Manual MIT RadLab {complete set} Vol 02 - Radar Aids to Navigation

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The Ultimate Weapon: the Atomic Bomb. Back Matter Pages About this book Introduction The latest advances in science were fully exploited in the Second World War. They included radar, sonar, improved radio, methods of reducing disease, primitive computers, the new science of operational research and, finally, the atomic bomb, necessarily developed like all wartime technology in a remarkably short time. Such progress would have been impossible without the cooperation of Allied scientists with the military.

The Axis powers' failure to recognise this was a major factor in their defeat. Fattorini Eds. Fatal Future? Suris auth. Kiguradze, et al. Li, Z. Chen, W. Ingle Eds. Manuel auth. Carroll, D. Oracle9i Database Reference Part No. A Release 9. Itakura, K. Brockhausen Shooting Action Sports. Dix auth. Baxter, R. David Logan auth. Gerson Ed.

The ASP. NET 2. Veenstra, J. Long Quantum electrodynamics Gribov's lectures on theoretical physics by Gribov V. Michael Conn Ed. Shriver Jr. Ivin, J. Wilcox Jr. Lonnie A. Real World modo: The Authorized Guide. Birkhoff's variety theorem with and without free algebras by Adamek J. Frontiers in Chemical Engineering.. Research Needs and Opportunities by Amundson N. The last stand of Fox Company: a true story of U. Masseras theorem for quasi-periodic differential equations by Ortega R.

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Henk Koop auth. Bradbury auth. Seminario Eds. Bloch auth. Peter C. Patrikis Fischer Weltgeschichte, Bd. Blanchard, J. Opera omnia IV. Schriften zur Naturwissenschaft. Matos Eds. Harper Jr. Development American Eras by Gerald J. Valley, H. Berberian Osprey Fortress - Fortifications in Wessex, c. Li PhD auth. Cooper, M. Objects [lg article] by F. Adams, G. Benson, B. Oceanography and marine biology. George Shillington Reaction Injection Molding. Polymer Chemistry and Engineering by Jiri E. Kresta Eds. Lebedev auth. In Appendix B, the problem of scattering by a conducting grating with rectangular groovesis ccrnsidered.

Analytical solutions are derived for the cases of TM- and TE-polarized incidentwaves by mode-matching between the free space and groove regions. Procedures for determiningthe validity of numerical results are described and the problem of determining the minimumnumber of modes required to accurately represent the fields in each region is studied. Animplementation of the analytical solutions as a pair of subroutines coded in Fortran 77 ispresented. In Appendix C, the problem of modifying a conventional trihedral corner reflector to presenta circular polarization selective response is considered.

It is shown that such a response cannotbe realized simply by using the techniques of Chapter 4 because the corresponding polarizationscattering matrix cannot be diagonalized. Alternative methods for obtaining such a responsebased on the addition of a suitable transmission polarizer to reflectors which present either alinear polarization selective or a twist polarizing response are proposed. In Appendix D, the experimental facility which was developed to measure the responseof prototype trihedral corner reflectors is briefly described.

Details of its physical layout, thedesign and implementation of the CW radar apparatus and digital pattern recorder, and theresults of tests used to verify its suitability for use in the measurement program are given. Recommendations for future modifications and improvements are offered.

References[1] F. New York: McGraw-Hill, , pp. Levanon, Radar Principles. New York, Wiley, , pp. Knott, J. Shaeffer, and M.


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Norwood, MA: Artech House, , pp. Bhattacharyya and D. Nor-wood, MA: Artech House, , pp. IEEE, vol. Macikunas and S. Haykin, Ed. Atlas, Ed. Ulaby and C. Elachi, Eds. Norwood,MA: Artech House, Brown, R. Newman, and J. Crispin, Jr.

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Siegel, Eds. New York: Academic Press, , pp. Ruck, Ed. New York: Plenum Press, , pp. Ulaby, R. Moore, and A. Brunfeldt and F. Remote Sensing, vol. GE, pp. Yueh, J. Kong, R. Barnes, and R. Waves Appi.

Freeman, Y. Shen, and C. Stiles, R. Hewitt, and C. TP E. Montreal: TransportationDevelopment Centre, Aug. Kalnicki and R. Lawrence River. Montreal: Transportation Development Centre, April Lanziner, D. Michelson, S. Lachance, and D. Michelson et al. Long, Radar Reflectivity of Land and Sea. Norwood, MA: Artech House, Ulaby and M. If the wave ismonochromatic, the tip of the electric field vector will trace an ellipse in the plane orthogonalto the direction of propagation.

