  As a parametric motion representation, B\'ezier curves have significant
applications in polynomial trajectory optimization for safe and smooth motion
planning of various robotic systems, including flying drones, autonomous
vehicles, and robotic manipulators. An essential component of B\'ezier curve
optimization is the optimization objective, as it significantly influences the
resulting robot motion. Standard physical optimization objectives, such as
minimizing total velocity, acceleration, jerk, and snap, are known to yield
quadratic optimization of B\'ezier curve control points. In this paper, we
present a unifying graph-theoretic perspective for defining and understanding
B\'ezier curve optimization objectives using a consensus distance of B\'ezier
control points derived based on their interaction graph Laplacian. In addition
to demonstrating how standard physical optimization objectives define a
consensus distance between B\'ezier control points, we also introduce geometric
and statistical optimization objectives as alternative consensus distances,
constructed using finite differencing and differential variance. To compare
these optimization objectives, we apply B\'ezier curve optimization over convex
polygonal safe corridors that are automatically constructed around a
maximal-clearance minimal-length reference path. We provide an explicit
analytical formulation for quadratic optimization of B\'ezier curves using
B\'ezier matrix operations. We conclude that the norm and variance of the
finite differences of B\'ezier control points lead to simpler and more
intuitive interaction graphs and optimization objectives compared to B\'ezier
derivative norms, despite having similar robot motion profiles.
