A Rational Finite Element Basis by Wachspress

By Wachspress

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Hence, property (5) is established and our candidates in Eq. ( 2 . 5. A quadrilateral wedge is sketched in Fig. 2 . 5 . (3;4) or (1;4) (2;3), and the construction lines indicate how this property is used in sketching the wedge surface. - Fig. 5. Quadrilateral wedge W 1' A m A L COORDINATES AS LIMITS OF RATIONAL WEDGES As one of the interior angles of a quadrilateral is increased to IT,the four rational wedges approach functions, only one of which in this ill-set limit 37 THE QUADRILATERAL is continuous.

It follows from J2 - q 1: k that s = m n , (k4p + m4q + n4r). 12) THE QUADRILATERAL It is not difficult to prove that for our convex quadrilateral, k4>l, m4<0, n4<0, and that within the quadrilateral s>l. Hence, the absolute value of the Thus the absolute determinant of J2 is k4m4n4/s3 value of the Jacobian of the transformation from (x,y) to (q,r) coordinates is . 14) where K567 is the area of the triangle of reference whose vertices are the diagonal points of the quadrangle. 15) ( 4 ; l ) = c (m k-k4m), 1 (5;6) = c5k.

To evaluate integrals in the projective coordinates and relate them to integrals in (x,y), we must find the Jacobian of the transformation. It is convenient to determine the Jacobian as the product of two Jacobians: J = J1J2, where J1 is for the transformation from (x,y) to the baryeentric coordinates of the diagonal-point-triangle of the quadrilateral and where J2 is for the transformation from barycentric to projective coordinates. The absolute value of the determinant of J1 is twice the area of the triangle of reference: ..

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