If only Elmer Fudd had known to apply the pesticides this week, not next!
But Elmer’s dilemma is that if he sprays too early, before new leaves have come out, those won’t be protected and if he waits too long the insects will have started their work of destruction.

If only Elmer had come to the Centre for Plant Architecture Informatics at the Department of Mathematics, University of Queensland. Here researchers with diverse backgrounds in Botany, Mathematics, Biology and Computer Science, have joined forces to develop techniques for creating virtual plants, i.e. plants that behave like real plants but live in the computer. Virtual plant research will help to reduce expensive and time consuming field tests by using computer simulated plant experiments.

Birgit is developing mathematical algorithms, which can generate a
3D representation of a leaf.

For instance if we model the leaf surface using the function

F(x,y) = ax+by+c

there are 3 unknown parameters a,b,c. Hence we require three data points, p1, p2, p3, on the leaf surface to generate three simultaneous equations and thus determine the unknowns.

So three data points on the leaf surface can be used to generate a plane (see diagram).

However, real leaves don’t have a triangular boundary and are not flat, so this simple planar model is not accurate enough for poor Elmer Fudd. Instead we collect a large number of points, then, using three points at a time, generate
small triangular surfaces which collectively model the leaf. This process is called triangulation.

Finding a triangulation is easy, but our goal is to find the best approximation to the surface of the leaf, which means we need an optimal triangulation; one in which the triangles are as equi-angular (i.e. having all the angles equal) and contain as few long, thin triangles as possible.

So we start with a triangulation, then each pair of adjoining triangles forms a quadrilateral. For each quadrilateral we measure the interior angles of the two triangles and find the minimum angle. Swap the diagonal of the quadrilateral and measure the angles of the two new triangles. If the minimum angle in this new triangulation is larger, then this is a better triangulation and we retain it. If not,
go back to the old triangulation. This is now our optimal triangulation which in
turn defines our surface.

As you can see, the surface is continuous but not smooth along the triangle edges. A smoother surface can be obtained by using a cubic polynomial instead
of the linear polynomial.

We have created a virtual leaf! It can now be added to the virtual plant!