A Quick Look at Some Math Topics Related to Bezier and Spline Curves.

• Linear Algebra
  1. Bernstein polynomials form a basis for linear spaces of polynomials.
  2. B-splines form a basis for linear spaces of splines.
  3. A cubic spline curve can be constructed that passes through specified points and has specified start and end conditions. To do this, you set up and solve a system of linear equations. The resulting matrix is tridiagonal.


• Calculus
  1. A space of splines is determined by specifying:
     • the degree of all the polynomial components of each spline in the space
     • a set of breakpoints at which the polynomial components join up
     • the levels of derivative continuity required at each breakpoint
  2. The tangent line at the start of a Bezier or spline curve is the line between its first and second control point. A similar property holds for the tangent line at the end of the curve.
  3. A powerful dual-basis formula for splines, developed by C. de Boor and G. Fix, makes use of derivative values for the splines.


• Geometry
  1. The shape of a Bezier or spline curve is governed by the location of the control points. This is demonstrated in the interactive web app Exploring Bezier and Spline Curves.
  2. Evaluation algorithms for Bezier and spline curves have geometric interpretations, which are also demonstrated in the interactive web app.
  3. Bezier and spline curves are contained within the convex hull of their control points. For a spline curve a stricter result is true; each span of a spline curve is contained within the convex hull of a particular set of degree + 1 control points.