## Curvature of Linear Interpolation

Most people are familiar with linear interpolation applied to a
two-dimensional table that gives discrete values of some function
z=f(x,y). Given a specific pair of coordinates x,y where x1 < x < x2
and y1 < y < y2 we interpolate using the four nearest table points
shown below:
y1 y2
x1 z11 z12
x2 z21 z22
In spite of its simplicity, the surface of interpolation (i.e., the
locus of interpolated points x,y,z) can be used to illustrate some
interesting aspects of differential geometry.
By a simple translation of the xyz coordinates the equation of this
surface becomes simply z=xy/R where R is a constant, and the
components of the metric are
g_xx = 1 + (y/R)^2
g_xy = g_yx = xy/(R^2)
g_yy = 1 + (x/R)^2
The determinant of the metric at any point (x,y) is g = 1 + (r/R)^2
where r = sqrt(x^2 + y^2), and the intrinsic curvature is
/ R \
K(x,y) = - ( ----------- )
\ R^2 + r^2 /
Since the curvature depends only on r, this shows that the lines of
constant curvature are circles when projected onto the xy plane. The
parametric equations of geodesics on this surface are
d^2 x 2y dx dy
------- + ------- ---- ---- = 0
d s^2 R^2+r^2 ds ds
d^2 y 2x dx dy
------- + ------- ---- ---- = 0
d s^2 R^2+r^2 ds ds
This shows that if either dx/ds or dy/ds equals zero, then the second
derivatives of x and y with respect to s must be zero, which means
that lines of constant x and lines of constant y are geodesics. Of
course, lines that are not parallel to either the x or the y axis can
also be geodesics, provided they satisfy the above equations.

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