## Problem statement

In this section the influence of the cost function t on the smoothness of a trajectory C is analyzed. Here C is the solution of the functional minimization problem:

where Q is the set of all the possible curves in Q, {c } is the set of all the possible curves in Q between xi and x2 and p is the continuous metric:

The Fast Marching method computes a derivable solution C associated with the continuous metric p. Therefore, tools from differential geometry can be used to examine the curvature properties of C.

Let us define the curvature parameters considered here.

d 2C d2s 1

• Lower bound on the curvature radius along a curve C: Rmin(C) = infse[01]R(C(s))

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