Carving ski sidecut

Why does a ski cut a clean arc when the skier only tips it on edge and never steers it sideways?

Watch a good skier from behind and the strange thing is what is missing. No push, no scrape, no sideways shove of the tails — the skis simply lie over onto their edges, and the slope behind is left with two pencil-thin curving grooves. Whatever turned the skier, it was not steering.

Which invites an uncomfortable question. If nothing was steered, what chose the radius? The arc has a definite size, and something must have set it.

Start with the ski's shape. From above it is not a rectangle: wide at the tip, narrow at the waist, wide again at the tail. Trace one edge and you are tracing a very shallow circular arc, whose radius — the sidecut radius — is stamped on most modern skis: roughly a dozen metres for a slalom ski, several times that for a downhill ski. (Those figures are recalled from equipment specifications; the argument does not depend on them.)

Put that ski flat on hard snow and only the tip and tail touch. The waist edge hangs above the snow by the sidecut depth, cutting nothing.

Tip it on edge by an angle, and press. Two things happen at once, and the whole answer is in how they add.

First, the geometry tilts. Call the sidecut depth at the waist s, measured across the ski in the ski's own base plane. Tip the ski by angle θ and that transverse offset splits: the waist edge is now s·sin θ above the snow, and s·cos θ across it.

Second, the ski bends. To close that gap it must bow out of its own base plane until the waist edge reaches the snow — reverse camber. Here is the step worth slowing down for: the bend pushes the waist along the base plane's normal, and that normal is tilted too, so it carries the waist partly downward and partly sideways, in the ratio cos θ to sin θ.

How much bend? Enough to drop s·sin θ vertically, so the displacement along the normal is s·sin θ / cos θ. Which drags the waist a further s·sin²θ / cos θ sideways — and crucially, sideways in the same direction the sidecut had already offset it.

So add the two sideways contributions:

s·cos θ + s·sin²θ / cos θ = s·(cos²θ + sin²θ) / cos θ = s / cos θ

The groove is a deeper arc than the one ground into the ski. Same chord, deeper bow, so a smaller radius: the turn radius is the sidecut radius multiplied by cos θ. Tip the ski further and the arc tightens, without anyone steering anything.

Sensible sizes? At 60 degrees cos θ is exactly one half, so a twelve-metre ski carves a six-metre arc — a real turn.

But notice what has not been said: nothing mentions speed. Does speed matter? It must, since something has to supply the sideways force. A skier leaning at angle θ from vertical is balanced in a turn of radius r when tan θ = v² / (g·r). Substitute r = R·cos θ and the cos θ cancels: sin θ = v² / (g·R). Speed does not choose the radius; it chooses the edge angle, and the geometry hands back the radius.

That has a limit built in. A sine cannot exceed one, so a skier whose ski and body lean as one unit can carve only up to about the square root of g·R — for a twelve-metre ski, roughly eleven metres per second. Racers go far faster, which is exactly why they angulate: bending at hip and knee lets the ski's edge angle exceed the body's inclination, breaking the assumption that the two are equal.

The cos θ relation is a first-order idealisation, and it is worth being clear about what it leaves out. It treats the ski as bending only enough to make contact, when in a real turn the ski is loaded by the turn's own forces and bends further, so carved arcs are typically tighter than the geometry alone predicts. It treats the snow as a rigid plane, when in fact the edge cuts a groove of finite depth and softer snow deforms rather than holds. It ignores torsional stiffness, which decides whether the tip and tail edge at the same angle as the waist. And the balance equation above treats the skier as a point mass, which no skier is.

It is also worth separating this from the other way a curved shape turns something. A sail or a keel deflects a fluid and takes lift from the pressure difference; nothing of that kind happens here. A ski carves because a solid edge cuts a slot in a solid and the slot's walls push sideways — which is why the technique needs snow firm enough to hold a wall, and fails in deep powder.

What would falsify the account? Its sharpest prediction is that the arc's size is set by shape and edge angle and not by speed. Film two carved tracks at very different speeds with the edge angle held equal and measure the grooves: the geometry says the radii should match. If the groove opened out as speed rose at fixed edge angle, the arc would be set by force balance and skidding rather than by geometry, and this explanation would be wrong.

• Why torsional stiffness decides whether tip, waist and tail edge equally.
• How angulation separates edge angle from body lean, and what that costs in balance.

Every term the collection defines is gathered in the glossary.