating cubic B-splines
    Bezier control points. Solving banded equations. Handling the terminal tangents.
• NURBS
  Weighted control points. Perspective projection. Conic sections. Circular arcs.

  3D Spline Surfaces (with interactive VRMLs)

• Tensor product spline surfaces
  Bezier, NURBS and Cardinal spline patches. Normals. SplineGlobe.
• Points, Nets, Patches... (mainly questions yet :)
  NURBS, "Hash" or triangular patches?
• Cubic Bezier patch with adaptive subdivision   WebGL

  3D Examples

• Spline based nonlinear morphing
  "How to teach shark to swim"
• Flight animation
• 3D gallery
• WebGL Demos

I used these courses and papers

• "Interactive Curves and Surfaces" by Alyn Rockwood, Peter Chambers
• "Introduction to Computer Graphics" University of Waterloo, Department of Computer Science (html copy)
• "Advanced Graphics Study Guide, 2000-2001. Splines" by Neil Dodgson
• Introduction to Computing with Geometry Notes by C.-K. Shene
• NURBS in VRML by Holger Grahn, Thomas Volk, Hans J. Wolters
• A complete algorithm for real-time spline-based character animation by Martin D. Hash

All sources are free. The whole Splines.zip (200 kb)   Evgeny Demidov