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For example, a "linear spline" is just a piecewise linear function that is continuous at the "knots"- where the linear pieces connect. Obviously, you can't require it to be differentiable there without making it just a single straight line.

A "quadratic spline" consists of quadratic functions between knots that are both continuous and differentiable where they connect- but not twice differentiable.

A "cubic spline" consists of piecewise cubics that are twice differentiable where they connect.

Cubic splines are most commonly used. In fact, the name "spline" comes from the use of "splines"- very thin flexible strips of wood used to draw complex curves (before computer design). One can show by stress arguments that they are cubic splines.

The term b-spline comes from "basis" spline. If you think of the set of all possible splines (of a particular type:quadratic, cubic, etc.), you can show that that forms a vector-space and so any such spline can be written in terms of some basis.

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simple def'n: different ways to create or store a curve or surface.

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