In the Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that... mathematical subfield of Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). This means it deals mainly with real variable or complex variable questions, numerical linear algebra over the real or complex fields, the solution of differential equations, and other related problems arising in... numerical analysis a Bézier curve is a In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. For example, circle in the plane can be defined as the curve γ where the vector γ... parametric curve important in Computer graphics (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. The first major advance in computer graphics was the development of the Sketchpad in 1962 by Ivan... computer graphics. A In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. It describes how errors in the input data propagate through the algorithm. In a stable method, the errors due to the approximations get damped out as the computation proceeds. In an unstable method, any errors... numerically stable method to evaluate Bézier curves is In the mathematical subfield of numerical analysis the de Casteljaus algorithm, named after its inventor Paul de Casteljau, is a recursive method to evaluate polynomials in Bernstein form or Bézier curves. Although the algorithm is slower for most architectures when compared with the direct approach it is... de Casteljau's algorithm.

Generalizations of Bézier curves to higher Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. (In common usage, the dimensions of an object are the measurements that define its shape and size. That usage is related to, but different from, what this article is about... dimensions are called A Bézier surface is a parametric tensor product surface defined by mathematical formulae, used in computer graphics, computer-aided design, and finite element modelling. It can be viewed as a generalization of a Bézier curve. Formula Bézier surfaces were first described in 1972 by... Bézier surfaces; the A cubic Bézier triangle is a surface with the equation where α3, β3, γ3, α2β, αβ2, β2γ, βγ2, αγ2, α2γ and αβγ are the control points of the triangle. An example... Bézier triangle is a special case.

Bézier curves are also formed by many common forms of string art, where strings are looped across a frame of nails.

History

Bézier curves were widely publicized in 1962 was a common year starting on Monday (link will take you to calendar). Events January January 1 - Western Samoa becomes independent from New Zealand January 3 - Pope John XXIII excommunicates Fidel Castro January 4 - New York City introduces a train that operates without a crew on-board January 5... 1962 by the The French Republic or France ( French: République française or France) is a country whose metropolitan territory is located in western Europe, and which is further made up of a collection of overseas islands and territories located in other continents. France is a democracy organised as a... French engineer Pierre Etienne Bézier ( September 1, 1910, - November 25, 1999) was a French engineer and creator of the Bézier curves and Bézier surfaces that are now the basis of most computer-aided design and computer graphics systems. Born in Paris, Bézier obtained a... Pierre Bézier who used them to design A small variety of cars, the most popular kind of automobile. An automobile is a wheeled vehicle that carries its own engine. Different types of automobile include cars, buses, vans and trucks, with cars being the most popular by far. Older terms include horseless carriage and motor car, with motor... automobile bodies. The curves were developed in 1959 was a common year starting on Thursday (link will take you to calendar). Events January-February January 1 - Cultivars of plants named after this date must be named in a modern language, not in Latin. January 1 - Cuba: Fulgencio Batista flees Havana when forces of Fidel Castro advance January... 1959 by Paul de Casteljau using In the mathematical subfield of numerical analysis the de Casteljaus algorithm, named after its inventor Paul de Casteljau, is a recursive method to evaluate polynomials in Bernstein form or Bézier curves. Although the algorithm is slower for most architectures when compared with the direct approach it is... de Casteljau's algorithm.

Definition

Given n+1 points P_i in R³ a Bézier curve of degree n is a In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. For example, circle in the plane can be defined as the curve γ where the vector γ... parametric curve

composed of In the mathematical subfield of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials. A numerical stable way to evaluate polynomials in Bernstein form is de Casteljaus algorithm. Polynomials in Bernstein form... Bernstein basis polynomials of degree n

with the Bernstein basis polynomials defined as

P_i is called control point for the Bézier curve. A A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. The straight line segments that make up the polygon are called its sides or edges and the points where the sides meet are... polygon can be constructed by connecting the Bézier points with A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i.e. a curve that is long and straight. Given two points, in Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection... lines, starting with P₀ and finishing with P_n. This polygon is called the Bézier polygon.

