Flight Crew Flight Engineer; Flight Crew Flight Instructor; Flight Crew Instrument; Flight Crew Multiengine
Flight Crew Last 90 Days 100; Flight Crew Total 6800; Flight Crew Type 1000
Communication Breakdown; Situational Awareness
Conflict NMAC; Deviation / Discrepancy - Procedural Other / Unknown
Flight Crew Took Evasive Action
Environment - Non Weather Related; Airspace Structure; Human Factors
While taking off from Runway X at ZZZ airport on up wind leg about 600 ft. AGL; a Cirrus flew roughly from my right side close to right angles to my flight path and about 50 ft. below me.  The Cirrus was about 310 degrees while my path was about 20 degrees.  Did not see the Cirrus until he was passing me. He passed about 100 ft. in front of me.  My student and I were in the traffic pattern at ZZZ Airport for the previous half hour announcing our positions.  We were not expecting an aircraft to be in our departure path 400 ft. below the pattern altitude near where we would be turning cross wind.  When we informed the Cirrus pilot of the near miss he said that he reported his position previously.  However that did not give him the right to basically enter the pattern on a cross wind 400 ft. below the pattern altitude.  By passing in front of us from our 'blind spot' he committed an act of reckless endangerment.  Either deliberately or by failing to pay attention to the aircraft in the traffic pattern and follow them in per AIM procedures.  I believe the Cirrus eventually landed on Runway XY at ZZZ.  With two intersecting runways and traffic often landing on both runways; plus traffic making instrument approaches at ZZZ; I believe that providing ZZZ with a control tower would greatly improve safety.
Cessna 172 Instructor reported a NMAC at 600 ft. during takeoff.