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In the mathematical subfield of numerical analysis a Bézier curve is a parametric curve important in computer graphics. A numerically stable method to evaluate Bézier curves is de Casteljau's algorithm. 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. ...
Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...
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. ...
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. ...
In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...
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. ...
Generalizations of Bézier curves to higher dimensions are called Bézier surfaces; the Bézier triangle is a special case. Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...
A Bézier surface is a parametric tensor product surface defined by mathematical formulae, used in computer graphics, computer-aided design, and finite element modelling. ...
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. ...
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 by the French engineer Pierre Bézier who used them to design automobile bodies. The curves were developed in 1959 by Paul de Casteljau using de Casteljau's algorithm. 1962 was a common year starting on Monday (link will take you to calendar). ...
France - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...
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. ...
A small variety of cars, the most popular kind of automobile. ...
1959 was a common year starting on Thursday (link will take you to calendar). ...
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. ...
Definition Given n+1 points Pi in R3 a Bézier curve of degree n is a parametric curve 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. ...
composed of Bernstein basis polynomials of degree n 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. ...
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with the Bernstein basis polynomials defined as -
Pi is called control point for the Bézier curve. A polygon can be constructed by connecting the Bézier points with lines, starting with P0 and finishing with Pn. This polygon is called the Bézier polygon. 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. ...
A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...
Notes - The starting point of the curve is P0 and the ending point is Pn
- The Bézier curve is completely contained in 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 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 railroad) 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.
In mathematics, the convex hull for an object or a set of objects is the minimal convex set containing the given objects. ...
In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ...
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. ...
For heuristics in computer science, see heuristic (computer science) Heuristic is the art and science of discovery and invention. ...
Examples
Linear Bézier curves Bézier "curve" of degree 1:
Given two control points P0 and P1 a linear Bézier curve is just a straight line between those two points. The curve is given by A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...
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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,
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TrueType fonts use Bézier splines composed of the quadratic Bézier curves. 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. ...
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. ...
Cubic Bézier curves Bézier curve of degree 3: 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. Diagram of a bézier curve. ...
The parametric form of the curve is: A parameter is a measurement or value on which something else depends. ...
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Modern imaging systems like PostScript, Metafont and GIMP use Bézier splines composed of cubic Bézier curves for drawing curved shapes. For information about the PostScript page description language, see PostScript. ...
METAFONT is a programming language used to produce rasterized outline fonts. ...
Often, the name GIMP is used erroneously for the Gimp-Print printer driver set. ...
Application in 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. In mathematics, the convex hull for an object or a set of objects is the minimal convex set containing the given objects. ...
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 Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...
In Euclidean geometry, scaling is an affine, linear transformation that can enlarge or diminish an object by certain factors. ...
This article is about rotation as a movement of a physical body. ...
The most important Bézier curves are quadratic and cubic curves. Higher degree curves are more expensive to evaluate and there is no 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 Bézier splines. Analytic may refer to analytic proposition or analytic philosophy, in philosophy analytic geometry, analytic function, analytic continuation, analytic set in 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. ...
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 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. 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. ...
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 Bernstein polynomials. 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. ...
Given n+1 control points Pi, the rational Bézier curve can be described by:
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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. ...
In the mathematical subfield of numerical analysis a spline is a special curve defined piecewise by polynomials. ...
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 surface is a parametric tensor product surface defined by mathematical formulae, used in computer graphics, computer-aided design, and finite element modelling. ...
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. ...
NURBS, short for nonuniform rational B-spline, is a computer graphics technique for drawing curves. ...
References Donald Knuth Donald Ervin Knuth (born January 10, 1938) is a renowned computer scientist and Professor Emeritus at Stanford University. ...
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