  Projections detect information about the size, geometric arrangement, and
dimension of sets. To approach this, one can study the energies of measures
supported on a set and the energies for the corresponding pushforward measures
on the projection side. For orthogonal projections, quantitative estimates rely
on a separation condition: most points are well-differentiated by most
projections. It turns out that this idea also applies to a broad class of
nonlinear projection-type operators satisfying a \textit{transversality
condition}. In this work, we establish that several important classes of
nonlinear projections are transversal. This leads to quantitative lower bounds
for decay rates for nonlinear variants of Favard length, including Favard curve
length (as well as a new generalization to higher dimensions, called Favard
surface length) and visibility measurements associated to radial projections.
As one application, we provide a simplified proof for the decay rate of the
Favard curve length of generations of the four corner Cantor set, first
established by Cladek, Davey, and Taylor.
