Surgeon needed to get the corners of my eyes fixed to look sharper instead of round. (Photos)

I believe you want more almond shaped eyes which involves elevating the lower eyelids, NOT just canthoplasty, in order to have natural, long-lasting results. Consultation is necessary to determine if you are a good candidate.

The outer half of the lower eyelid margin is bow around the curvature of the eye.  This cannot be fixed by simply doing a lateral cathoplasty to tighten the lower eyelid.  Doing a lateral canthoplasty with no not other measures will result in the lower eyelid falling lower under the curvature of the eye.  You will not like that.  Some might offer you a cosmetic orbital decompression.  I personally do not think the risk of blindness is worth having this done.  The risk of permanent visual loss from an orbital decompression is estimated to be between 1 in 750 and 1 in 1500.  To me, for a cosmetic surgery, that risk is too high.  A better alternative is the use of a spacer graft to lift the lower eyelid in conjunction with the canthoplasty.  Ultimately there is no substitute for a personal consultation.

Hello, you should visit your local oculoplastic surgeon to get a full evaluation, you may be a candidate for a surgical lower lid tightening to lift the lower lid and produce a sharper corner. 

Thank you for your question and photo. I do see the issue that is bothering you. Fortunately this can be corrected. Oculoplastic surgeons are specialists at addressing what is called the lateral canthal angle. This is the outer corner where your two eyelids come together. Achieving a sharp lateral canthal angle is a youthful and happy look and can likely be achieved in your situation. 
Consider seeing an ASOPRS oculoplastic surgeon for a consultation. Best of luck!