A Geometric Approach to Saddle Points of Surfaces

What is a saddle point of a surface in 3-space? A reasonable answer is: a saddle point is like the center point of a horse saddle or the low point of a ridge joining two peaks. In other words, a saddle point is that peculiar point on the surface which is at once a peak along a path on the surface and a dip along another path on the surface. Another answer that is mundane but more likely to fetch points in a Calculus test is as follows. A saddle point of a real-valued function of two real variables is a critical point (that is, a point where the gradient vanishes) which is not a local extremum. The first answer gives an intuitive description of a saddle point, while the second is the mathematical definition commonly given in most texts on Calculus. (See, e.g., [1, §9.9] or [6, §3.3].) A typical example is the hyperbolic paraboloid given by z = xy or by (the graph of) the function f : R 2 → R defined by f (x, y) := xy. Here the origin is a saddle point. Indeed if we look at the paths along the diagonal lines y = -x and y = x in the plane, then we readily notice that the origin is at once a peak and a dip. Also, the origin is the only critical point of f and clearly f does not have a local extremum at the origin.

The aim of this article is first, to point out that there is a significant disparity between the two answers, and second, to suggest an alternative approach to saddle points which may take care of this. The first point is easy to illustrate. There are surfaces or rather, functions of two variables where the conditions in the second answer are met but the geometric picture is nowhere close to the description in the first answer. For example, if f : R 2 → R is defined by f (x, y) := x 3 or by f (x, y) := x 2 + y 3 , then the origin is a saddle point according to the usual mathematical definition, but the corresponding surface (Figure 1) hardly looks like a saddle that you might want to put on a horse for any rider! Another unsatisfactory aspect is the a priori assumption that the saddle point is a critical point, that is, a point at which the gradient exists and is zero. This is quite unlike the usual definitions of analogous concepts in one variable calculus, such as local extrema or points of inflection, where one makes a clear distinction between a geometric concept and its analytic characterization (See, for instance, [3] and its review [8].) The definition we propose here seems to fare better on these counts in the case of functions of two variables. The basic idea is quite simple and, we expect, scarcely novel. However, we have not seen in the literature an exposition along the lines given here. For this reason, and with the hope that the treatment suggested here could become standard, we provide a fairly detailed discussion of the definition, basic results and a number of examples in the next three sections. Alternative 2000 Mathematics Subject Classification. Primary 26B12, 00A05; Secondary 53A05.

and moreover, γ ′ 1 (t 1 ) and γ ′ 2 (t 2 ) are not multiples of each other. In other words, the two paths pass through p and their tangent vectors at p are not parallel.

Examples 2.1. (i) γ : [-1, 1] → R 2 defined by γ(t) := (t, t 2 ) is a regular path, while γ : [-1, 1] → R 2 defined by γ(t) := (t 2 , t 3 ) is not a regular path.

(ii) If γ 1 , γ 2 : [-1, 1] → R 2 are defined by γ 1 (t) := (t, -t) and γ 2 (t) := (t, t), then γ 1 and γ 2 are regular paths in R 2 which intersect transversally at the origin. Further, the path γ 3 : [-1, 1] → R 2 defined by γ 3 (t) := (2t + t 2 , 2t -t 2 ), is also regular and passes through the origin. The paths γ 1 and γ 3 intersect transversally at the origin, whereas the paths γ 2 and γ 3 do not.

Let D ⊆ R 2 , p ∈ D and γ : [a, b] → D be a regular path in D passing through p so that γ(t 0 ) = p for some t 0 ∈ (a, b). Now, any f : D → R can be restricted to (the image of) γ so as to obtain a real-valued function of one variable φ : [a, b] → R defined by φ(t) := f (γ(t)). We shall say that f has a local maximum (resp: local minimum) at p along γ if φ has a local maximum (resp: local minimum) at t 0 . Definition 2.2. Let D ⊆ R 2 and p be an interior point of D. A real-valued function f : D → R has a saddle point at p if there are regular paths γ 1 and γ 2 in D intersecting transversally at p such that f has a local maximum at p along γ 1 , while f has a local minimum at p along γ 2 .

The above definition is a faithful abstraction of the idea that a saddle point is the point at which the graph of the function is at once a peak along a path and a dip along another path. The condition that the two paths intersect transversally might seem technical. But its significance will be clear from Example 2.3(iii) below. It may be remarked that in our definition of a saddle point, we have permitted ourselves as much laxity as is usual while defining local extrema. To wit, if a function is locally constant at p, then it has a local maximum as well as a local minimum at p. In the

…(Full text truncated)…

This content is AI-processed based on ArXiv data.