Why is a ski shaped... like a ski.

Are you talking about any ski with sidecut?

yeah

not versus flat skis however

Well skis with sidecut help you turn. I know first hand. I was on a old pair of straight skis and you don't use your edge to turn. You just hop around. Super tiring. Skis with sidecut just help you turn with your edge.

Your question is a tad bit broad without me giving out LOADS of detail on ski shapes.

Yes but what I am asking is why, from a carving a turn on hardpack situation, is a ski wider up front than in a tail and not just a perfect arc, or even narrower in the tip and wider in the tail. I know why skis are shaped, but what determines this specific, consistent formula of a wider tip than tail.

03gadeYes but what I am asking is why, from a carving a turn on hardpack situation, is a ski wider up front than in a tail and not just a perfect arc, or even narrower in the tip and wider in the tail. I know why skis are shaped, but what determines this specific, consistent formula of a wider tip than tail.

There's a lot of reasons for that. Consistency, weight, type of ski, shape/geometry, etc. just a bunch of different factors

The reason the center of sidecut is back has to to with the way your weight is positioned when you're in a carve, shins against the fronts of your boots. Most mounts are back from true center of the ski as well, hence the larger tips if your center of sidecut stays around the ball of your foot...

It's all about turn initiation. A wide tip, narrow waist, and a gradually widening tail is the best shape for a ski if your goal in construction is primarily carved turns. Rocker adds to this by bringing the contact point toward the skier and still allowing the ski to take advantage of its hour glass shape when edged.
I remember skiing on some old elans from the beginning of shaped skis, the tails were the same width as the tip maybe 100 and the underfoot was maybe 65. They carved one turn great but to get the skis to come into the next turn was very difficult. Because the ski was symmetric it just wanted to do donuts.