studies have shown that a straighter ski is better for high speeds, hence DH skis are very straight. Slalom skis are quite shapped, and the tail and tip are both significantly wider than the waist.
For park, the common shape is moderate to straight. Some sidecut for turning ease. Also a mid-fat width for stability. Park skis range from 72mm under foot to 80mm under foot. Mogul skis and slalom skis are in the low 60's, All mountain skis are in the low, mid and high 60's, carving skis are in the mid 60's and DH skis are in the high 60's to low 70's. Powder skis have a large range, and usually in the mid 80's on up to the high 90's.
the tip and tail widths are based off of the sidecut radius and the length of the ski. The sidecut radius is the radius of the circle that would be made if the arc of the sidecut was completed to form a perfect circle. If you havnt take geometry you may get confused.
The sidecut radius is different on all models of skis. It is based on flex, camber, strength, intended use (type of skiing), and the customer skill level target. And from there the arc of the sidecut, the binding mount is placed.
Hope this answers some questions about sidecut.
'now the rule of thumb here....'
'RULE OF THUMB! In the early 1900's it was legal for a man to beat his wife, if he used a stick no wider than his thumb!'
'That sucks, you cant do much with that, should have been the rule of wrist!'