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In this question it was stated that a B-spline is just a representation of a Bézier curve. I wonder if this also holds true for Bézier curves. Splines are defined piecewise by polynomials and so are Bézier curves. Is it correct to say that Bézier curves are splines? Is the term spline like a umbrella term for B-splines, Bézier curves, composite Bézier curves, NURBS etc.? Are they all splines at the end? Or are they all just representatives?

Jean-Claude Arbaut
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AliceTheCat
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1 Answers1

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A spline is a piecewise continuous curve. Each piece (or segment) of the spline could be represented by polynomial functions, trigonometric functions, exponential functions,...., etc with polynomial function being the most popular choice.

A B-spline curve (sometimes referred as B-splines for simplification) is a polynomial spline and each segment of a B-spline curve is geometrically equivalent to a Bezier curve (of the same degree). A NURBS curve is basically a rational B-spline curve where each piece is a rational polynomial function.

A Bezier curve is also a polynomial curve and we can analytically convert a polynomial curve into a Bezier curve or a Bezier curve into a polynomial curve. However, a Bezier curve is not really a spline and it would be confusing or even inappropriate to use terms like "Bezier spline".

fang
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