Saddle (landform)

The saddle between two hills (or mountains) is the region surrounding the highest point of the lowest point on the line tracing the drainage divide (the col) connecting the peaks. When, and if, the saddle is navigable, even if only on foot, the saddle of a (optimal) pass between the two massifs, is the area generally found around the lowest route on which one could pass between the two summits, which includes that point which is a mathematically when graphed a relative high along one axis, and a relative low in the perpendicular axis, simultaneously; that point being by definition the col of the saddle.
Topography.
A saddle is the lowest area between two highlands (prominences or peaks) but has two wings which span the divide (the line between the two prominences) by crossing the divide at an angle, so is concurrently the local highpoint of the land surface which falls off in the down direction. That is, the drainage divide is a ridge along the high point of the saddle, as well as between the two peaks so defines the major reference axis. A saddle can vary from a sharp narrow gap to a broad comfortable sway-backed shallow valley so long as it is both the high point in the sloping faces descending to lower elevations and the low area between the two (or three or four.) flanking summits. Concurrently, along a different axis, it is the low point between two peaks, so is the likely 'optimal' high point in a pass if the saddle is traversed by a track, road or railway.
Saddles and cols.
The relationship between saddles and cols is not universally agreed. A col is sometimes defined as the lowest point on a saddle co-linear with the drainage divide that connects the peaks. Whittow describes a saddle as "low point or col on a ridge between two summits", whilst the Oxford Dictionary of English implies that a col is the lowest point on the saddle. Monkhouse describes a saddle as a ""broad, flat col in a ridge between two mountain summits."
The term col tends to be associated more with mountain, rather than hill, ranges.
The height of a summit above its highest saddle (called the key saddle) is effectively a measure of a hill's prominence, an important measure of the independence of its summit. Saddles lie on the line of the watershed between two hills.
Structural geology.
In structural geology, a saddle is a depression located along the axial trend of an anticline.
Mathematical saddles.
A 'saddle point' in mathematics derives its name from the fact that the prototypical example in two dimensions is a surface that "curves up" in one direction, and "curves down" in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle.