John O'Connor and Edmund Robertson

We have implemented some of these ideas by allowing users to alter pictures in real time by varying parameters in mathematical functions and seeing the pictures alter "before their eyes". This allows much greater flexibility than Thompson was able to achieve, since he was forced to do his transformations laboriously by hand.

We have experimented to show that although Thompson used a wide variety of transformations, most of his affects can be achieved by quadratic maps from the plane to itself. That is, maps of the form f(x,y) = (p(x,y), q(x,y)) where p and q are polynomials of degree 2 in two variables. This means that the user has the freedom to vary 10 parameters.

Under these circumstances, working with so many degrees of freedom, the ability to vary the parameters continuously and see the results of such variation, is vital to keep control of how the picture is changing. Since the way one believes that natural selection has worked to produce changes in organisms is in the continuous fashion, it is very satisfying to feel that one has the ability to transform things in the same sort of way.

We believe that this application of interactive computer technology gives insights into biological processes which are not achievable in other ways. The fact that D'Arcy Thompson held a chair here in St Andrews in Scotland for a record-breaking period gives us particular satisfaction.