Notes

• The starting point of the curve is P₀ and the ending point is P_n
• The Bézier curve is completely contained in the In mathematics, the convex hull for an object or a set of objects is the minimal convex set containing the given objects. It is the minimal convex set because the convex hull is a subset of any convex set which contains the given objects. If X is some set, the... convex hull of the control points.
• If and only if all control points lie on the curve it is a straight line.
• The start (end) of the curve is In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. Geometry In plane geometry, a straight line is tangent to a curve, at some point, if both line and curve pass through the point with the same direction; such a line... 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.
• A circle cannot be exactly formed by a Bézier curve. Not even a circular arc. (However, often a Bézier curve is an adequate approximation to a small enough circular arc).
• The curve at a fixed offset from a given Bézier curve ("parallel" to that curve, like the offset between tracks in a This is the top-level page of WikiProject trains Rail tracks Rail transport refers to the land transport of passengers and goods along railways or railroads. These consist of two parallel rails, usually of steel, generally mounted upon cross-sectional beams (termed sleepers or ties) of timber, concrete or other... railroad) cannot be exactly formed by a Bézier curve (except in some trivial cases). However, there are For heuristics in computer science, see heuristic (computer science) Heuristic is the art and science of discovery and invention. The word comes from the same Greek root (`ευρισκω) as eureka, meaning to find. A heuristic is a way of directing your attention fruitfully. The... heuristic methods that usually give an adequate approximation for practical purposes.

Examples

Linear Bézier curves

Bézier "curve" of degree 1:

Given two control points P₀ and P₁ a linear Bézier curve is just a A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i.e. a curve that is long and straight. Given two points, in Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection... straight line between those two points. The curve is given by

Quadratic Bézier curves

Bézier curve of degree 2:

A quadratic Bézier curve is the path traced by the function B(t). For points A, B, and C,

TrueType is an outline font standard originally developed by Apple Computer in the late 1980s as a competitor to Adobes Type 1 fonts used in PostScript. The primary strength of TrueType is that it offers font developers a high degree of control over precisely how their fonts are displayed... TrueType fonts use In the mathematical subfield of numerical analysis and in computer graphics a Bézier spline is a spline curve where each polynomial of the spline is in Bézier form. A Bézier spline is sometimes called bezigon as Bézier splines are like polygons but... Bézier splines composed of the quadratic Bézier curves.

Cubic Bézier curves

Bézier curve of degree 3:

Diagram of a bézier curve. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. Click on date to download the file or see the image uploaded on that date. (del) (cur) 00:28, 9 Oct 2004 . . DrBob...

Four points A, B, C and D in the plane or in three-dimensional space define a cubic Bézier curve. The curve starts at A going toward B and arrives at D coming from the direction of C. In general, it will not pass through B or C; these points are only there to provide directional information. The distance between A and B determines "how long" the curve moves into direction B before turning towards D.

The A parameter is a measurement or value on which something else depends. Example For example, a parametric equaliser is a tone control circuit that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings... parametric form of the curve is:

Modern imaging systems like For information about the PostScript page description language, see PostScript. A postscript (from post scriptum, a Latin expression meaning after writing and abbreviated P.S.) is a sentence, paragraph, or occasionally many paragraphs added, often hastily and incidentally, after the signature of a letter or (sometimes) the main body of... PostScript, METAFONT is a programming language used to produce rasterized outline fonts. This programming language was devised by Donald Knuth as counterpart to his TEX typesetting system. One of the characteristics of METAFONT is that all the outlines of the glyphs are defined with powerful geometrical equations, e.g., you can... Metafont and Often, the name GIMP is used erroneously for the Gimp-Print printer driver set. Screenshot of the GIMP version 2 The GNU Image Manipulation Program or The GIMP is a bitmap graphics editor, a program for creating and processing raster graphics. It also has some support for vector graphics. The... GIMP use Bézier splines composed of cubic Bézier curves for drawing curved shapes.

