Macro Elements on the Worsey-Farin Split

The Worsey-Farin Macro Panel lets you impose symmetric super smoothness conditions on the Worsey-Farin split and then check if a macro element can be built with those conditions.

Because of the split faces of the macro tetrahedron the panel works somewhat differently than the other panels. We refer to the vertices of the macro tetrahedron as its corners and the centroids of its faces as the face centroids. Note that both corners and face centroids are boundary vertices of the macro tetrahedron.

To obtain the macro panel select the configuration "Worsey-Farin", and clock on the button "macro elements" in the control panel. The buttons on the macro panel have the following effects:

Hide cause the macro panel to disappear. To make it reappear click again on "macro elements".
Draw causes the current smoothness conditions to be drawn initially, or if the linear algebra in the drawing window has been initialized. (Otherwise the smoothness conditions are drawn automatically whenever they are changed.)
Init initializes the linear algebra (and exits super selection mode) in the drawing window.
V initializes the linear algebra if necessary and attempts to set the data in the corner balls. If is the smoothness around the corners, then the corresponding number of data is

$\displaystyle M_V=4{R+3 \choose 3}.$
(The code actually also includes any inactive points (that do not enter any smoothness conditions) since these must be imposed.)
E imposes the natural data for the six boundary edges connecting the corners, given the vertex data. Of these there are

$\displaystyle M_E = 6\sum_{\rho=0}^E (d-2R+\rho-1)(\rho+1)$
where is the degree of smoothness imposed around those edges. This number is the same as for the Clough-Tocher and Morgan-Scott Splits.
F imposes additionally all points in the boundary faces of the macro tetrahedron and the first layers moving inwards, given the vertex and edge data. A general formula for the natural number of such data is currently unknown, and the code will simply report the total number of conditions that have been imposed.
Clear removes all super smoothness conditions.
The status window gives information about the current analysis in some situations.

Super smoothness conditions can be imposed using the text fields in the second row of the macro panel, and the buttons to either side of those text fields. The text fields lists the additional degree of smoothness. So if all the text fields show zero then the spline is the ordinary space without super smoothness conditions at all.

The following groups of super smoothness conditions are available:

$V_{0}$ corner
$V_{4}$ Centroid
$V_{5}$ face centroid
$E_{01}$ edges connecting corners
$E_{04$ edges connecting corners and the centroid
$E_{05}$ edges connecting corners and face centroids.
$E_{45}$ edges connecting the centroid and face centroids
$F_{014}$ a face containing two corners and the centroid
$F_{045}$ a face containing a corner, the centroid, and a face centroid