From space angles to point position using Sylvester

Joseph L. Awange1 and Erik W. Grafarend2

From space angles to point position using Sylvester resultant

Using the Sylvester resultant which is accessible in MAPLE and MATHEMATICA software, it is here demonstrated how point positioning can be performed directly from the dimensionless space angles. The proposed approach could be applied to obtain the position of a point in the three-dimensional intersection problem vital in Photogrammetry and Computer vision. By converting the unknown distances to space angles, we demonstrate that the three nonlinear system of equations for distances can be decomposed to three quadratic polynomials for position that are solvable with the help of MATLAB-solve command to give position without actually solving for the distances themselves. The advantage here is that when one is faced with the minimum number of known points, three in this case, and only direction observations leading to space angles, one can still carry out an intersection to obtain the coordinates of the desired point. In Photogrammetry, the procedure could be used to obtain the coordinates of pass points where known stations are limited only to the minimum number.

1 Introduction In Awange et al. (submitted), the reduced Groebner basis approach was proposed for the solution of point positioning directly from space angles without going through the distance determination step. The present contribution provides an alternative approach to solving the problem

1

2

Maseno University, Dept. of Environment and Earth Sciences, Private Bag, Masena, Kenya. Department of Geodesy and GeoInformatics, Geschwister-Schollstr. 24D, 70174 Stuttgart, Germany,Tel.: þ49-7 11-1 21 33 89, E-mail: [email protected]

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using Sylvester resultant approach (Awange and Grafarend 2002). Thanks to the Global Positioning System (GPS: Global Problem Solver) classical geodetic and photogrammetric positioning techniques have reached a new horizon. Geodetic and photogrammetric direction observations (Machine Vision, “Total Observing Stations”) have to be analyzed in a three-dimensional Euclidean Space. The pair of tools called “Resection and Intersection” has to operate three-dimensionally. Unlike the resection problem, the intersection problem has not been well endowed with exact solution procedures. One reason for the rare existence of the closed form solution of the three-dimensional intersection problem is the nonlinearity of the directional observational equations, partially caused by the external orientation parameters. One target of our contribution is accordingly to address the problem of orientation parameters by presenting you the reader with the Sylvester resultant approach that solves for the position directly from the dimensionless space angles. The alternate procedure of Groebner basis has already been treated elsewhere (e.g. Awange et al. submitted). The key to overcome the problem of nonlinearity caused by orientation parameters is taken from the Baarda Doctrine. Baarda (1967, 1973) proposed to use dimensionless quantities in geodetic and photogrammetry networks: Angles in a three-dimensional Weitzenboeck space shortly called space angles as well as distance ratios are the dimensionless structure element which are equivalent under the action of the seven parameter conformal group, also called similarity transformation. For the two-dimensional intersection problem (Awange, 2003), the closed form solution in terms of angles has a long tradition. Consult Figure (1) where we introduce the angles w12 and w21 in the planar triangle D: P0P1P2. P0,P1,P2 are the nodes: The Cartesian coordinates (x1,y1) and (x2,y2) of the points P1 and P2 are given, the Cartesian coordinates (x0,y0) of the point P0 are unknown. The angles w12 ¼ a and w21 ¼ b are derived from direction observations by differencing horizontal or vertical directions. w12 ¼ T12 ÿ T10 or w21 ¼ T21 ÿ T20 are examples for observed horizontal directions T10 and T12 from P1 to P0 and P1 to P2 or T21 to T20 from P2 to P1 and P2 to P0. By means of taking differences we map direction observations to angles and eliminate orientation unknowns. The solution of the two-dimensional intersection problem in terms of angles, a classic in analytical surveying, is given by equation (1-1) and (1-2). Note the Euclidean distance between the nodal points, namely qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s12 ¼ ðx2 ÿ x1 Þ2 þ ðy2 ÿ y1 Þ2 . 265

Awange, Grafarend – From space angles to point position using Sylvester resultant

2 From dimensionless space angles to point position

Box 1-1 Closed form solution of the two-dimensional intersection problem in terms of angles: x0 ¼ s12

cosasinb sinða þ bÞ

ð1 ÿ 1Þ

y0 ¼ s12

sinasinb sinða þ bÞ

ð1 ÿ 2Þ

For the three-dimensional intersection problem, the problem of transferring observed horizontal and vertical directions to space angles or of image of coordinates in a photogram to space angles has already been solved. Equations (1-3) and (1-4) are the analytical versions of the map of directions or image coordinates to space coordinates. Indeed, the map eliminates the external orientation parameters.

