spray patterns
Hi. You ask a question that takes some explaination. I am not certain if I am clear on your question, but here goes.
First, spray heads come in several fixed patterns. Full (F) , half (H), third (T) and quarter (Q): named in reference to a circle. Additionally, there any number of "strip" sprayers. One covers a more or less rectangular pattern, which may satisfy your request for an eliptical coverage. Other strip heads sit on one side of a square or rectangle. Hard to describe, but if you navigate around the Rainbird site, you should find diagrams.
Each head is designed for a certain pattern (as the prior paragraph explains) and a coverage distance. Referred to as the radius for the F,H,T,Q patterns. So, for example, the various companies manufacture them in 8,10,12,15 feet covering sizes. The strip types also come in a variety of ranges. Check out sprinklerwarehouse.com, the sponser of this site for ideas.
Matching the pattern and "radius" of the spray is usually planned before installation. Retrofiting is a little more difficult. Plan to have the spray coverage "head to head" which mean the base of one head is being splattered with water from the next head.
Water conservation also deals with overspray within the covered area, and not just what hits the pavement. Designers try to match the paercipitation rate, so IDEALLY, all the grass gets the same amount of water during the water cycle. You can check this by placing empty straight sided cans (cat food) at various places in the yard and measuring how much water is in each after a given period of time. Notice I said IDEALLY.
You stated your lawn is gettting dry spots in the midle. A catch test will tell you if it is from lack of water, or maybe if it is getting ample moisture, another problem.
Hey, hope that helps, and if not write again, and I'll do my best. Jeff