Application in computer graphics

Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the In mathematics, the convex hull for an object or a set of objects is the minimal convex set containing the given objects. It is the minimal convex set because the convex hull is a subset of any convex set which contains the given objects. If X is some set, the... convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively. An affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. In a geometric setting, these are precisely the functions that map straight lines to straight lines. A linear transformation is a function that preserves all... Affine transformations such as In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. Each translation is an... translation, In Euclidean geometry, scaling is an affine, linear transformation that can enlarge or diminish an object by certain factors. See also homothety. A scaling can be represented by a scaling matrix. To scale an object by a vector v = (vx, vy, vz), each point p = (px, py, pz) would need... scaling and This article is about rotation as a movement of a physical body. For other meanings, see rotation (disambiguation). Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant. The fixed point can... rotation can be applied on the curve by applying the respective transform on the control points of the curve.

The most important Bézier curves are quadratic and cubic curves. Higher degree curves are more expensive to evaluate and there is no Analytic may refer to analytic proposition or analytic philosophy, in philosophy analytic geometry, analytic function, analytic continuation, analytic set in mathematics. the use of analytic expressions, or periphrasis, in linguistics See also: analysis analytical chemistry analytical engine Analytical Society Analytical Thomism This is a disambiguation page — a navigational aid... analytic formula to calculate the roots of polynomials of degree 5 and higher. When more complex shapes are needed low order Bézier curves are patched together (obeying certain smoothness conditions) in the form of In the mathematical subfield of numerical analysis and in computer graphics a Bézier spline is a spline curve where each polynomial of the spline is in Bézier form. A Bézier spline is sometimes called bezigon as Bézier splines are like polygons but... Bézier splines.

The following code is a simple practical example showing how to plot a cubic Bezier curve in C. Note, this simply computes the coefficients of the polynomial and runs through a series of t values from 0 to 1 - in practice this is how it is usually done, even though neat algorithms such as deCasteljau's are often cited in graphics discussions, etc. This is because in practice a linear algorithm like this is faster and less resource-intensive than a recursive one like deCasteljau's. The following code has been factored to make its operation clear - an optimization in practice would be to compute the coefficients once and then re-use the result for the actual loop that computes the curve points - here they are recomputed every time, which is less efficient but helps to clarify the code.

The resulting curve can be plotted by drawing lines between successive points in the curve array - the more points, the smoother the curve.

 /****************************************************** Code to generate a cubic Bezier curve Warning - untested code *******************************************************/ typedef struct float x; float y; Point2D; /****************************************************** cp is a 4 element array where: cp[0] is the starting point, or A in the above diagram cp[1] is the first control point, or B cp[2] is the second control point, or C cp[3] is the end point, or D t is the parameter value, 0 <= t <= 1 *******************************************************/ Point2D PointOnCubicBezier( Point2D* cp, float t ) float ax, bx, cx; float ay, by, cy; float tSquared, tCubed; Point2D result; /* calculate the polynomial coefficients */ cx = 3.0 * (cp[1].x - cp[0].x); bx = 3.0 * (cp[2].x - cp[1].x) - cx; ax = cp[3].x - cp[0].x - cx - bx; cy = 3.0 * (cp[1].y - cp[0].y); by = 3.0 * (cp[2].y - cp[1].y) - cy; ay = cp[3].y - cp[0].y - cy - by; /* calculate the curve point at parameter value t */ tSquared = t * t; tCubed = tSquared * t; result.x = (ax * tCubed) + (bx * tSquared) + (cx * t) + cp[0].x; result.y = (ay * tCubed) + (by * tSquared) + (cy * t) + cp[0].y; return result; /***************************************************************************** ComputeBezier fills an array of Point2D structs with the curve points generated from the control points cp. Caller must allocate sufficient memory for the result, which is <sizeof(Point2D) * numberOfPoints> ******************************************************************************/ void ComputeBezier( Point2D* cp, int numberOfPoints, Point2D* curve ) float t, dt; int i; dt = 1.0 / ( numberOfPoints - 1 ); for( i = 0, t = 0; i < numberOfPoints; i++, t += dt) curve[i] = PointOnCubicBezier( cp, t ); 

