3D-position determination method and device

Abstract

A device and method for determining the three-dimensional position of an object. The device comprises at least one transmitter that is adapted to emit a signal; at least three receivers, wherein the at least three receivers and the at least one transmitter are preferably arranged within a first plane, wherein a first receiver and a second receiver are preferably arranged along a first straight line, and a third receiver is preferably arranged at a distance from the first straight line; and a processor that is configured to determine at least three propagation times, wherein the respective propagation time is a time required by the signal from the transmitter via the object to the respective receiver, and wherein the processor is further configured to determine the three-dimensional position of the object on the basis of the determined propagation times as well as on the basis of the arrangement of the transmitter and the receivers.

Claims

1. A device for determining a three-dimensional position of an object, wherein the device comprises: a first transmitter that is adapted to emit a first signal; at least three receivers comprising a first receiver, a second receiver, and a third receiver, wherein the at least three receivers and the first transmitter are arranged within a first plane, wherein the first receiver and the second receiver are arranged along a first straight line, and the third receiver is arranged at a distance from the first straight line; and a processor that is configured to determine a first propagation time, a second propagation time, and a third propagation time of the first signal from the first transmitter to the object and then to the first, second, or third receiver, respectively, wherein the processor is further configured to determine the three-dimensional position of the object on the basis of the determined first, second, and third propagation times as well as on the basis of the arrangement of the first transmitter and the first, second, and third receivers, wherein the processor is configured to determine a first elliptical orbit or a first semielliptical orbit within a second plane on the basis of the first propagation time and to determine a second elliptical orbit or a second semielliptical orbit within the second plane on the basis of the second propagation time, wherein the position of the object in three-dimensional space is calculated using a position of an intersection of the first and second elliptical orbits or semielliptical orbits in the second plane.

2. The device according to claim 1, wherein the first transmitter is arranged on the first straight line.

3. The device according to claim 1, wherein a first focal point of a first elliptical orbit coincides with a second focal point of the first elliptical orbit and a first focal point of the second elliptical orbit coincides with a second focal point of the second elliptical orbit.

4. The device according to claim 3, wherein the processor is further configured to determine an angle between the first and second planes on the basis of a third propagation time that the first signal requires from the first transmitter via the object to the third receiver.

5. The device according to claim 1, wherein the processor is further configured to determine an extension and/or a size and/or a condition of the surface of an object on the basis of a shape and/or a propagation time of one or more signals received by one or more of the at least three receivers.

6. The device according to claim 2, wherein the position of the first transmitter is identical to the position of one of the first, second, or third receiver.

7. The device according to claim 6, wherein one of the first, second, or third receiver also serves as the first transmitter.

8. The device according to claim 2, wherein the first transmitter is arranged between the first and second receivers.

9. A method for determining a three-dimensional position of an object, comprising the steps of: emitting a first signal from a first transmitting position and at least partly reflecting the first signal by the object; receiving the at least partly reflected first signal at three or more receiving positions comprising a first receiving position, a second and receiving position and a third receiving position, wherein the three or more receiving positions are arranged within a first plane comprising the first transmitting position, wherein the first receiving position and the second receiving position are arranged along a first straight line, and the third receiving position is arranged at a distance from the first straight line; determining a first propagation time, a second propagation time, and a third propagation time of the first signal from the first transmitting position to the object and then to the first, second, or third receiving position, respectively; and detecting the three-dimensional position of the object using the determined first, second, and third propagation times as well as the arrangement of the first transmitting position and the first, second, and third receiving positions, wherein the detection of the position of the object in the three-dimensional space is performed using an intersection of a first and a second elliptical orbit or a first and a second semielliptical orbit within a second plane, wherein the first propagation time of the first signal defines the first elliptical orbit or the first semielliptical orbit in the second plane and the second propagation time of the first signal defines the second elliptical orbit or the second semielliptical orbit in the second plane.

