Alternatively, a

polar coordinate system can be used for targeting, with the needle angled slightly upwards as opposed to perpendicular in the Cartesian coordinate system.

Similarly, in Table II, there are relative changes of parameters of the considered indices when SE is realized in the

polar coordinate system. The mentioned changes of the parameters of the considered indices characterizing properties of PSSE are calculated using formula: [DELTA][p.sub.X] = 100([p.sub.X,S] - [p.sub.X,S-])/ [p.sub.X,S-], where [p.sub.X,S], [p.sub.X,S-] are parameters of index X, when SE is performed for PS with or without PhS, respectively.

*

polar coordinate system [tau], [theta] in a two-dimensional space);

Polar coordinate transformation is similar to Cartesian coordinate transformation, it firstly establishes the

polar coordinate system with the center point as the midpoint and the direction of the center point as the positive direction, then divides the fingerprint image into many segments, and each segment is divided into many small blocks.

[18], as the original data is in the

polar coordinate system, an interpolation method was utilized to transform the

polar coordinate system into the three-dimensional Cartesian coordinate system.

Firstly, a transformation of the rectangular coordinate geometry into a series of local

polar coordinate systems is devised as depicted in Figure 2, in which the first local

polar coordinate system is set as in Figure 1; thus, every geometric center of machines is set as a local polar origin, and their local polar axes are directed as x-axis.

Bear [1] proved that if r = [K.sub.q.sup.1/2] or r = [K.sub.j.sup.-1/2] in the

polar coordinate system, mapping can form an ellipse (under the condition of three-dimensional ellipsoid), where r represents radius vector, [K.sub.q] represents permeability coefficient in the direction of the flow line, and [K.sub.j] represents permeability coefficient in the direction of the hydraulic gradient.

Then we turn to the

polar coordinate system such that [x.sup.*.sub.[phi]] = 0 (Figure 3) and we assume that values of all angles satisfy -[pi] < [a.sup.i.sub.[phi]] [less than or equal to] [pi] (the case [a.sup.i.sub.[phi]] = 0 is excluded since [x.sup.*.sub.[phi]] [not equal to] [a.sup.i.sub.[phi]]).

From the periodicity of limit cycle and the property of

polar coordinate system, we know that the conclusion of the lemma is true.

This block treats [[??].sub.pi,l(n)] as amplitude variation [alpha] and phase shift [DELTA][theta] in a

polar coordinate system in order to enable microvariations to be easily calculated.

The resulting coordinate values can be stored in a table and drawn in the plane equipped with the

polar coordinate system. GeoGebra is a very convenient tool with which to do so.

The radius e of the trajectory of centre [O.sub.2] of roundness measure and polar angle [alpha] in

polar coordinate system are described by the equations