  We propose the concepts of intersection distribution and non-hitting index,
which can be viewed from two related perspectives. The first one concerns a
point set $S$ of size $q+1$ in the classical projective plane $PG(2,q)$, where
the intersection distribution of $S$ indicates the intersection pattern between
$S$ and the lines in $PG(2,q)$. The second one relates to a polynomial $f$ over
a finite field $\mathbb{F}_q$, where the intersection distribution of $f$
records an overall distribution property of a collection of polynomials
$\{f(x)+cx \mid c \in \mathbb{F}_q\}$. These two perspectives are closely
related, in the sense that each polynomial produces a $(q+1)$-set in a
canonical way and conversely, each $(q+1)$-set with certain property has a
polynomial representation. Indeed, the intersection distribution provides a new
angle to distinguish polynomials over finite fields, based on the geometric
property of the corresponding $(q+1)$-sets. Among the intersection
distribution, we identify a particularly interesting quantity named non-hitting
index. For a point set $S$, its non-hitting index counts the number of lines in
$PG(2,q)$ which do not hit $S$. For a polynomial $f$ over a finite field
$\mathbb{F}_q$, its non-hitting index gives the summation of the sizes of $q$
value sets $\{f(x)+cx \mid x \in \mathbb{F}_q\}$, where $c \in \mathbb{F}_q$.
We derive bounds on the non-hitting index and show that the non-hitting index
contains much information about the corresponding set and the polynomial. More
precisely, using a geometric approach, we show that the non-hitting index is
sufficient to characterize the corresponding point set and the polynomial when
it is close to the lower and upper bounds. Moreover, we employ an algebraic
approach to derive the intersection distribution of several families of point
sets and polynomials, and compute the sizes of related Kakeya sets in affine
planes.
