On Double Interpolation in Polar Coordinates
Codruţa Vancea, Florin Vancea and Antoniu Nicula
* Department of Electrical Engineering, Electrical Measurements and Electric Power Use,
University of Oradea, Faculty of Electrical Engineering and Information Technology,
1 Universităţii street, 410087 Oradea, Romania,
Abstract – Interpolation is an important tool in
numerical modeling of real-life systems. The Lagrange
interpolation is frequently used, due to particular
advantages in implementation. The bi-dimensional
version may be implemented with Cartesian or with
polar coordinate system. Choice of the coordinate
system is important in order to obtain accurate results.
The polar case has particular properties that can be
exploited to minimize some of the common
disadvantages of polynomial interpolation.
Keywords: Lagrange, interpolation, polar coordinates.
The attention given to function interpolation is due to
many situations when we know the values of a function
over a set of points and we need to evaluate the function
for other values in the domain. The main concern is to
find functions than offer good approximations and the
most appropriate are interpolation polynomials. For an
interpolation polynomial it is important to know its
expression, its accuracy and the graphical representation
of the curve that approximates the tabulated function. By
integrating the interpolation one derives the numerical
integration schemes and by deriving the interpolation
polynomials one builds finite differences
approximations that are the base of finite differences
The study presented in the paper is about double
Lagrange interpolation, the Lagrange polynomial being
one of the most general interpolation polynomials.
Unlike the power series interpolation, where one
should solve a linear system of equations, the Lagrange
interpolation has the advantage of a more efficient
What should be specified when we talk about the
interpolation accuracy is that on one hand the domain
chosen should be as small as possible and the number of
known points should be as high as possible.
The domain should be chosen as small as possible
because on shout intervals the function y=f(x) can be
approximated very well with algebrical polynomial
The errors arising from interpolation can be reduced
by increasing the number of given points, but with
caution. The reason is that if the known values are
tainted by errors or uncertainty then the polynomial
interpolation error becomes significant and is rising with
number of data points. In other words, the interpolation
accuracy is rising with the number of known points, but
only within a certain range .
In order to obtain a more precise approximation we
will have to impose certain conditions over the
interesting domain. It is important to choose correctly
the coordinate system. We should use a cartesian
coordinate system for rectangular domains and a polar
coordinate system for circular domains.
We have to mention that the interpolation over a bi-
dimensional domain can be performed in several ways.
Among others: bi-linear interpolation, double Lagrange
interpolation and transfinite interpolation
The transfinite interpolation is an interpolation
method for a bi-dimensional space where the
functional’s values over the exterior frontier and along
the interior grid lines are known.
The double Lagrange interpolation or bi-dimensional
Lagrange interpolation is applicable when the values of
the functional are known only at the intersection of the
grid lines, unlike the transfinite interpolation that “fits”
the continuous functions along the horizontal and
vertical lines .
II. BI-DIMENSIONAL LAGRANGE INTERPOLATION IN
The Lagrange interpolation can be used to fit a
polynomial curve to a set of known data points. For the
single-dimensional case with n+1 data points we know a
set of yi=f(xi) values for a set of xi points and according
to well-known formula we can obtain a set of n+1