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Such a wave is said to be completely polarized. If the wavecoiltains a random component in amplitude or phase, it will occupy a finite bandwidth andthe polarization ellipse will tend to change shape and orientation with time. Such a wave issaid to be partially polarized. In the extreme case, the tip of the electric field vector will traceout a figure which is totally random in shape and the wave is said to be randomly polarized or unpolarized.

The fundamental aspects of wave polarization have been reviewed by severalauthors, e. Transformation of Polarization Descriptors Between Coordinate Frames 12Since polarization is a vector quantity, its description must be referred to a particularcoordinate frame.

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If the radiation or scattering characteristics of an object can be determined moreefficiently in a different frame or if the object is free to rotate about one or more axes as inthe case of airborne or spaceborne platforms, it may be preferable to define the polarizationcharacteristics of the object with respect to a local or body-fixed coordinate frame instead,as suggested by Figure 2. In turn, the response of the device is measured in yet anothercoordinate frame which is defined by the antenna.

However, with the exception of the specialcases considered by Mott [6] and Krichbaum [7], the problem of transforming polarizationdescriptors between coordinate frames has received little attention in the literature. In section 2. Two methodsfor determining the angle of rotation for the case in which the local vertical is defined by thedirection in each frame are derived using spherical trigonometry and vector algebra, respectively. Although the matrix can be determined from either the relative directions ofthe three principal axes in each coordinate frame or the Euler angles which define a seriesof rotations which will transform one coordinate frame into the other, in practice it may bedifficult to obtain these parameters.

A third method is derived which permits the elements ofthe transformation matrix to be determined from any pair of arbitrary directions which havebeen expressed in terms of both coordinate frames. Chapter 2. The coordinate frame I, , k can be specified in terms of the triad i,, defined bya spherical coordinate system such thatk. The definitions of I and I given by 2. In cases where the direction of propagation coincides with the z axis, e. The shape, sense of rotation, and orientationof the effipse are sufficient to specify the polarization state of the wave.

Consider a polarizationellipse with semi-major axis OA and semi-minor axis OB, as depicted in Figure 2. Transformation of Polarization Descriptors Between Coordinate Frames 15A polarization state with ellipticity angle E and tilt angle T corresponds to a point havinglongitude 2T and latitude 2c.

Linear and circular polarization states map onto the equator andpoles, respectively, while left and right elliptical polarization states map onto the upper andlower hemispheres. Figure 2. The limitations areparticularly apparent in cases where polarization state is described with respect to an ellipticallyor circularly polarized basis or where the wave is partially polarized. A more general approachis suggested by considering transformation of the corresponding polarization ellipse betweencoordinate frames. The ellipticity angle c is invariant under either translation or rotation sinceit depends only on the magnitude of the axial ratio and the polarization sense of the wave.

Although the tilt angle r is invariant under translation, it will not be preserved under rotationxYZm1 m2 m3ni 2 fl3Chapter 2. In general, it can be shown that transformation of any polarization descriptorbetween coordinate frames may be regarded as a change of basis transformation correspondingto rotation of the polarization basis by an angle a about the direction of propagation. This conditionwill be satisfied only if the coordinate transformation matrix is of the form 2.

Otherwise, the difference betweenthe tilt angles of the polarization ellipse in the two coordinate frames is given by6Chapter 2. Thisvariation is apparent in Figures 2. Since the Mercator projection is conformal, the angles between the parallels are accurately depicted at all points on the grid and the variation in the angle with the direction of propagationcan be easily visualized. In the context of radar cross section measurement, Krichbaum [7] hasderived an expression for this angle 1 for the special case in which the direction of propagation iscoincident with the z axis and the coordinate transformation corresponds to rotation about thex and y axes.

Here, two methods for determining the angle cr for any direction of propagationand coordinate transformation are derived using spherical trigonometry and vector algebra,respectively. Expressions for transforming direction expressed in terms of the elevation angle 6 andazimuth angle 4 between coordinate frames can be derived from the coordinate transformations1which he refers to as the polarization angle T.

