Snap - Together Mathematics

The mathematical techniques of the previous chapters permit controlling graphical objects by specifying the derivatives of functions of their parameters. In this chapter, we consider techniques for defining and evaluating these functions. The challenge stems from the dynamic nature of interactive systems: objects change in response to system actions as the system runs. This means that the functions that define controls must be created on the fly, in response to user actions. In order to effectively implement the mathematical calculations, we must evaluate the functions and their derivatives rapidly. This chapter presents Snap-Together Mathematics, a toolkit for dynamically defining functions and rapidly evaluating them and their derivatives. With the differential approach, objects provide their attributes as output connectors for other objects to use, and as attachment points for controllers that will control the objects. These connectors compute functions of the objects’ parameters and input dependencies, and must support the operations of attaching other object inputs and controllers. Snap-Together Mathematics provides a mechanism for realizing the connectors. “Wiring” connector outputs to inputs builds new, more complicated functions from the elements being assembled. Building a new function may happen any time a new object, constraint or control is defined. It would be unacceptable if building a function required an extensive symbolic math computation or for the program to be recompiled and re-linked. Snap-Together Mathematics explicitly represents the expression graph of connected functional elements as C++ data structures. A connector is simply the output of a node in the expression graph. We will call the nodes blocks. To wire an input, it merely needs to be given a reference to some output. Snap-Together Mathematicshas efficient mechanisms for evaluating the values and derivatives of nodes by traversing the graphs. For example, consider an expression graph that represents connecting the endpoints of two line segments together with an attachment constraint, as shown in schematically Figure 5.1. In this figure, the line segments have a state vector to store their parameters (x; y; ; l), and provide the positions of their endpoints as connectors. The inputs to the attachment constraint are plugged into these output connectors. The output of the