Box 1-2 Three-dimensional intersection: “space angles in terms of horizontal and vertical directions”: cosw12 ¼ cosB1 cosB2 cosðT2 ÿ T1 Þ þsinB1 sinB1

ð1 ÿ 3Þ

“space angles in terms of image coordinates/ perspective coordinates (x1,y1),(x2,y2) and the focal length f” x1 x2 þ y1 y2 þ f 2 cosw12 ¼ pffiffiffi2ffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffipffiffiffi2ffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffi x1 þ y1 þ f 2 x2 þ y2 þ f 2

ð1 ÿ 4Þ

This problem is formulated as follows; In Figure (1) given six space angles {w12,w23,w31,w21,w32,w13} obtained from the spherical coordinates of type horizontal directions Ti and vertical directions Bi for i ¼ 1,...,3, obtain the position of the unknown point P 2 E3. In the three other known stations Pi 2 E3 j i ¼ 1,2,3 are converted to space angles. In the second step, the obtained system of equations are solved by Sylvester resultant to determine the position {X,Y,Z} of the unknown point P 2 E3 (Awange and Grafarend 2002). When only three known stations are used to determine the position of the unknown station in three-dimension, the problem reduces to that of 3d closed form solution. First, we develop the algebraic form of the problem by converting the nonlinear system of equations into polynomial. We point out here that the unknowns {X,Y,Z} are expressed directly in terms of the space angles. From (1), the nonlinear system of equations for the three dimensional 3-point positioning is given by (2-1) as 2 2 x2 ¼ x21 þ S212 ÿ 2S212 cosðw12 Þx1 4 x2 ¼ x2 þ S2 ÿ 2S2 cosðw Þx2 ð2 ÿ 1Þ 23 3 2 23 23 x21 ¼ x23 þ S231 ÿ 2S231 cosðw31 Þx3 which can be expressed in terms of distances a ¼ S1 ¼ x1, b ¼ S2 ¼ x2, c ¼ S3 ¼ x3 of Fig. (1) as " a ¼ x23 þ S231 ÿ 2S231 cosðw31 Þx3 ð2 ÿ 2Þ b ¼ x21 þ S212 ÿ 2S212 cosðw12 Þx1 c ¼ x22 þ S223 ÿ 2S223 cosðw23 Þx2 The distances (a,b,c) in equation (2-2) can be solved from the space angles and sine rule as 2 sinðw21 Þ a ¼ S12 sinðw 12 þw21 Þ 6 6 b ¼ S23 sinðw32 Þ ð2 ÿ 3Þ sinðw23 þw32 Þ 4 sinðw13 Þ c ¼ S31 sinðw þw Þ 31

Here, we present you with a closed form solution of the three-dimensional intersection problem where a triple of three points P1,P2,P3 are given by their three-dimensional Cartesian coordinates X1,Y1,Z1, X2,Y2,Z2, X3,Y3,Z3, but the coordinates of the zero point X0, Y0,Z0 are unknown. The dimensionless quantities w12,w23,w31 are space angles w12 ¼ L P0P1P2, w23 ¼ L P0P2P3, w31 ¼ L P1P3P0 which are derived from the measurements as outlined above. Section 2 outlines the quadratic observational equations for space angles which are converted to the distances P0P1,P0P2,P0P3 and finally solved for the unknown position by means of Sylvester resultant approach (Awange and Grafarend 2004, Awange et al. in press a) in Section 3. Section 4 presents an Test example for the solution of the three-dimensional intersection problem where the space angles are converted to distances and station coordinates by the three-dimensional ranging Awange-Grafarend Sylvester algorithm. Our contribution of solving the three-dimensional intersection problems extends the earlier results of Grafarend (1990), Grafarend and Mader (1993), Grafarend and Shan (1997) and Awange et al. (2003 b). 266

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Fig. 1: 3-point intersection AVN 7/2005

Awange, Grafarend – From space angles to point position using Sylvester resultant

With the distances obtained via (2-2) and (2-3) respectively, the observation equation relating the distances and the unknown point P 2 E3 and the unknown stations Pi 2 E3 j i ¼ 1,2,3 is now written as " a ¼ ðX1 ÿ XÞ2 þ ðY1 ÿ YÞ2 þ ðZ1 ÿ ZÞ2 ð2 ÿ 4Þ b ¼ ðX2 ÿ XÞ2 þ ðY2 ÿ YÞ2 þ ðZ2 ÿ ZÞ2 c ¼ ðX3 ÿ XÞ2 þ ðY3 ÿ YÞ2 þ ðZ3 ÿ ZÞ2 Equation (2-4) is now solved using Sylvester resultant as discussed in Section 3.