Since the projection is conformal, the angles between the parallels are accuratelydepicted at all points on the grid. Consider twocoordinate frames which are related by pure rotation as shown in Figure 2. Since both the sine and cosine of the angle are known, it is a simple matter to determine theangle a using a four-quadrant arctangent function, e. In suchcases, the direction of the horizontal plane with respect to the direction of propagation mustbe defined arbitrarily. This can be accomplished if either the relative directionsof the basis vectors defined by the three principal axes in each coordinate frame or a series ofEuler angle rotations which will transform one coordinate frame into the other are known [9],[ In the second case, the transformation is described in terms a seriesof angles through which the first frame can be rotated in order to bring it into coincidence withthe second.

A maximum of three Euler angle rotations is sufficient to bring any two framesChapter 2. The product of the three rotation matrices given in 2. In practice, it may be difficult to obtain the parameters required by the basis vector andEuler angle methods for determining the elements of the coordinate transformation matrix. Analternative method is derived here which allows the matrix to be determined from two arbitrarydirections 8k, qf1 and 02, 2 which have been expressed in terms of both coordinate frames.

Transformation of Polarization Descriptors Between Coordinate Frames 25The unit vectors which correspond to these two directions can be determined by applying 2. Methods for rotatingthe basis of the Stokes vector and the coherency basis are derived. Some of the results presentedhere have recently been confirmed using a different approach by Mott [6].

For example, a sphericalcoordinate system can be devised in which a polarization state W with ellipticity angle E andtilt angle r is represented by a point having longitude 2r and latitude 2e, as shown in Figure2. If the basis vectors Ii. The quantities aH and av are the magnitudes of EH and Ev,respectively, and 6H and 5v are their phase angles. Transformation of Polarization Descriptors Between Coordinate Frames 28In each case, a polarization state W with polarization angle y and phase angle is represented by a point having elevation 27 and azimuth 5.

The orientation of the axes from whichand cc are measured is shown in Figure 2. The phase reference for orthogonal circularcomponents is defined in Figure 2. Expressions for transforming the polarization coordinates 7L, 6L , 7D, 6D , or 7c, 6c into thepolarization coordinates e, r and back can be derived using spherical trigonometry.

Considerthe right spherical triangle defined by the points corresponding to horizontal polarization, thepolarization state W, and the polarization state 0, T. Each polarization state is represented by a complex number which isreferred to as the complex polarization ratio p. Rotation of the basis of the complex polarizationratio is easily accomplished if the ellipticity angle e and tilt angle i- of the correspondingChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 30polarization ellipse are known.

Together with 2. Complex Polarization VectorsThe pair of complex amplitudes which arise from the representation of a plane wave as theweighted sum of orthogonally polarized basis states may be arranged to yield a complex polarization vector, e. Transformation of Polarization Descriptors Between Coordinate Frames 31If the basis of the complex polarization vector is linear, the corresponding rotation operator[RU is simply given bycosa —sina[RL]. After it has beenrotated, the polarization basis can be restored to its original effipticity by the reverse transformation.

Let be the ellipticity angle of the polarization basis state whichdefines the first element in the polarization vector. Since the basis states are orthogonal,the ellipticity angle of the polarization basis state which defines the second element in thepolarization vector is given by -e. Transformation of Polarization Descriptors Between Coordinate Frames 33Since all the elements of the Stokes vector are expressed in units of power, polarimetric datawhich are expressed in Stokes format can be spatially and temporally averaged with relativeease.

Also, the elements of the Stokes vector are always expressed in real numbers so recourse tocomplex arithmetic is not required. Unlike the complex polarization vector, the Stokes vectorcan also represent the polarization state of quasi-monochromatic or partially polarized wave. Transformation of Polarization Descriptors Between Coordinate Frames 34Consider rotation of the polarization basis about the propagation vector by an angle a.

From 2. LetSm represent the modified Stokes vector in the original coordinate frame and let S representChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 36the modified Stokes vector in the new coordinate frame. Transformation of Polarization Descriptors Between Coordinate Frames 37Consider rotation of the polarization basis about the propagation vector k by an angle a.

Radar cross section refers to that portion of the scatteringcross section which is associated with a specified polarization component of the scattered waveand is a function of the size, shape, composition, and orientation of the target, the frequency ofthe incident wave, and the polarization state of the radar transmitting and receiving antennas. The relationship between the polarization states of the incident and scattered fields canbe described by a polarization scattering operator expressed in matrix form.

