# Spline

## Bézier Curve

### A Cubic Bezier Spine has four control points, two of which are knots.

Some of theses properties may sound somewhere obvious, but they are presented here because they are still valid for Bézier curves of higher degree

### A Bézier cure is a polynomial.

The degree of the polynomial is always one less than the number of control points. In computer graphics, we generally use degree 3. Quadratic curves are not flexible enough and going above degree 3 gives rises to complications and so the choice of cubics is the best compromise for most computer graphics applications.

### The control points do not exert 'local' control.

Moving any control point affects all of the curve to a greater or lesser extent. All the basis functions are everywhere non-zero except at the point u = 0 and u = 1

### Moving the control points alters the magnitude and direction of the tangent vectors

This is the basis of the intuitive 'feel' of a Bézier curve interface.

### Variation diminishing property

The curve does not oscillate about any straight line more often than the control point polygon

### Affine transformation compatibility

The curve is transformed by applying any affine transformation (that is, any combination of linear transformations) to its control point representation. The curve is invariant (does not change shape) under such a transformation.

### Non localness

As soon as you move one control point, you affect the entire curve

### There exist some other way to construct Bézier curves...

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