Types of Saddles

Saddles are common topographic features that occur on drainage divides. Simple saddles are surfaces that can be represented mathematically by the function:

     z(x,y) = x² - y².

A simple saddle, of course, has two "places that go down" and these are where your legs go when you are riding a horse. However, there are many other kinds of saddle surfaces. The monkey saddle is the kind of saddle that a monkey would need in order to ride a horse (comfortably anyway), because he needs a third "place that goes down" for his tail. Mathematically, the standard monkey saddle surface can be expressed as:

     z(x,y) = x³ - 3 x y².

While the monkey saddle is fairly well-known, there is a hierarchy of "higher-order" saddles that you never seem to hear about. If we change variables to polar coordinates, then we see that the simple saddle has two minimums and two maximums as we move along a circle that is centered on the origin. Similarly, a monkey saddle has three mins and three maxes. A saddle surface with n mins and n maxes can be constructed using Euler's identity and the binomial formula (2 famous math theorems):

     e^{(i n t)} = cos(n t) + i sin(n t) 
             = [cos(t) + i sin(t)]ⁿ
             = sum[k=0:n] (n choose k) *
                 cos(t)^(n-k) [i sin(t)]^k.

Here, (n choose k) stands for a mathematical object that involves factorials. The function cos(n t) has n mins and n maxes as t varies from 0 to 2 pi. Multiplying by r^n and recalling that x = r cos(t) and y = r sin(t), we get an equation for the n-saddle:

     z(x,y) = sum[k=0:n; k even] (n choose k) x^(n-k) y^k.

Notice that the sum is over the even values of k so as to exclude the imaginary (complex number) terms. For example, the 4-saddle is given by:

     z(x,y) = x⁴ - 6 x² y² + y⁴.

We'll leave it to you, the reader, to work out the equations for the starfish and octopus saddles!

By the way, the slope at each point on a surface z(x,y) can be computed as:

     S(x,y) = sqrt[(dz/dx)² + (dz/dy)²].

where dz/dx and dz/dy are partial derivatives (from calculus). Applying this to the saddle surfaces just described, and recalling that r = sqrt(x^2 + y^2) in polar coordinates, it can be shown that the slope on such a surface is a function of r. For the simple and monkey saddles we have:

     S(x,y) = S(r) = 2 r      (simple saddle)
     S(x,y) = S(r) = 3 r²     (monkey saddle).

Saddle surfaces can be used as a stringent test of how well computer algorithms can compute contributing area and slope on divergent surfaces, somewhat similar to headwater hillslopes. As can be seen from these figures, the D8 algorithm doesn't do too badly on the slopes, but doesn't do well at all on the contributing areas as compared to the D-Infinity algorithm proposed by Tarboton (1997). This is a bit misleading, though, because the D8 algorithm does a good job for convergent parts of a surface and so it computes contributing areas to channels just as well as the D-Infinity method does. Both of these methods are available in RiverTools 2.4.

The simple and monkey saddle are included with several other mathematical surfaces (such as a pyramid and a Gaussian hill) on the RiverTools Data CD in a folder called Test_Surfaces.