Example 9.  What is the nature of the  "spline fit"  constructed by Mathematica ?

Solution 9.

Let's look at  .  

Let's look at  .  

Observe that the second through fourth elements of   agree with the first through third elements of  .  
Hence the spline is a continuous curve.  

Observe that the second through fourth elements of   agree with the first through third elements of  .  
Hence the spline is a smooth curve.  

Observe that the second through fourth elements of   agree with the first through third elements of  .   
Hence the second derivatives agree at all of the interpolation knots.  Also, the second derivative is zero in both coordinates at both endpoints.   

Thus, Mathematica is using a "Natural Bézier curve" which is a "natural cubic spline" in each coordinate.

(c) John H. Mathews 2004