Saddle point

In mathematics, a saddle point is a specific type of critical point on a surface. A critical point is a location where the slopes, or derivatives, are all zero in orthogonal directions. This means if you were standing at that point, the surface would appear flat for a moment. However, a saddle point is unique because it is not a local extremum. A local extremum is a point that is either a local maximum, like a mountain peak, or a local minimum, like the bottom of a bowl. Instead, a saddle point represents a location where the surface behaves differently depending on which direction you move.

The name comes from the shape of a riding saddle. In a two-dimensional example, the surface curves upward in one direction and downward in another. This creates a shape that looks just like the seat of a saddle.

Mathematically, this can be described using contour lines on a map. Contour lines represent points of equal height on a surface. At a saddle point, these lines can intersect. In a perfect model, you might see a pair of lines crossing at the point. However, real-world maps like ordnance survey maps use discrete intervals for height. Because of this, the saddle point often appears as a blank space. This space is surrounded by four sets of contour lines that approach the point and then veer away. These sets usually appear in pairs, with one pair representing high ground and the other representing low ground.

To identify a saddle point in a function with two variables, mathematicians use the Hessian matrix. The Hessian is a matrix of second-order partial derivatives. If the Hessian is indefinite at a stationary point, then that point is a saddle point. This is a sufficient condition, meaning it proves the point is a saddle point. However, it is not the only way to find one. For some functions, the Hessian might be a null matrix, which is not indefinite, yet the point remains a saddle point. In a more general sense, a saddle point on a smooth surface is a point where the surface does not stay entirely on one side of its tangent space.

There are different types of surfaces that contain these points. A saddle surface is any smooth surface that contains one or more saddle points. One classic example is the hyperbolic paraboloid. This is often called the standard saddle surface. Another example is the hyperboloid of one sheet.

These surfaces have negative Gaussian curvature. This distinguishes them from convex or elliptical surfaces, which have positive Gaussian curvature. There are also more complex versions, such as the third-order monkey saddle.

Saddle points appear in many different scientific fields. In game theory, specifically in a two-player zero-sum game played in a continuous space, the equilibrium point is a saddle point. In the study of linear autonomous systems, a critical point is a saddle point if its characteristic equation has one positive and one negative real eigenvalue. Optimization also uses this concept. When solving problems with equality constraints, the first-order conditions describe a saddle point of the Lagrangian. In dynamical systems, a saddle point is a type of hyperbolic periodic point. This occurs when the stable and unstable manifolds have a dimension that is not zero.

Even in simple one-dimensional math, we can find these points. In a single dimension, a saddle point is a point that is both a stationary point and a point of inflection. Because it is a point of inflection, it cannot be a local extremum. This shows that the concept of a saddle point is a fundamental way to describe how surfaces and functions change direction without reaching a peak or a valley.