3 Sylvester resultant approach Sylvester resultant approach has already been treated in Awange and Grafarend (2002) and Awange et al. (2003 a). Here, we refer to Awange et al. (2003, Boxes 3-1, 3-5) to solve the point positioning problem from space angles. As already discussed by Awange et al. (2003), starting from three nonlinear 3d Pythagorus distance observation equations (3-5) in Box (3-1) relating to the three unknowns {X,Y,Z} and the distances {S1,S2,S3} corresponding to {a,b,c} respectively, two equations with three unknowns are derived. Equation (3-5) is expanded in the form given by (3-6) and differenced in (3-7) to eliminate the quadratic terms {X2,Y2,Z2}. Collecting all the known terms of equation (3-7) to the right hand sides and those relating to the unknowns on the left hand side leads to equation (3-8) with the terms {a,b} given by (3-9). The solution of the unknown terms {X,Y,Z} now involves solving equation (3-8), which has two equations with three unknowns. To circumvent the problem of having more unknowns than the equations, two of the unknowns are sought in terms of the third unknown (e.g. X ¼ g(Z),Y ¼ g (Z)).

The problem is solved in four steps as illustrated in Box (3-2). In the first step, we solve for the first variable X in (3-10) by hiding it as a constant and homogenizing the equation using a variable W as in (3-10). In the second step, the Sylvester resultant (Awange and Grafarend, 2002) or the Jacobian determinant is obtained as in (3-11). The resulting determinant (3-12) is solved for X ¼ g(Z) and presented in (3-13). The procedure is repeated for steps three and four as in equations (3-14) to (3-17) to solve for Y ¼ g(Z). The obtained values of X ¼ g(Z) and Y ¼ g(Z) are subsstituted in the first equation of (3-5) to give a quadratic equation in Z. Once this quadratic has been solved for Z. The values of X and Y can be obtained from (3-13) and (3-17) respectively.

4 Example Using the sine formula (2-3) we determine the distances Si ¼ xi 2 R,þ i ¼ {1,2,3} 2 Z3þ between the unknown station P 2 E3 and the known stations Pi 2 E3 for the test network „Stuttgart Central” in Awange (2002). The unknown point P0 in this case is the pillar K1 on top of the University building at Kepler Strasse 11. Points P1,P2,P3 of the tetrahedron {PP1P2P3} in Figure (1) correspond to the chosen known GPS stations Schlossplatz, Liederhalle, and Eduardpfeiffer. The distance from K1 to Schlossplatz. is designated a ¼ S1 ¼ x1 2 R,þ K1 to Liederhalle b ¼ S2 ¼ x2 2 R,þ while that of K1 to Eduardpfeiffer is designated c ¼ S3 ¼ x3 2 R.þ The distances between the known stations {S12, S23, S31} 2 Rþ are computed from their respective GPS coordinates in Table (1). Their corresponding space angles w12, w23, w31, w21, w32, w13 are as given in Table (2). From (2-4) in section (2),a,bandcarecomputedasS1 ¼566.8635,S3 ¼430.5286, and S2 ¼ 542.2609.

