What is this curve formed by latticing lines from the $x$ and $y$ axes? Consider the following shape which is produced by dividing the line between $0$ and $1$ on $x$ and $y$ axes into $n=16$ parts. 
Question 1: What is the curve $f$ when $n\rightarrow \infty$?


Update: According to the answers this curve is not a part of a circle but with a very similar properties and behavior. In the other words this fact shows that how one can produce a "pseudo-cycle" with equation $x^{\frac{1}{2}}+y^{\frac{1}{2}}=1$ from some simple geometric objects (lines) by a limit construction. 
Question 2: Is there a similar "limit construction by lines" like above drawing for producing a circle?

The OP's curve is (a portion of) the parabola with $(1/2,1/2)$ for its focus and $x+y=0$ for its directrix.  With a slight modification (see below), the lines in the OP's drawing are tangent lines to the parabola, which can be thought of as (origami) crease lines created when the focus is "folded" to lie atop various points along the directrix.  
The slight modification is this:  The tangent lines for the parabola run from $(0,t)$ to $(1-t,0)$, but it appears from the drawing that the OP is connecting $(0,{k\over16})$ to $({17-k\over16},0)$ rather than $(1-{k\over16},0)$.  In the limit, this doesn't matter, but it does make a small difference along the way.
