A319
Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument
Situational Awareness; Human-Machine Interface
Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert; Flight Crew Became Reoriented; Flight Crew Regained Aircraft Control; Flight Crew Took Evasive Action
Human Factors; Procedure
We were flying the VOR 13L approach into JFK. I was the pilot flying; it was a night VMC approach. As I crossed the MAP inbound to the runway; I disconnected the AP (autopilot) and instructed the FD (Flight Display) to be turned off; and the FPV (Flight Path Vector) to be enabled. While turning for the runway and following the approach light system around the corner I erroneously continued the descent below DDA (Derived Decision Altitude). I noticed I was low; around 500 ft. AGL; and corrected our altitude back to a proper glide path angle.  ATC notified us of a low altitude alert just after I became aware of my mistake.In the approach set-up I had initially considered conducting an RNP (Required Navigational Performance) approach to that runway instead; until realizing that the aircraft was not capable of it. During the briefing I had discussed a notation from the FAF (Final Approach Fix) to the MAP (Missed Approach Point) that indicated a proper 3 degree glide path equated to 2.2 nm from the MAP.  Subconsciously I interpreted this as a 3 degree path to the runway; and did not do the math on the distance from the MAP to the runway. Therefore; when crossing the MAP I continued the descent without consideration that I needed to hold DDA altitude longer. Apart from better airmanship; I think there could be a note about this in the company pages. (If one exists; I didn't see it during the set-up).  We did a thorough briefing and discussed a lot of threats; but this one got overlooked.
We were concentrating on the flight path; sighted the runway; and started a visual descent to the runway. At some point; the Tower controller gave us an altitude alert. That caused us to recognize we were low on the VASI/PAPI; so we adjusted the descent to correct that; and landed normally.We were both trying to be conscientious about the horizontal flight path; and allowed the vertical path to go somewhat lower than was optimal.I think it was just a long time since either of us had flown this approach; much less at night; and we over-emphasized the route to the detriment of the glide path. Both of us used to do JFK routinely; but that was years ago. So this was (for me) a heads-up reminder to keep the big picture in mind.
A319 flight crew reported descending below final approach path into JFK.