Macro Elements on the double Clough-Tocher Split

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

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

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 types of smoothness conditions that can be imposed are listed below. To understand the labels on the macro panel recall the labeling of the vertices: $ V_0$, $ V_1$, $ V_2$, and $ V_3$ are the boundary vertices,

$\displaystyle V_4 = \frac{V_0 + V_1 + V_2 + V_3}{4}$

is the centroid, and $ V_{5+i}$ is the subcentroid opposite $ V_i$, $ i
= 0,1,2,3$. For example,

$\displaystyle V_5 = \frac{V_4 + V_1+ V_2 + V_3}{4}.$

The phrase "$ V_5$" below refers to super smoothness at all four subcentroids, and $ E_{45}$ refers to super smoothness along edges for the centroid the subcentroid since $ V_4$ is the centroid, and $ V_5$ is a subcentroid.

The following groups of super smoothness conditions are available:

Natural Data

The natural data are the same as for the single Clough-Tocher split.

C2 Macro Schemes

The following links let you explore the possibility of constructing a C2 macro scheme of degree 9.