The radius of a turn is different, and it varies as the turn is being executed with the edge angle being used by the skier and with the amount of reverse camber in the ski during the turn. However, it is not too hard to calculate an approximation of the the radius of a "pure carved turn".
Geometrically, a carved turn is the composition of two curves, the sidecut of the ski and the deformed sidecut of the ski when its camber is reversed as it is pressed onto the snow. To find these curve, I assumed that the carving ski looks like a segment of a cylinder, and that the ski hill is an inclined plane. The arc the carving ski follows is the curve at the intersection of the cylinder and the inclined plane.
It will help you understand the formula if you'll do the following simple geometric demonstration. Take two identical rectangular pieces of paper. With scissors, make a large radius (shallow) sidecut on one of the long sides of piece 1. Make shorter radius (deeper) sidecut on the corresponding side of piece 2. Now hold each piece with the sidecut against a tabletop so that the edge of the sidecut is in full contact with the table. Notice the decambering effect of pushing the edge into contact with the table, and notice how changing the angle changes the radius of the approximate cylinder.
To use this approximation, you have to calculate the radius of the cylinder, and this involves the geometry of the ski. The approximation I make is not too bad if you assume that the ski does not twist, but it does calculate the radius of the carved turn when you are traversing the hill.
To approximate the cylinder radius, I computed the distance from the ski edge to a chord extended from the widest point on the ski tip to the widest point on the ski tail. Then I projected that point to the inclined plane given the inclination angle of the pressing force. If you think about this, its easy to see that there will be a minimum angle where this actually can be done. Thus all these calculations should be only considered for angles above that minimum.
The variables are:
- phi = angle swept out by the sidecut of the ski, i.e. Rsc*phi is the length of the edge from the widest point on the tip to the widest point on the tail.
- d_rc = reverse camber distance, distance from the flat ski to the edge when the ski is pushed into the snow
- alpha = inclination angle of the skiers leg.
- theta = inclination angle of the slope.
- Rcyl = cylinder radius.
- Rturn = turn radius
Here is the result
- d = Rsc * (1-cos(phi/2))
- d_rc = d * cot(alpha-theta)
- Rcyl = (L^2 + 4*d_rc^2)/(8*d_rc)
- Rturn = Rcyl * sin(alpha+theta)
I would say that the arms change the d_rc with this ski