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2013.08.18 00:14

# 베지어 곡선 (Bezier curve)

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# Bezier curve

In the mathematical field of numerical analysis, a Bezier curve is a parametric curve important in computer graphics. Generalizations of Bezier curves to higher dimensions are called Bezier surfaces, of which the Bezier triangle is a special case.

Bezier curves were widely publicised in 1962 by the French engineer Pierre Bezier, who used them to design automobile bodies. The curves were first developed in 1959 by Paul de Casteljau using de Casteljau's algorithm, a numerically stable method to evaluate Bezier curves.

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## Examination of cases

### Linear Bezier curves

Given points P0 and P1, a linear Bezier curve is just a straight line between those two points. The curve is given by and is equivalent to linear interpolation.

A quadratic Bezier curve is the path traced by the function B(t), given points P0P1, and P2, TrueType fonts use Bezier splines composed of the quadratic Bezier curves.

### Cubic Bezier curves Cubic Bezier curve

Four points P0P1P2 and P3 in the plane or in three-dimensional space define a cubic Bezier curve. The curve starts at P0 going toward P1 and arrives at P3 coming from the direction of P2. Usually, it will not pass through P1 or P2; these points are only there to provide directional information. The distance between P0 and P1 determines "how long" the curve moves into direction P2 before turning towards P3.

The parametric form of the curve is: Modern imaging systems like PostScriptAsymptote and Metafont use Bezier splines composed of cubic Bezier curves for drawing curved shapes.

## Generalization

The Bezier curve of degree n can be generalized as follows. Given points P0P1,..., Pn, the Bezier curve is For example, for n = 5: This formula can be expressed recursively as follows: Let denote the Bezier curve determined by the points P0P1,..., Pn. Then In words, the degree n Bezier curve is an interpolation between two degree n ? 1 Bezier curves.

### Terminology

Some terminology is associated with these parametric curves. We have where the polynomials are known as Bernstein basis polynomials of degree n, defining 00 = 1.

The points Pi are called control points for the Bezier curve. The polygon formed by connecting the Bezier points with lines, starting with P0 and finishing with Pn, is called the Bezier polygon(or control polygon). The convex hull of the Bezier polygon contains the Bezier curve.

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