Box 3-1 (differencing of the nonlinear distance equations): 2 S2 ¼ ðX1 ÿ XÞ2 þ ðY1 ÿ YÞ2 þ ðZ1 ÿ ZÞ2 6 12 4 S2 ¼ ðX2 ÿ XÞ2 þ ðY2 ÿ YÞ2 þ ðZ2 ÿ ZÞ2 S23 ¼ ðX3 ÿ XÞ2 þ ðY3 ÿ YÞ2 þ ðZ3 ÿ ZÞ2 2 2 S1 ¼ X12 þ Y12 þ Z12 þ X 2 þ Y 2 þ Z 2 ÿ 2X1 X ÿ 2Y1 Y ÿ 2Z1 Z 4 S2 ¼ X 2 þ Y 2 þ Z 2 þ X 2 þ Y 2 þ Z 2 ÿ 2X2 X ÿ 2Y2 Y ÿ 2Z2 Z 2 2 2 2 S23 ¼ X32 þ Y32 þ Z32 þ X 2 þ Y 2 þ Z 2 ÿ 2X3 X ÿ 2Y3 Y ÿ 2Z3 Z differencing above " S21 ÿ S22 ¼ X12 ÿ X22 þ Y12 ÿ Y22 þ Z12 ÿ Z22 þ 2XðX2 ÿ X1 Þ þ 2YðY2 ÿ Y1 Þ þ 2ZðZ2 ÿ Z1 Þ S22 ÿ S23 ¼ X22 ÿ X32 þ Y22 ÿ Y32 þ Z22 ÿ Z32 þ 2XðX3 ÿ X2 Þ þ 2YðY3 ÿ Y2 Þ þ 2ZðZ3 ÿ Z2 Þ  

ð3 ÿ 5Þ

ð3 ÿ 6Þ

ð3 ÿ 7Þ

2XðX2 ÿ X1 Þ þ 2YðY2 ÿ Y1 Þ þ 2ZðZ2 ÿ Z1 Þ ¼ f1 2XðX3 ÿ X2 Þ þ 2YðY3 ÿ Y2 Þ þ 2ZðZ3 ÿ Z2 Þ ¼ f2

ð3 ÿ 8Þ

f1 ¼ S21 ÿ S22 ÿ X12 þ X22 ÿ Y12 þ Y22 ÿ Z12 þ Z22 f2 ¼ S22 ÿ S23 ÿ X22 þ X32 ÿ Y22 þ Y32 ÿ Z22 þ Z32

ð3 ÿ 9Þ

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Awange, Grafarend – From space angles to point position using Sylvester resultant

Box 3-2 (Sylvester resultant approach): Step 1: Solve for X in terms of Z f1 :¼ ða02 X þ c02 Z þ f02 ÞW þ b02 Y f2 :¼ ða12 X þ c12 Z þ f12 ÞW þ b12 Y

ð3 ÿ 10Þ

Step 2: Obtain the Sylvester resultant " #   qf1 qf1 b02 ða02 X þ c02 Z þ f02 Þ qY qW ¼ det JX ¼ det qf 2 qf2 b12 ða12 X þ c12 Z þ f12 Þ qW qW

ð3 ÿ 11Þ

JX ¼ b02 a12 X þ b02 c12 Z þ b02 f12 ÿ b12 a02 X ÿ b12 c02 Z ÿ b12 f02

ð3 ÿ 12Þ

from (3-12) X¼

fðb12 c02 ÿ b02 c12 ÞZ þ b12 f02 ÿ b02 f12 g ðb02 a12 ÿ b12 a02 Þ

ð3 ÿ 13Þ

Step 3: Solve for Y in terms of Z f3 :¼ ðb02 Y þ C02 Z þ f02 ÞW þ b02 X f4 :¼ ðb12 Y þ c12 Z þ f12 ÞW þ a12 X

ð3 ÿ 14Þ

Step 4: Obtain the Sylvester resultant " #   qf3 qf3 a02 ðb02 Y þ c02 Z þ f02 Þ qX qW JY ¼ det qf4 qf4 ¼ det a12 ðb12 Y þ c12 Z þ f12 Þ qW qW

ð3 ÿ 15Þ

JY ¼ a02 b12 Y þ a02 c12 Z þ a02 f12 ÿ a12 b02 Y ÿ a12 c02 Z ÿ a12 f02

ð3 ÿ 16Þ

from (3-16) Y¼

fða12 c02 ÿ a02 c12 ÞZ þ a12 f02 ÿ a02 f12 g ða02 b12 ÿ a12 b02 Þ

ð3 ÿ 17Þ

Once the distances have been established, the position is then determined from first for Z from the quadratic equation resulting after the substitution of X(Z) and Y(Z) in the first equation of (3-5). Once this quadratic has been solved for Z(m) ¼ 4774879.3704. The values of X and Y are obtained from (3-13) and (3-17) respectively as X(m) ¼ 4157066.1116 and Y(m) ¼ 671429.6655 in the Global

Reference Frame. Figures (2), (3) and (4) indicate that each component of the coordinates of the unknown point P 2 E3 i.e., X,Y,Z has two solutions. The correct solution from the quadratic curve can be obtained with help of prior information. The critical configuration of the three-dimensional ranging problem is presented in Awange et al. (2003b).

