Modelling plants can provide examples in data fitting, geometry, trigonometry and calculus, as I will try to show.  Many of these images come from the Centre for Plant Architecture Informatics, University of Queensland web page whose URL is  www.cpai.uq.edu.au .  More details of this interesting area of research can be found there.
 
 

    Light penetration through the canopy

        Pesticide Deposition

  Schematic and actual plant

Doing the measuring

Doing the theory

We take the scattered data points and triangulate them
 

DesirableUndesirable
A good triangulation is one in which we don’t get long thin triangles.

To get a good triangulation we do a best least squares fit to the data thereby getting them roughly in the same plane.  We rotate the points so that the vertical direction  is the normal to the plane.

So we have the example Given the points

         x            y               z

    1.1000    1.0000       9.1000
    2.0000    3.9000       17.9000
    1.0000    2.4000       11.0000
     0            3.0000       9.9000
    4.1000    5.0000      26.0000

find the best fit by the function
z = ax + by + c .

Solution.  We find the best least squares fit to the data.  So we get the matrix equation
    1.1000    1.0000    1.0000
    2.0000    3.9000    1.0000
    1.0000    2.4000    1.0000
         0    3.0000    1.0000
    4.1000    5.0000    1.0000

9.1000
17.9000
11.0000
9.9000
 26.0000

and solve it by multiplying by A^{T  to get}

  ^{23.0200   31.8000    8.2000
   31.8000   55.9700   15.3000
    8.2000   15.3000    5.0000     as coefficient matrix and}

^{163.4100
  265.0100
   73.9000   as the right side and the system has solution  (a b c ) = 2.9471    2.0877    3.5583}
 
 

A good triangulation has the smallest angle of all triangles as large as possible.  One way of doing this is to take the quadrilateral formed by four points and put in a diagonal.  We then swap the diagonals and look at the angles again.  When we've done this we have to check the adjoining triangles.  So we have the example  The quadrilateral ABCD has angle A equal to 60 degrees, AB = 7,  AC = 9,  BD = 11, and CD = 10.  Which diagonal will  give two triangles with their smallest angle greatest?

CB is the correct diagonal,  ACB = 47.8  ABC = 72.2

BC = 8.8154, angle CDB = 45.57,  BCD = 73.3,  CBD = 60.75,  then we find that DAB = 29.0  which is smaller than any of the angels found so far so ABD will contain the smallest angle.
 

We can use piecewise linear functions but the surface is then has sharp corners so we use instead piecewise cubics

Suppose we have the leaves on a plant how can we use the knowledge.

We have to do 2 dimensional problems, then we can model the plant with the simple scheme

Applying a pesticide or just watering the plant

Quantifying Sunshine or Rainfall

To make our model reasonable the size of the line segments should reflect the size of the leaves.  So each stick length should be an average leaf length.  Suppose we measure a leaf,  we can work out its average length by fitting a curve to its perimeter and then integrating to get the area, then we divide by the width to get an average length.

Thus we have the example:

Given the data points which are the perimeter of a certain leaf determine an average length for the leaf.
 

From the diagram it can be seen that the two sides of the leaf are roughly parabolic, so we will fit a parabola to the points on each side .

x                       0            0.2000      0.4900     0.7300       1.0000 y(upper)            0             0.1600     0.2499     0.1971         0
y(lower)      -0.0955       - 0.2264   - 0.2999   - 0.2567     - 0.0955

We could also use Simpson's rule on the points we have or use piecewise linears, i.e. the Trapezoidal rule.  We get interesting answers.
 
 

So now we can take a simple model of a nozzle with a fixed spray angle and ask how much of the material emanating from the nozzle gets on the leaves.  We have the following situation:

So the model problems become:

 Given the dimensions of the plant, the lengths GC, GT, GM, and GL and the angles LGT, TGM and the nozzle spraying angle ANE,
(i)   Plot the ratio of the lengths  AB+DE to BD  as the height of the nozzle is varied.
(ii)  What is the height at which one of the leaves prevents the material from reaching the ground ?
(iii)  What is the height at which both leaves intercept the material before it reaches the ground?