AN APPROACH TO DESIGN A CURVE

Bezier curve is a parametric curve important in computer graphics and related fields.

Generalizations of Bezier curves to higher dimensions are called Bezier surfaces.

In vector graphics, Bezier curves are an important tool used to model smooth curves that can be scaled indefinitely.

Bezier curves were widely publicized 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.

Bezier curves are commonly referred to in image manipulation programs such as InAdobe Illustrator, Adobe Photoshop, and GIMP are combinations of Bezier curves.

Bezier curves are also used in animation as a tool to control motion in applications such as Adobe Flash, Adobe After Effects, and Autodesk 3ds max.

Bezier curves are widely used in computer graphics to model smooth curves.

In animation applications, such as Adobe Flash and Adobe Shockwave, or in applications like Game Maker, Bézier curves are used to outline for example movement. Users outline the wanted path in Bezier curves, and the application creates the needed frames for the object to move along the path.

Control points Degree of curve/polynomial Hull

Linear Bezier curves Given points P0 and P1, a linear Bezier curve is simply a

straight line between those two points. The curve is given by

Constructing Linear curves Animation of a linear Bezier curve, t in [0,1] The t in the

function for a linear Bezier curve can be thought of as describing how far B(t) is from P0 to P1. For example when t=0.25, B(t) is one quarter of the way from point P0 to P1. As t varies from 0 to 1, B(t) describes a curved line from P0 to P1.

Quadratic Bezier curves A quadratic Bezier curve is the path traced by the function

B(t), given points P0, P1, and P2,

Constructing Quadratic curves For quadratic Bezier curves one can construct

intermediate points Q0 and Q1 such that as t varies from 0 to 1:

Point Q0 varies from P0 to P1 and describes a linear Bezier curve.

Point Q1 varies from P1 to P2 and describes a linear Bezier curve.

Point B(t) varies from Q0 to Q1 and describes a quadratic Bezier curve.

Cubic Bezier curves Four points P0, P1, P2 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:

Generalization The Bezier curve of degree n can be generalized as follows.

Given points P0, P1,..., Pn, the Bezier curve is.

For example, for n = 5:

The basis function are real. Bezier curve always passes through the first and last control

points. The degree of polynomial defining the curve segment is one

less then the number of defining polygon points. Therefore , for 4 control points degree is 3, i.e. cubic polynomial.

The curve lies entirely within the convex hull formed by four control points.

The convex hull property for Bezier curve ensure that polynomial smoothly follow the control points.

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