Polymer ammunition and cartridge having a convex primer insert

1. A primer insert for a polymer ammunition cartridge comprising:

a top surface;

a bottom surface opposite the top surface;

a coupling element that extends from the bottom surface, wherein the coupling element comprises an interior surface and an exterior surface, wherein the interior surface comprises:

a transition region that transitions from the bottom surface to a second segment wherein the transition region comprises a first transition segment 512 a extending from the second segment 508 and having a concave shape, a second transition segment 512 c extends from the first transition segment 512 a and having a convex shape and a third transition segment 512 e that extends from the second transition segment 512 c having a concave shape and extends to the bottom surface 34 ;

a first segment extending from the second segment and terminates at a tip, wherein the first segment has a first segment distance from 0.05 to 0.09 inches and the second segment has a second segment distance from 0.05 to 0.09 inches, wherein the second segment has a second segment angle from +3 to −3 degrees relative to the first segment angle and the first segment has a first segment angle from +6 to −6 degrees from perpendicular to the top surface;

a primer recess in the top surface that extends toward the bottom surface;

a primer flash aperture positioned in the primer recess through the bottom surface; and

a flash aperture groove in the primer recess and positioned around the primer flash aperture and adapted to receive a polymer overmolding through the primer flash aperture.

2. The primer insert of claim 1 , wherein the convex transition region comprises one or more straight segments that form a convex segment that transitions from the bottom surface to the second segment.

3. The primer insert of claim 1 , wherein the convex transition region comprises one or more curved segments that form a convex segment that transitions from the bottom surface to the second segment.

4. The primer insert of claim 3 , wherein the second segment has a second segment angle is about 1 degree and the first segment has a first segment is about 3 degrees.

5. The primer insert of claim 3 , wherein the second segment has a second segment angle is about 3 degrees and the first segment has a first segment is about 0 degrees.

6. The primer insert of claim 3 , further comprising an insert height that extends from the top surface to the tip, wherein the insert height is about 0.36 inches.

7. The primer insert of claim 3 , further comprising an insert height that extends from the top surface to the tip, wherein the insert height is about 0.38 inches.

8. The primer insert of claim 7 , wherein the second segment has a second segment angle is about 1 degree and the first segment has a first segment is about 3 degrees.

9. The primer insert of claim 7 , wherein the second segment has a second segment angle is about 3 degrees and the first segment has a first segment is about 0 degrees.

10. The primer insert of claim 7 , further comprising an insert height that extends from the top surface to the tip, wherein the insert height is about 0.36 inches.

11. The primer insert of claim 7 , further comprising an insert height that extends from the top surface to the tip, wherein the insert height is about 0.38 inches.

12. The primer insert of claim 11 , wherein the second segment has a second segment angle is about 1 degree and the first segment has a first segment is about 3 degrees.

13. The primer insert of claim 11 , wherein the second segment has a second segment angle is about 3 degrees and the first segment has a first segment is about 0 degrees.

14. The primer insert of claim 7 , further comprising an insert height that extends from the top surface to the tip, wherein the insert height is about 0.4 inches.

15. The primer insert of claim 3 , further comprising an insert height that extends from the top surface to the tip, wherein the insert height is about 0.4 inches.

16. The primer insert of claim 1 , wherein the convex transition region comprises one or more straight segments and one or more curved segments that form a convex segment that transitions from the bottom surface to the second segment.

17. The primer insert of claim 1 , wherein the first segment has a first segment distance is about 0.07 inches, the second segment has a second segment distance is about 0.07 inches and the convex transition region is convex with a radius of about 0.05.

18. The primer insert of claim 1 , wherein the first segment has a first segment distance is about 0.068 inches, the second segment has a second segment distance is about 0.073 inches and the convex transition region is convex with a radius of about 0.05.

19. The primer insert of claim 1 , wherein the first segment has a first segment distance is about 0.072 inches, the second segment has a second segment distance is about 0.068 inches and the convex transition region is convex with a radius of about 0.05.

20. The primer insert of claim 1 , wherein the first segment has a first segment distance is about 0.066 inches, the second segment has a second segment distance is about 0.075 inches and the convex transition region is convex with a radius of about 0.05.

21. The primer insert of claim 1 , wherein the first segment has a first segment distance is about 0.065 inches, the second segment has a second segment distance is about 0.074 inches and the convex transition region is convex with a radius of about 0.05.