I imagine you mean circular arcs.
Here are (Fig. 1) two ovals close to ellipses with (only) four circular arcs (an arc centered in $A$ with radius $AB$ ; another one centered $C$ and radius $CD$, and their symmetrical arcs). The left oval with $\pi/2$ "openings", the right one with $\pi/3$ and $2 \pi/3$ "openings".
An important feature of both ovals is that there is a smooth connection of the arcs at meeting points like $D$ : the tangent in $D$ considered as a endpoint coincides with the tangent in $D$ at the beginning of the other arc. This is due to the following property : two circular arcs with centers $C_1$ and $C_2$ are smoothly joined (in the sense given above) in a point $P$ iff $C_1, \ C_2, $ and $P$ are aligned.
These solutions can be traced back to Renaissance, but probably were known since Antiquity. One finds them well described in the works of Serlio, an italian architect working in the 1550s.
The proximity of the first oval with the ellipse represented by a dotted pink line (with foci featured as black points) is very good as one can see on Fig. 2 :
One can object that that these constructions are for a fixed eccentricity. A way to palliate this drawback is by extending or shrinking the radii of circular arcs by a same amount in the way described on Fig. 3.
See the following references :
About Serlio's methods :
About the kind of oval "plazzas" one finds in Washington DC or in front of Saint Peter's cathedral in Rome, see :