Tangent-line finite-field Kakeya construction ($K=\{(x,y):x^2-y\text{ is a square}\}\cup \{x=0\}$)

Work in the affine plane over a finite field ${\mathbb{F}}_q$ of odd characteristic. The union of the affine lines $y=mx-m^2/4$, for $m\in {\mathbb{F}}_q$, contains a line of every finite slope. Completing the square givesFor a fixed $x$, the union therefore contains exactly the $(q+1)/2$ values of $y$ for which $x^2-y$ is a square, including zero. Adding the vertical affine line $x=0$ completes all directions. The resulting finite-field Kakeya set has cardinalityIts density tends to one half as $q$ grows through odd prime powers. Geometrically the finite-slope lines are the tangents to the parabola $y=x^2$, because the tangent at $x=m/2$ has the displayed equation.