Rational Bézier curves

Some curves that seem simple, like the See The Circle for the distributed file storage system, and see Ring (diacritic) for the diacritic mark. In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. Circles are simple closed curves... circle, cannot be described by a Bézier curve or a piecewise Bézier curve (though in practice the difference is small and may be tolerable). To describe some of these other curves, we need additional degrees of freedom.

The rational Bézier curve adds weights that can be adjusted. The numerator is a weighted Bernstein form Bézier curve and the denominator is a weighted sum of In the mathematical subfield of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials. A numerical stable way to evaluate polynomials in Bernstein form is de Casteljaus algorithm. Polynomials in Bernstein form... Bernstein polynomials.

Given n+1 control points P_i, the rational Bézier curve can be described by:

or simply

See also

• In the mathematical subfield of numerical analysis the de Casteljaus algorithm, named after its inventor Paul de Casteljau, is a recursive method to evaluate polynomials in Bernstein form or Bézier curves. Although the algorithm is slower for most architectures when compared with the direct approach it is... de Casteljau's algorithm
• In the mathematical subfield of numerical analysis a spline is a special curve defined piecewise by polynomials. The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar... Spline (mathematics)
• In the mathematical subfield of numerical analysis and in computer graphics a Bézier spline is a spline curve where each polynomial of the spline is in Bézier form. A Bézier spline is sometimes called bezigon as Bézier splines are like polygons but... Bézier spline
• A Bézier surface is a parametric tensor product surface defined by mathematical formulae, used in computer graphics, computer-aided design, and finite element modelling. It can be viewed as a generalization of a Bézier curve. Formula Bézier surfaces were first described in 1972 by... Bézier surface
• A cubic Bézier triangle is a surface with the equation where α3, β3, γ3, α2β, αβ2, β2γ, βγ2, αγ2, α2γ and αβγ are the control points of the triangle. An example... Bézier triangle
• NURBS, short for nonuniform rational B-spline, is a computer graphics technique for drawing curves. A NURBS curve is defined by a set of weighted control points, the curves order and a knot vector. NURBS are generalizations of both B-splines and Bézier curves, with the primary difference... NURBS

References

• Paul Bourke: Bézier curves, http://astronomy.swin.edu.au/pbourke/curves/bezier/
• Donald Knuth Donald Ervin Knuth (born January 10, 1938) is a renowned computer scientist and Professor Emeritus at Stanford University. Knuth (pronounced Ka-NOOTH [1]) is best known as the author of the multi-volume The Art of Computer Programming, one of the most highly respected references in the computer... Donald Knuth: Metafont: the Program, Addison-Wesley 1986, pp. 123-131. Excellent discussion of implementation details; available for free as part of the TeX distribution.
• Dr. Thomas Sederberg, BYU Bézier curves,http://www.tsplines.com/tutorials/ch2.pdf

External links

• Living Math Bézier applet (http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Bezier/bezier.html)
• Living Math Bézier applets of different spline types, JAVA programming of splines (http://www.ibiblio.org/e-notes/Splines/Bezier.htm) in An Interactive Introduction to Splines (http://ibiblio.org/e-notes/Splines/Intro.htm)
• Don Lancaster's Cubic Spline Library (http://www.tinaja.com/cubic01.asp) describes how to approximate a circle (or a circular arc, or a hyperbola) by a Bézier curve; using cubic splines for image interpolation, and a explanation of the math behind these curves.