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Extra resources for A Rational Finite Element Basis
The linear forms in the denominator are appropriate combinations of the polygon exterior diagonals, these combinations being different for each wedge. 5. They may be renormalized to yield degree zero approximation over the polygon, but degree one has not been achieved with these wedges. This wedge construction is illustrated for the pentagon in Fig. 9.
To obtain a system of projective coordinates, we first select four points, no three of which are collinear. We then choose three of these points as vertices of a "triangle of reference". The projective coordinates of these three points are equal to their barycentric coordinates. The fourth point is the "unit" point with projective coordinates defined 41 THE QUADRILATERAL as (l,l,l). The barycentric coordinates of the unit point are determined uniquely (up to a common multiplier, of course) by the location of the other three points.
In this connection, Coxeter's (1961) "Introduction to Geometry" is invaluable. We quote *This section may be skipped on a first reading. Results obtained here are referred to in Chapter 9. 40 RATIONAL FINITE ELEMENT BASIS a few definitions from it: "If four points in a plane are joined in pairs by six distinct lines, they are called the vertices of a complete quadrangle, and the lines are its six sides. Two sides are said to be opposite if they have no common vertex. 234-237 of Coxeter's book. " The homogeneous coordinates (k,m,n) linear in x and y which assume the values (a,O,O), (O,b,O) and (O,O,c) at the vertices of a triangle, where a, b, and c are arbitrary, are called the "barycentric" coordinates of the triangle.