10. The method according to claim 9, wherein the first transmitting position is arranged along the first straight line.

11. The method according to claim 9, wherein determination of a first coordinate of the three-dimensional position of the object is performed on the basis of the first and the second propagation time that the first signal requires from the first transmitting position via the object to the first and the second receiving positions, and the distances of the first and the second receiving positions to the first transmitting position, wherein the first coordinate defines the position of the object along the first straight line.

12. The method according to claim 9, wherein a first focal point of a first elliptical orbit coincides with a second focal point of a first elliptical orbit and a first focal point of a second elliptical orbit coincides with a second focal point of a second elliptical orbit.

13. The method according to claim 11, wherein calculation of the position of the object in the three-dimensional space is performed using an angle between the first and second planes, wherein the third propagation time that the first signal requires from the first transmitting position via the object to the third receiving position defines the angle.

14. The method according to claim 9, wherein the method comprises a step of calculating distances of the first, second, and third receiving positions to the first transmitting position, wherein the calculation of the distances is based on a propagation time of the first signal transmitted directly from the first transmitting position to the respective receiving position using a predetermined propagation speed.

15. The method according to claim 9, wherein the method comprises a step of calculating a propagation speed of the emitted first signal, wherein the calculation of the propagation speed is based on the first signal transmitted directly from the first transmitting position to one of the three or more receiving positions using a predetermined distance between the first transmitting position and the one of the three or more receiving positions.

16. The method according to claim 9, wherein the method comprises a step of calculating an extension and/or a size and/or a condition of a surface of the object using a shape and/or a propagation time of one or more signals received by one or more of the three or more receiving positions.

17. The method according to claim 9, wherein the method is carried out repeatedly.

18. The method according to claim 10, wherein the first transmitting position is identical to one of the first, second, or third receiving position.

19. The method according to claim 10, wherein the first transmitting position is between the first and second receiving positions.

20. The method according to claim 9, wherein the method is carried out repeatedly and the first transmitting position alters between two or more of the three or more receiving positions.

Description

(1) In the following, the present invention is explained in more detail by means of exemplary embodiments and the Figures, in which:

(2) FIG. 1 shows a schematic top view of a sensor arrangement according to a first embodiment of the invention,

(3) FIG. 2 shows a schematic 3D view of the geometric construct for three propagation time measurements by reflection according to the first embodiment of the invention,

(4) FIG. 3a shows a schematic top view of the geometric construct for three propagation time measurements by reflection according to the first embodiment of the invention,

(5) FIG. 3b shows a schematic geometric construct of two auxiliary distances according to the first embodiment of the invention,

(6) FIG. 4 shows a schematic illustration of a propagation time measurement using two receivers according to the first embodiment of the invention,

(7) FIG. 5a shows a schematic geometric construct for two propagation time measurements by reflection according to the first embodiment of the invention,

(8) FIG. 5b shows a schematic illustration for the determination of the positions of the object on two elliptical orbits according to the first embodiment of the invention,

(9) FIG. 6 shows a schematic illustration of a possible structure of a semicircular transmitter according to a second embodiment of the invention,

(10) FIG. 7 shows a schematic 3D view of the geometric construct for three propagation time measurements by reflection using a semicircular transmitter according to the second embodiment of the invention,

(11) FIG. 8a shows a schematic top view of the geometric construct for three propagation time measurements by reflection using a semicircular transmitter according to the second embodiment of the invention,

(12) FIG. 8b shows a schematic illustration of the geometric construct of two auxiliary distances according to the second embodiment of the invention,

(13) FIG. 9 shows a schematic illustration of the propagation time measurement using two receivers and a semicircular transmitter according to the second embodiment of the invention,

(14) FIG. 10a shows a schematic illustration of the geometric construct for two propagation time measurements by reflection using the semicircular transmitter according to the second embodiment of the invention,

