Katarina, Possible new build path?

So i was curious about which is better, AD or AP. I decided i'd do the math to the best of my ability and also did Hybrid Build to. Keep in mind i chose the most optimal build not the build for getting the most stats, as in if the item isn't useful i didn't use it in the example.

Full AP Build: Gunblade(40 AD, 80 AP), DeathCap(120+35% Total AP), Zhonya's(70 AP), Abyssal Scepter(60 AP) and Void Staff(80 AP)

40 AD and 553 AP

Passive Damage Per Dagger Pick Up: 287+50+553= 890 Q Damage: 195+165= 360 E Damage:90+138+99.4= 327.4 R Damage: 750+1576.05+132= 2458 One Auto Attack: 153 Gunblade: 250+165= 415

Total: 4603.4

Full AD Build: Edge of Night(60 AD), Mercurial Scimitar(65 AD), Maw of Malmortius(50 AD), Youmu's Ghost Blade(60 AD), Gunblade(40 AD, 80 AP).

275 Bonus AD(388 Total) and 80 AP

Passive Damage Per Dagger Pick Up: 287+343.74+80= 710 Q Damage: 195+24=209 E Damage: 90+20+252= 360 R Damage: 750+228+907= 1885 One Auto Attack: 388 Gunblade: 250+24=274

Total: 3826

Hybrid Build: Gunblade(40 AD, 80 AP), Deathcap(120+35% AP), Zhonya's(70 AP), Mercurial(65 AD), Edge of Night(60 AD)

165 Bonus AD(278 Total), 364 AP

Passive Damage Per Dagger Pick Up: 287+206+364= 857 Q Damage: 195+109.2= 304.2 E Damage: 90+91+180= 360 R Damage: 750+1037.4+544.5= 2331.9 One Auto Attack: 278 Gunblade: 250+109.2= 359.2

Total: 4130.3 Note: If i used something like Luden's instead of Zhonya's she'd have an extra 30 AP which might make up for the difference in the Hybrid and AP builds a little better.

Conclusion: Hybrid Build offer's the most defenses, not much can bust through a Zhonya's, Scimitar and Edge of Night but you'll lose a total of 497 damage, but that's pretty negligible considering you can make up for that with auto attacks on each E if you practice it.

Also, for boots i'd most likely go Sorc Shoes, but i'v been thinking of taking Tabi instead because i don't like being shredded by ADC's.

So which do you guys think is better?