Following thedefinition of scattering cross section presented in 2. Transformation of Polarization Descriptors Between Coordinate Frames 40The Mueller matrix [L] relates incident and scattered fields which have been expressed as Stokesvectors, i. In practice, polarization scattering matrices andMueller matrices are often normalized by factoring out the scattering cross section of the targetand the range dependence of the response.


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Other polarization scattering operators which arederived from the polarization scattering matrix and the Mueller matrix, such as the covariancematrix and the Stokes scattering operator, are used in computationally efficient methods forsynthesizing arbitrary polarization responses from experimental data [4]. The local coordinate systems used to define the polarization state ofthe incident and scattered fields are specified in a manner similar to that presented in section 2. However, it is also necessary to specify the relationship between the local coordinate systems. According to the forward scatter alignment FSA convention, the propagation vectors of the incident and scattered fields are aligned with the direction of propagation while according to backscatter alignment BSA convention, they alwayspoint towards the scatterer.

While the expressions for the incident field are identical underboth conventions, the expressions for the scattered field and, by extension, the correspondingpolarization scattering operator, are not. The subscripts i and s refer to fields expressed with respect to the FSA convention whilethe subscripts t and r refer to fields expressed with respect to the BSA convention.

The unitChapter 2. Unless otherwise stated, the BSA convention will be theconvention used in the remainder of this study. Once the local coordinate systems for the incident and scattered fields have been defined,it is a simple matter to apply the results derived earlier in this section to the problem ofrotating the basis of either the polarization scattering matrix or the Mueller matrix about theradial vector i by an angle c. Multiplying both sides of 2. Two methods for determining the angle of rotation for the case in which thelocal vertical is defined by the direction in each frame have been derived using sphericaltrigonometry and vector algebra, respectively.

Both methods are robust and will yield thecorrect result but the method based on vector algebra is more compact and would be easierto implement in software. Although the elements of the coordinate transformation matrix canbe determined from either the relative directions of the three principal axes in each coordinateframe or the Euler angles which define a series of rotations which will transform one coordinateframe into the other, in practice it may be difficult to obtain these parameters. A third methodhas been derived which overcomes this limitation by allowing the elements of the coordinatetransformation matrix to be determined from any pair of directions which have been expressedin terms of both coordinate frames.

Algorithms for rotating the basis of several commonly usedpolarization descriptors, including polarization coordinates, the complex polarization ratio, thecomplex polarization vector, the Stokes vector and several of its variants, the coherency matrix,the polarization scattering matrix, and the Mueller matrix have been derived. References[1] R. Azzam and N. Bashara, Ellipsometry and Polarized Light. Amsterdam: North-Holland, Kraus, Antennas, 2nd ed.

Norwood,MA: Artech House, , pp. Stutzman, Polarization in Electromagnetic Systems. Norwood, MA: Artech House, NewYork: Wiley, Ruck et al. Thomson, Introduction to Space Dynamics. New York: Wylie, , pp. Bate, D. Mueller, and J. White, Fundamentals of Astrodynamics. New York:Dover, , pp. Chandrasekhar, Radiative Transfer. New York: Dover, , pp. Born and E. Wolf, Principles of Optics. New York: Pergamon, , pp. Chapter In general, a ray incident upon one of its interior surfaces willundergo reflection from each of the others in succession and will be returned to the source.

Although other scattering mechanisms contribute to the response, triple-bounce reflections fromthe interior of the reflector dominate over most directions of incidence. Since trihedral cornerreflectors present a large scattering cross section over a wide angular range, are mechanicallyrugged, and can be manufactured with relative ease, they are widely used in radar navigationand remote sensing as location markers and calibration targets. The relative sizes of trihedralcorner reflectors in common use are compared in Figure 3.

Truncation and Compensation of Trihedral Corner Reflectors 48The dependence of the scattering cross section and angular coverage of a trihedral cornerreflector on the size and shape of its reflecting panels has been recognized since the adventof radar. Although it was apparent that awide variety of response characteristics could be obtained by appropriate shaping of the reflecting panels, a procedure which Robertson [4], [5] referred to as truncation and compensation,work in this area was not pursued due to the lack of either suitable methods for determining theresponse of a trihedral corner reflector with panels of completely arbitrary shape or a need forphysically large targets which would benefit from such modifications.

