In the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.
Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier, 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 Bézier curves.
In vector graphics, Bézier curves are an important tool used to model smooth curves that can be scaled indefinitely. "Paths," as they are commonly referred to in image manipulation programs such as Inkscape, Adobe Illustrator, Adobe Photoshop, and GIMP are combinations of Bézier curves patched together. Paths are not bound by the limits of rasterized images and are intuitive to modify. Bézier curves are also used in animation as a tool to control motion in applications such as Adobe Flash, Adobe After Effects, Microsoft Expression Blend, Blender, Autodesk Maya and Autodesk 3ds max.
Bezier curves are also commonly used over the time domain, particularly in animation and interface design. Thus, a bezier curve is often used to describe or control the velocity over time of an object moving from A to B. For example, an icon might "ease-in-out" or follow a "cubic bezier" in moving from A to B, rather than simply moving at a fixed number of pixels per step. Indeed, when animators or interface designers discuss the "physics" or "feel" of an operation, they often are referring to the particular bezier curve used to control the velocity over time of the move in question.
Applications
Computer graphics
Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively. Affine transformations such as translation, scaling and rotation can be applied on the curve by applying the respective transform on the control points of the curve.
Quadratic and cubic Bézier curves are most common; higher degree curves are more expensive to evaluate. When more complex shapes are needed, low order Bézier curves are patched together. This is commonly referred to as a "path" in programs like Adobe Illustrator or Inkscape. These poly-Bézier curves can also be seen in the SVG file format. To guarantee smoothness, the control point at which two curves meet must be on the line between the two control points on either side.
The simplest method for scan converting (rasterizing) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments. However, this does not guarantee that the rasterized output looks sufficiently smooth, because the points may be spaced too far apart. Conversely it may generate too many points in areas where the curve is close to linear. A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line segment to within a small tolerance. If not, the curve is subdivided parametrically into two segments, 0 ≤ t ≤ 0.5 and 0.5 ≤ t ≤ 1, and the same procedure is applied recursively to each half. There are also forward differencing methods, but great care must be taken to analyse error propagation. Analytical methods where a spline is intersected with each scan line involve finding roots of cubic polynomials (for cubic splines) and dealing with multiple roots, so they are not often used in practice.
Animation
In animation applications, such as Adobe Flash and Synfig, or in applications like Game Maker, Bézier curves are used to outline, for example, movement. Users outline the wanted path in Bézier curves, and the application creates the needed frames for the object to move along the path. For 3D animation Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation.
Examination of cases
Linear Bézier curves
Given points P 0 and P 1 , a linear Bézier curve is simply a straight line between those two points. The curve is given by
and is equivalent to linear interpolation.
Quadratic Bézier curves
A quadratic Bézier curve is the path traced by the function B ( t ), given points P 0 , P 1 , and P 2 ,
A quadratic Bézier curve is also a parabolic segment.
TrueType fonts use Bézier splines composed of quadratic Bézier curves.
Cubic Bézier curves
Four points P 0 , P 1 , P 2 and P 3 in the plane or in three-dimensional space define a cubic Bézier curve. The curve starts at P 0 going toward P 1 and arrives at P 3 coming from the direction of P 2 . Usually, it will not pass through P 1 or P 2 ; these points are only there to provide directional information. The distance between P 0 and P 1 determines "how long" the curve moves into direction P 2 before turning towards P 3 .
The parametric form of the curve is:
Modern imaging systems like PostScript, Asymptote and Metafont use Bézier splines composed of cubic Bézier curves for drawing curved shapes.
Generalization
The Bézier curve of degree n can be generalized as follows. Given points P 0 , P 1 ,..., P n , the Bézier curve is
For example, for n = 5 :
This formula can be expressed recursively as follows: Let
denote the Bézier curve determined by the points P 0 , P 1 ,..., P n . Then
In other words, the degree n Bézier curve is a linear interpolation between two degree n − 1 Bézier 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 t 0 = 1 and (1 - t) 0 = 1. The binomial coefficient,
, has the alternative notation,
The points P i are called control points for the Bézier curve. The polygon formed by connecting the Bézier points with lines, starting with P 0 and finishing with P n , is called the Bézier polygon (or control polygon ). The convex hull of the Bézier polygon contains the Bézier curve.
- The curve begins at P 0 and ends at P n ; this is the so-called endpoint interpolation property.
- The curve is a straight line if and only if all the control points are collinear.
- The start (end) of the curve is tangent to the first (last) section of the Bézier polygon.
- A curve can be split at any point into 2 subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.
- Some curves that seem simple, such as the circle, cannot be described exactly by a Bézier or piecewise Bézier curve; though a four-piece cubic Bézier curve can approximate a circle, with a maximum radial error of less than one part in a thousand, when each inner control point (of offline point) is the distance
horizontally or vertically from an outer control point on a unit circle. More generally, an n -piece cubic Bézier curve can approximate a circle, when each inner control point is the distance
from an outer control point on a unit circle, where t is 360/ n degrees, and n > 2.
- The curve at a fixed offset from a given Bézier curve, often called an offset curve (lying "parallel" to the original curve, like the offset between rails in a railroad track), cannot be exactly formed by a Bézier curve (except in some trivial cases). However, there are heuristic methods that usually give an adequate approximation for practical purposes.
- Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, eve
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