A bivariate polynomial *p* of degree *d* is a
function of the form

(1)

where *x* and *y* are the independent
variables and the *a _{ij}* are the
coefficients.

There are many alternative ways to write *p*. If the
relevant domain of *p* is a ** triangle** (as it
is in many applications (like the finite element method or
the design of surfaces) then the Bernstein-Bézier form is
particularly appropriate.

Let's denote the relevant triangle by *T*. Denote the
vertices of *T* by *V _{1 }*,

(2)

The Bernstein-Bézier form of *p* is given by

(3)

The *c _{ijk}* are the

Note that the sum of the subscripts *i*, * j*,
and *k* is equal rather than less than or equal to
*d*, and that there are as many coefficients in (3) as
there are in (1).

It can be easily verified that (3) does indeed define a
polynomial of degree *d* and that any polynomial of
the form *(1)* can be written uniquely in the
form *(3)*.

With each Bézier ordinate we associate a * domain point
*

(4)

These points form a regular triangular array of *
(d+2)(d+1)/2* points in the triangle *T*. They are
illustrated for *d=4* and *d=6* in the Figures
nearby (which show the domain points as small green dots).

The domain points are combined with the Bézier ordinates for
form ** Control Points** *(P _{ijk},
c_{ijk} )*

A major application ( **the** major application in these
pages) of the Bernstein--Bézier is the analysis of
smoothness conditions between polynomials on neighboring
triangles. The power of the Bernstein--Bézier form stems
from the fact that algebraic (smoothness) conditions can be
expressed, interpreted, and analyzed **geometrically.**

Suppose we are given two triangles, *T* with vertices
*V _{1}*,

We write the polynomial *p* on *T* in the form
(3) and the polynomial *q* on *U* as

(4)

where the bar indicates barycentric coordinates and Bézier
ordinates with respect to the triangle * U.*

We ask under what conditions do the polynomials * p*
and *q* coincide along the edge *e _{12}
*. Along that edge

Thus

(6)

It follows that equating the two restrictions along e_{
12} gives the condition

(7)

which is true if and only if

(8)

We have thus obtained the geometric criterion:

** The polynomials on the two triangles will join
continuously if and only if the control points above the
common edge coincide.**

Since we always want at least continuity between neighboring polynomial pieces we do not distinguish the domain points along an edge shared by two neighboring triangles.

Differentiability conditions are more subtle. We ask for
conditions under which first order derivatives of *p*
and *q* join continuously across *e _{12}
*. Thus we differentiate both

Instead we consider *directional derivatives*. Let
*e* denote a *direction*, i.e., a vector in
the plane. (Often this is an edge of a triangle which
accounts for the notation *e* .) We denote
differentiation in the direction *e* with *D*.
Thus

(9)

Reorganizing the sum gives the derivative in Bernstein-Bézier form:

(10)

where

(11)

Note that since the barycentric coordinates are linear functions their derivatives are constants.

Doing the same on the triangle *U* and then equating
coefficients gives rise to the algebraic conditions

(12)

For each *i* and *j*, equation (12) relates four
coefficients whose domain points form a quadrilateral in the domain
and whose control points form a quadrilateral in space.

The quadrilaterals of domain points are illustrated in the
nearby figure for the case *d=3*. The quadrilaterals
themselves are gray, the domain points corresponding to the
right hand side of (12) are red, those on the left hand
side are green, and domain points not relevant to the
smoothness conditions are blue.

The coefficients *A _{1}*,

The coefficients in (12) are independent of the polynomial
degree *d*. Consider them in particular for the case
*d=1* where the polynomials *p* and *q*
are * linear.* Two linear functions join
differentiably if and only if they are identical. Note that
for general *d*

(13)

Thus the control points at the vertices of *T* lie on
the graph of the function *p*. It follows that the
piecewise linear function will be differentiable if and only
if the control points at the vertices of *T* and *
U* lie in the same plane!

If *d>1* we still have the same algebraic
relationships between the control points at the vertices
of the gray quadrilaterals. However, the small
quadrilaterals are *similar* to the quadrilateral
formed by *T* and * U*. The geometric
interpretation of the algebraic conditions is therefore the
same: The control points at the vertices of the gray
quadrilaterals must be in the same plane for each
quadrilateral!. The planes may differ from quadrilateral to
quadrilateral, but each quadrilateral of control points
(in three dimesnional space must be planar. We have thus
obtained the geometric
criterion:

** The polynomials on the two triangles will join
differentiably if and only if for each quadrilateral there
is a plane in 3-space that contains the four control points at
the vertices of the quadrilateral.**

It is now also clear that the *A _{i}* in
(12) are the barycentric coordinates of

(14)

Higher order differentiability does not possess a geometric
interpretation that is quite as striking as the first order
differentiability conditions. Algebraically the
*r*-th
order conditions are given by

(15)

They also involve quadrilaterals, but they are larger than
in the first order ( *C ^{1}* ) case and they
overlap.

The nearby figure illustrates these conditions for the case
where the polynomials *d=5* and the degree of
smoothness *r=3*.

The overlapping quadrilaterals are again drawn in gray. The coefficients entering the Right hand side of (14) are marked in red. Coefficients not entering any smoothness conditions are marked in blue. Coefficients determined by the first order conditions are green, those determined by the second order conditions are cyan, and those determined by the third order conditions, magenta.

[15-Mar-1999]