The software described in these pages lets you explore the ** Bernstein-B
ézier form** of a bivariate polynomial. In
particular it allows you to construct **minimal determining
sets** for bivariate spline spaces defined on
triangulations. It's great for teaching the Bernstein-B
ézier form, and it can also serve as a research tool.

You do need a Java compatible browser.

If you can't wait to read the instructions and documentation click on the applet at the top of the page. Then start clicking on the control panel and the drawing window that will pop up. In particular click on the points in the triangles.

To whet your appetite, here is an example. The colorful
triangle nearby, like most of the figures on these pages,
were obtained with the applet. It shows the well known **
Clough-Tocher finite element** as it can be used for
interpolation. You see one large (macro) triangle that has
been divided into three smaller micro triangles. Bold black
lines indicate the boundaries of the triangles. Each of the
micro triangles is covered by a triangular mesh drawn with
thin black lines. The points marked with colored squares
and circles indicate coefficients of a bivariate cubic
polynomial defined on each micro triangle. The gray
quadrilaterals indicate smoothness conditions across the
boundaries between micro triangles, each involving four of
the coefficients. The red crosses are determined by the
interpolation conditions, the green crosses are determined
by the requirement to maintain differentiability across the
boundaries of individual macro triangles, and the filled red
and green circles are determined by internal smoothness
conditions.

This is only the tip of a large iceberg. If you want to find out more follow the links below. If you are new to the Bernstein-Bézier form, splines, and minimal determining sets, follow the links in the given sequence, otherwise, or if you are impatient, skip around as you wish!

[15-May-2020]