Many sensor-based techniques, such as those in sections 2.3.3 and 5.2.5, are essentially curve-tracing algorithms. In both cases, the robot is, in a sense, "determining" the curve as it is being traced. Such techniques relied on two fundamental principles: the curve being traced exists and under the "right conditions," the curve can be traced with simple predictor-corrector techniques. These principles rested on two theorems, the implicit function theorem, and the Newton-Raphson convergence theorem, described below.

D.1 Implicit Function Theorem

Consider a smooth function of multiple variables, f( x, y), and consider the surface that is defined by the equation f( x, y) = z ₀ for some fixed z ⁰. Under certain conditions, this surface can be used to write a new function that defines the y variables in terms of the x variables, i.e., y = g( x, z ₀). The theorem that states these conditions is called the implicit function theorem.

THEOREM D.1.1: (Implicit Function Theorem)

Let be a smooth vector-valued function, f( x, y). Assume that D _yf( x ₀, y ₀) is invertible for some . Then there exist neighborhoods X ₀ of x ₀ and Z ₀ of f( x ₀, y ₀) and a unique, smooth map such that