In recent years, interestin altering the response of trihedral corner reflectors in this manner has been renewed by arequirement for physically large targets to serve as location markers and calibration targetsin radar navigation and remote sensing [6]—[9]. However, very little design data and relatedmaterial to guide the development of such reflectors are available in the literature. In section 3. Design curves for bilaterally symmetricreflectors which are composed solely of triangular, elliptical, or rectangular reflecting panels aregiven. The response characteristics of a selected set of bilaterally symmetric reflectors whichare composed of combinations of panels with various shapes including triangular, circular, andsquare are compared.

A related problem, the design of top hat reflectors with specified responsecharacteristics, is considered in Appendix A. Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors Numerical techniques such as the finite-difference time-domain FD-TD and the shooting and bouncing ray SBR methods have been successfully applied tothe problem and can account for most contributions to the response. However, calculating theresponse of a large target is extremely demanding and access to some type of supercomputeror massively parallel processor is generally required [10], [11].

In the case of an ideal reflectorwith reflecting panels which are perfectly flat and mutually orthogonal, the problem can besimplified considerably. A reasonably complete solution can be obtained by using physicaloptics P0 to account for the contribution of reflections from the interior of the reflector whileusing the method of equivalent currents MEC to account for first order diffraction from theedges [12]. Alternatively, a hybrid approach which permits application of the Uniform Theoryof Diffraction UTD to the problem can be employed [13].

However, neither of these techniquescan be easily applied to reflectors with panels of completely arbitrary shape. For the purposes of designing trihedral corner reflectors with specified response characteristics, it is usually sufficient to account for the contribution of triple-bounce reflections from theinterior since they completely dominate the response for most directions of incidence. If thereflector is ideal, a ray which is incident upon one of the interior surfaces will generally undergoreflection from each of the others in succession and will be returned to the source.

However,the reflecting panels are of finite extent and some rays will fail to intercept one or more of thepanels and will be lost. The equivalent flat plate area A of the reflector can be determined bylaunching a set of parallel rays towards the target, tracing each ray as it is reflected by eachof the interior surfaces, and projecting that portion of the reflector which contributes to thebackscatter response onto a view plane which is normal to the direction of incidence, as suggested by Figure 3.

The scattering cross section a of the reflector is related to its equivalentChapter 3. It is convenient to describe scattering by a trihedral corner reflector with respect to thecoordinate frame shown in Figure 3. Spencer [1] empirically derived a simple geometricmodel for predicting the equivalent flat plate area of an ideal trihedral corner reflector based onexperiments that he conducted with reflectors fabricated from optical mirrors.

In the model,the polygon which defines the outside edges of the reflecting panels and its inverted image areprojected onto a view plane which is normal to the direction of incidence. The inverted imageis obtained by projecting the original polygon through the apex of the reflector. The projectionand its inverted image are referred to as the entrance pupil and exit pupil of the reflector,respectively. An example for the case of a reflector with triangular panels is shown in Figure3. According to the model, the equivalent flat plate area of the reflector is the area commonto the two pupils.

Using this model, Spencer derived closed-form expressions for the response oftrihedral corner reflectors composed of triangular and square panels with equal corner lengths. Truncation and Compensation of Trihedral Corner Reflectors 52Later, these were extended to the case of trihedral corner reflectors composed of triangular,elliptical, or rectangular panels with unequal corner lengths by Siegel et al. Thevalues given to p, q, and r are then reassigned in order of increasing magnitude such thatjpj JqJ jr.

Truncation and Compensation of Trihedral Corner Reflectors 53These closed-form expressions permit rapid and efficient computation of the equivalent flatplate area of trihedral corner reflectors with panels having certain specific shapes. This was first noticed by Robertson [4]who proposed an alternative geometric model which will always yield the correct solution.

Consider the trihedral corner reflectorwith panels of arbitrary shape which is shown in Figure 3. First, the panels of the reflector are replaced by complementary apertures which are derived from each reflecting panel byreflection about the trihedral axes as shown in Figure 3. The optical model which resultsis shown in Figure 3. To the observer, the polygons defined by the three complementaryapertures are projected onto a view plane which is normal to the direction of incidence. Thearea common to all three polygons is the equivalent flat plate area of the reflector.