Tab. 1: GPS Coordinates in the Global Reference Frame Fl(X,Y,Z),(Xi,Yi, Zi), i ¼ 1,2,3 Station Name

X(m)

Y(m)

Z(m)

rX(m)

rY(m)

rZ(m)

Schlossplatz Liederhalle Eduardpfeiffer

4157246.5346 4157266.6181 4156748.6829

671877.0281 671099.1577 671171.9385

4774581.6314 4774689.8536 4775235.5483

0.0008 0.00129 0.00193

0.0008 0.00128 0.00184

0.0008 0.00134 0.00187

Tab. 2: Ideal spherical coordinates of the relative position vector in the Local Horizontal Reference Frame F*: Spatial distance, horizontal direction, vertical direction

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Observation from

Space angle (gon)

K1-Schlossplatz-Liederhalle w12 K1-Liederhalle-Eduardpfeiffer w23 K1-Eduardpfeiffer-Schlossplatz w31 K1-Liederhalle-Schlossplatz w21 K1-Eduardpfeiffer-Liederhalle w32 K1-Schlossplatz-Eduardpfeiffer w13

35.84592 49.66335 14.19472 49.61464 37.71933 13.56892

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Awange, Grafarend – From space angles to point position using Sylvester resultant

Fig. 2: Solution for coordinate X

Fig. 3: Solution for coordinate Y

5 Conclusion It has been demonstrated how the coordinates of a single station can be obtained. By converting the nonlinear observation equations of the three-dimensional intersection into algebraic (polynomials), the Sylvester resultant proceeds to determine the position of a point directly from the dimensionless space angles, which is a vital step in photogrammetry. References [1] Awange, J. L.: Groebner bases, Multipolynomial resultants and the Gauss-Jacobi Combinatorial algorithms-adjustment of nonlinear GPS/LPS observations. Dissertation, Technical Reports, department of Geodesy and GeoInformatics, Report No. 2002(1).

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Fig. 4: Solution for coordinate Z [2] Awange, J. L., Grafarend, E.: Sylvester resultant solution of planar ranging problem, Allgemeine VermessungsNachrichten, 108, 2002, 143 – 146. [3] Awange, J. L.: Buchberger algorithm applied to planar lateration and intersection problems. Survey Review, No. 290, 2003, vol. 37. [4] Awange, J. L., Grafarend, E., Fukuda, Y., Takemoto, S.: Direct Polynomial approach to nonlinear distance (ranging) problems. Submitted to the Earth, Planets and Space, 55, 2003 a, 231 – 241. [5] Awange, J. L., Fukuda, Y., Takemoto, S.: (2004) B. Strumfel’s resultant solution of planar resection problem. Allgemeine Vermessungs-Nachrichten 128. [6] Awange, J. L., Grafarend, E. W., Fukuda Y.: (2004) Closed form solution of the triple three-dimensional intersection problem. Zeitschrift fu¨r Geoda¨sie, Geoinformation und Landmanagement. [7] Awange, J. L., Fukuda, Y., Grafarend, E. W.: (submitted) Algebraic optimization of space angles for point positioning. Submitted to Journal of Photogrammetry and Remote sensing. [8] Baarda, W.: A generalization of the concept strength of the figure. Publications on Geodesy, New Series, Vol. 2, No. 4, Delft, 1967, 528 – 543. [9] Baarda, W.: S-transformation and criterion matrices, Netherlands Geodetic Commission. Publications on Geodesy, New Series, Vol. 5, No. 1, Delft, 1973. [10] Grafarend, E.: Dreidimensionaler Vorwa¨rtsschnitt. Zeitschrift fu¨r Vermessungswesen 115, 1990, 414 – 419. [11] Grafarend, E., Mader, A.: Robot vision based on an exacst solution of the threedimensional resection-intersection. Applications of Geodesy of Engineering. In K. Linkwitz, V. Eisele and H.-J. Moenicke, Symposium No. 108, Springer-Verlag, Berlin – Heidelberg – NewYork – London – Paris – Tokyo – HongKong – Barcelona – Budapest, 1993. [12] Grafarend, E., Shan, J.: Closed form solution to the twin P4P or the combined threedimensional resection-intersection problem in terms of Moebius barycentric coordinates. Journal of Geodesy 71, 1997, 232 – 239.

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