A cubic Bézier triangle is a In mathematics, a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has... surface with the equation

where α³, β³, γ³, α²β, αβ², β²γ, βγ², αγ², α²γ and αβγ are the 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... control points of the triangle.

An example Bézier triangle with control points marked

The corners of the triangle are the points α³, β³ and γ³. The edges of the triangle are themselves 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 Casteljaus algorithm. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces... Bézier curves, with the same control points as the Bézier triangle.

It is also possible to create In mathematics, a quadratic equation is a polynomial equation of the second degree. The generalized form is The numbers a, b and c are called coefficients: a is the coefficient of x2, b is the coefficient of x, and c is the free term or constant. A quadratic equation with... quadratic or other degrees of Bézier triangles, by changing the exponent in the original equation, in which case there will be more or less control points. With the exponent 1, the resulting Bézier triangle is actually a regular flat For alternate meanings, such as the musical instrument, see triangle (disambiguation). A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. Types of triangles Triangles can be classified according to the lengths of their sides... triangle. In all cases, the edges of the triangle will be Bézier curves of the same degree.

By removing the γu term, a regular Bézier curve results. Also, while not very useful for display on a physical computer screen, by adding extra terms, a Bézier A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or equilateral, and is one of the Platonic solids. The area A and the volume V of a regular... tetrahedron or Bézier In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. Beyond that, the term is used for a variety of related mathematical concepts. This is analogous to the way the term square may be used to refer to a square... polytope results.

Due to the nature of the equation, the entire triangle will be contained within the volume surrounded by the control points, and 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 of the control points will correctly transform the whole triangle in the same way.

An advantage of Bézier triangles in computer graphics is, they are smooth, and can easily be approximated by regular triangles, by In mathematics and computer science, recursion is a particular way of specifying (or constructing) a class of objects (or an object from a certain class) with the help of a reference to other objects of the class: a recursive definition defines objects in terms of the already defined objects of... recursively dividing the Bézier triangle into two separate Bézier triangles, until they are considered sufficiently small, using only addition and division by two, not requiring any A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. In particular, it represents an integer or fixed-point number (the significand or, informally, the mantissa) multiplied by... floating point arithmetic whatsoever.

• The following computes the new control points for the half of the full Bézier triangle with the corner α³, a corner halfway along the Bézier curve between α³ and β³, and the third corner γ³.

equivalently, using addition and division by two only,

where := means to replace the vector on the left with the vector on the right.

Note that halving a bézier triangle is similar to halving Bézier curves of all orders up to the order of the Bézier triangle.

See also:

• 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 Casteljaus algorithm. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces... Bézier curve
• (square) 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
• In mathematics, a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has... surface