The four geometric primitives used in this algorithm are defined as follows [17]—[19j: A pointis specified by its coordinates P x, y, z. A line segment is specified by giving its end pointsPi xi, yi, zi and P2 x,Y2, z2. A polyline is a chain of connected line segments which isspecified by giving a list of the vertices F1,. The firstvertex is called the initial or starting point while the last vertex is called the final or terminalpoint. A polygon is a closed polyline in which the initial and terminal points coincide.

The linesegments P1 F2,F2 F3,. The vertex list for theexterior boundary of the polygon is traversed in a counterclockwise direction and the enclosedregion has a positive vector area. If the polygon contains interior boundaries or holes , thecorresponding vertex lists are traversed in a clockwise direction and the enclosed regions havea negative vector area. Once the polygons which represent the panels of the reflector and thedirection of incidence have been specified, the prediction algorithm is executed in four steps The polygons which represent the x-y, y-z, and z-x reflecting panels are converted intocorresponding aperture polygons by reflection about the principal axes of the trihedral.

The x-y, y-z, and z-x aperture polygons are projected onto a view plane which containsthe origin and is normal to the direction of incidence. The polygon which represents the region that is common to the projection of all threeaperture polygons is determined. This is accomplished by calculating the intersection ofthe projection of the x-y aperture polygon and the projection of the y-z aperture polygonthen calculating the intersection of the result and the projection of the z-x polygon. The area of the polygon which represents the region that is common to the projection ofall three aperture polygons is calculated.

Truncation and Compensation of Trihedral Corner Reflectors 58The first step in the prediction algorithm, conversion of the polygons which represent thereflecting panels into aperture polygons by reflection about the principal axes of the trihedral,can be performed by inspection. The second step, projection of the aperture polygons ontoa view plane which contains the origin and is normal to the direction of incidence, may beaccomplished by a transformation of coordinates through pure rotation.

The projection of each of the aperture polygonsonto the view plane can be determined by transforming the coordinates of each vertex from thereflector frame into the view plane frame using 3. The third step in the prediction algorithm, determining the region of the view plane whichis common to the projection of all three aperture polygons, is more difficult. A variety ofalgorithms for determining the intersection of overlapping polygons have been developed for usein computer graphics applications and are widely used. However, the Weiler-Atherton polygon-clipping algorithm overcomes this limitationand is capable of clipping a concave polygon with interior holes to the boundaries of anotherconcave polygon with interior holes [18]—[21j.

In the Weiler-Atherton polygon-clipping algorithm, the subject and clip polygons are described by circular lists of vertices S1, 52,. Before theactual clipping is performed, the points at which the subject and clip polygons intersect aredetermined. The coordinates of the intersection points are inserted into both the subject andclip polygon vertex lists in the appropriate sequence.

In order to establish a bidirectional linkbetween the vertex lists, each intersection point in the subject polygon vertex list is given apointer to the location of the same intersection point in the clip polygon list and vice versa. Theactual clipping is performed as follows: The subject polygon is traversed in a counterclockwisedirection until an intersection is reached.

If this series of points lies in the interior of the clippolygon, they are added to the result list. If the next vertex of the subject polygon lies insideChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 61the clip polygon, the subject polygon vertex list is followed. Otherwise, the algorithm jumps tothe clip polygon vertex list and follows it to the next intersection.

This process continues untilall the intersections have been traversed and the algorithm has returned to the first point inthe result polygon. Methods for implementing the algorithm and enhancing its efficiency androbustness have beendiscussed in the literature [18 —[21]. Application of the Weiler-Atherton polygon-clipping algorithm to the problem of determining the equivalent flat plate area of a trihedral corner reflector with triangular panels of equalcorner length for incidence along the symmetry axis is demonstrated in Figure 3.

The projections of the x-y, y-z, and z-x aperture polygons onto the view plane are shown in Figure 3. The shaded region represents the area which is common to the projections of all three aperturepolygons. In Figure 3. The polygon which definesthe region common to the projection of all three aperture polygons is shown in Figure 3. The last step in the prediction algorithm is calculation of the area of the polygon whichdefines the region common to the projection of all three aperture polygons.

Ineach case, S refers to points in the subject polygon, C refers to points in the clip polygon,and I, refers to the points at which the polygons intersect. In each case, the contribution of triple-bounce reflections to the scatteringcross section of the reflector was calculated over the entire quadrant defined by the axes of thetrihedral.

The resulting array of values in 8 and were then converted to contours expressedin decibels with respect to the maximum response of the reflector.