Кривые Безье: Основы и применение в компьютерной графике

Curve used in computer graphics and related fields

A Bézier curve (/'//b//E//z//.//i//.//ei/ BEHzeeay) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. The Bézier curve is named after French engineer Pierre Bézier (1910–1999), who used it in the 1960s for designing curves for the bodywork of Renault cars. Other uses include the design of computer fonts and animation. In animation applications such as Adobe Flash, Adobe After Effects, Microsoft Expression Blend, Blender, Autodesk Maya and Autodesk 3ds Max. user interface design and smoothing cursor trajectory in eye gaze controlled interfaces. For example, a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators or interface designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question. This also applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear.

The mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but the polynomials were not applied to graphics until some 50 years later when mathematician Paul de Casteljau in 1959 developed de Casteljau's algorithm, a numerically stable method for evaluating the curves, and became the first to apply them to computer aided design at French automaker Citroën. De Casteljau's method was patented in France but not published until the 1980s while the Bézier polynomials were widely publicised in the 1960s by the French engineer Pierre Bézier, who discovered them independently and used them to design automobile bodies at Renault.

A Bézier curve is defined by a set of control points P0 through Pn, where n is called the order of the curve (n = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points generally do not lie on the curve. The sums in the following sections are to be understood as affine combinations – that is, the coefficients sum to 1.

Given distinct points P0 and P1, a linear Bézier curve is simply a line between those two points. The curve is given by

This is the simplest and is equivalent to linear interpolation. The quantity represents the displacement vector from the start point to the end point.

Bézier curves can be defined for any degree n.

A recursive definition for the Bézier curve of degree n expresses it as a point to point linear combination (linear interpolation) of a pair of corresponding points in two Bézier curves of degree n − 1. Let denote the Bézier curve determined by any selection of points P0, P1,  , Pk. Then to start,

This recursion is elucidated in the animations below.

The curve begins at and ends at ; this is the so called endpoint interpolation property. The curve is a line if and only if all the control points are collinear. The start and end of the curve is tangent to the first and last section of the Bézier polygon, respectively. A curve can be split at any point into two 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 (see composite Bézier curve), with a maximum radial error of less than one part in a thousand, when each inner control point (or 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 (i. e. ), and Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points is equivalent (including the parametrization) to the degree n + 1 curve with control points , where , and define , Bézier curves have the variation diminishing property. What this means in intuitive terms is that a Bézier curve does not "undulate" more than the polygon of its control points, and may actually "undulate" less than that. There is no local control in degree n Bézier curves—meaning that any change to a control point requires recalculation of and thus affects the aspect of the entire curve, "although the further that one is from the control point that was changed, the smaller is the change in the curve". A Bézier curve of order higher than two may intersect itself or have a cusp for certain choices of the control points.

A quadratic Bézier curve is also a segment of a parabola. As a parabola is a conic section, some sources refer to quadratic Béziers as "conic arcs". Tangents to the parabola at the endpoints of the curve (A and B) intersect at its control point (C). If D is the midpoint of AB, the tangent to the curve which is perpendicular to CD (dashed cyan line) defines its vertex (V). Its axis of symmetry (dash dot cyan) passes through V and is perpendicular to the tangent. E is either point on the curve with a tangent at 45° to CD (dashed green). If G is the intersection of this tangent and the axis, the line passing through G and perpendicular to CD is the directrix (solid green). The focus (F) is at the intersection of the axis and a line passing through E and perpendicular to CD (dotted yellow). The latus rectum is the line segment within the curve (solid yellow).

The curve at a fixed offset from a given Bézier curve, called an offset or parallel curve in mathematics (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). In general, the two sided offset curve of a cubic Bézier is a 10th order algebraic curve and more generally for a Bézier of degree n the two sided offset curve is an algebraic curve of degree 4n − 2. However, there are heuristic methods that usually give an adequate approximation for practical purposes. In the field of vector graphics, painting two symmetrically distanced offset curves is called stroking (the Bézier curve or in general a path of several Bézier segments).

The concept of degree elevation can be repeated on a control polygon R to get a sequence of control polygons R, R1, R2, and so on. After r degree elevations, the polygon Rr has the vertices P0,r, P1,r, P2,r, , Pn+r,r given by A number of approximation algorithms have been proposed and used in practice.

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 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 computationally expensive to evaluate. When more complex shapes are needed, low order Bézier curves are patched together, producing a composite Bézier curve. A composite Bézier curve is commonly referred to as a "path" in vector graphics languages (like PostScript), vector graphics standards (like SVG) and vector graphics programs (like Artline, Timeworks Publisher, Adobe Illustrator, CorelDraw, Inkscape, and Allegro). In order to join Bézier curves into a composite Bézier curve without kinks, a property called G1 continuity suffices to force the control point at which two constituent Bézier curves meet to lie on the line defined by 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 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. The rasterisation algorithm used in Metafont is based on discretising the curve, so that it is approximated by a sequence of "rook moves" that are purely vertical or purely horizontal, along the pixel boundaries. To that end, the plane is first split into eight 45° sectors (by the coordinate axes and the two lines ), then the curve is decomposed into smaller segments such that the direction of a curve segment stays within one sector; since the curve velocity is a second degree polynomial, finding the values where it is parallel to one of these lines can be done by solving quadratic equations. Within each segment, either horizontal or vertical movement dominates, and the total number of steps in either direction can be read off from the endpoint coordinates; in for example the 0–45° sector horizontal movement to the right dominates, so it only remains to decide between which steps to the right the curve should make a step up. There is also a modified curve form of Bresenham's line drawing algorithm by Zingl that performs this rasterization by subdividing the curve into rational pieces and calculating the error at each pixel location such that it either travels at a 45° angle or straight depending on compounding error as it iterates through the curve. This reduces the next step calculation to a series of integer additions and subtractions.

In animation applications, such as Adobe Flash and Synfig, 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. In 3D animation, Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation. Bézier curves are now very frequently used to control the animation easing in CSS, JavaScript, JavaFx and Flutter SDK. Font engines, like FreeType, draw the font's curves (and lines) on a pixellated surface using a process known as font rasterization. Typically font engines and vector graphics engines render Bézier curves by splitting them recursively up to the point where the curve is flat enough to be drawn as a series of linear or circular segments. The exact splitting algorithm is implementation dependent, only the flatness criteria must be respected to reach the necessary precision and to avoid non monotonic local changes of curvature. The "smooth curve" feature of charts in Microsoft Excel also uses this algorithm. Because arcs of circles and ellipses cannot be exactly represented by Bézier curves, they are first approximated by Bézier curves, which are in turn approximated by arcs of circles. This is inefficient as there exists also approximations of all Bézier curves using arcs of circles or ellipses, which can be rendered incrementally with arbitrary precision. Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves (or surfaces) into NURBS, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition. This approach also preserves the curve definition under all linear or perspective 2D and 3D transforms and projections.