(15) FIG. 10b shows a schematic illustration of the possible positions of the object on four tilted semielliptical orbits according to the second embodiment of the invention,

(16) FIG. 11 shows a schematic top view of a sensor arrangement according to the third embodiment of the invention,

(17) FIG. 12 shows a schematic 3D view of the geometric construct for three propagation time measurements by reflection according to the third embodiment of the invention,

(18) FIG. 13a shows a schematic top view of the geometric construct of FIG. 12 according to the third embodiment of the invention,

(19) FIG. 13b shows a schematic illustration of the geometric construct of FIG. 12 according to the third embodiment of the invention,

(20) FIG. 14 shows a schematic illustration for the propagation time measurement using two receivers according to the third embodiment of the invention,

(21) FIG. 15a shows a schematic illustration of the geometric construct for two propagation time measurements by reflection according to the third embodiment of the invention,

(22) FIG. 15b shows a schematic illustration of the possible positions of the object on two circular orbits according to the third embodiment of the invention,

(23) FIG. 16 shows a schematic illustration for switching through the transmission functionality of the sensors according to the third embodiment of the invention,

(24) FIG. 17 shows a schematic illustration of the received signals according to a further embodiment of the invention,

(25) FIG. 18 shows a schematic illustration of the reflection at two extensive objects,

(26) FIG. 19 shows a schematic illustration of the signals received by the two objects according to FIG. 18,

(27) FIG. 20 shows a schematic illustration of the signals received by the two objects according to FIG. 18,

(28) FIG. 21 shows a schematic illustration of the reflection at two extensive objects and

(29) FIG. 22 shows a schematic illustration of a rough surface.

(30) FIG. 1 shows the schematic structure of a first embodiment of the present invention comprising a first receiver 10, a second receiver 20, a third receiver 30 and a transmitter 40. The relationships between the individual components are also shown and explained in more detail in the following.

(31) According to this embodiment, the transmitter 40 and the three receivers 10, 20, 30 are preferably at the same height (i.e., preferably within a first plane which is characterized here by z=0). The receiver 10 and the receiver 20 are preferably positioned in a row (i.e., preferably along a first straight line) with the transmitter 40 (preferably on the dashed line), i.e., here preferably on the x-axis (i.e., y=0 and z=0). The first receiver 10 has a distance 2f.sub.1 from the transmitter 40 and the second receiver 20 has also a distance 2f.sub.2 from the transmitter 40. The distances can have different sizes, i.e. 2f.sub.12f.sub.2, as well as have the same size, i.e., 2f.sub.1=2f.sub.2. In the x-direction the receiver 30 can have the distance b and in the y-direction the distance a from the coordinate origin O (here the position of the transmitter 40). The distances a and b can have any value. However, a should preferably be unequal to zero, i.e., the third receiver should preferably not be arranged on the x-axis. In other words, the receiver 30 should preferably be arranged at a distance from the first x-axis. Even if in FIG. 1 a mathematically correct positioning within one plane or along one straight line is assumed, it will be clear to the person skilled in the art that small deviations thereof, which may result from manufacturing tolerances etc., cannot entail the exclusion from the scope of protection of the invention. Even though the accuracy of the positioning of the individual system components has an influence on the accuracy of the determined position, the position of an object can even be determined with sufficient accuracy if the components 10, 20, 30 and 40 are arranged essentially within a plane and/or if the components 10, 20 and 40 are arranged essentially along a straight line.

(32) FIG. 2 shows a schematic illustration of the first embodiment in a three-dimensional view. According to FIG. 2, the transmitter 40, the receivers 10, 20, 30 as well as the object 50 are schematically illustrated. FIG. 2 further shows the distance s that the signal covers from the transmitter 40 to the object 50. The distances e.sub.1, e.sub.2, e.sub.3 according to FIG. 2 are the distances that the signal reflected at the object 50 travels to the respective receivers. The further indicated distances x.sub.2D, y.sub.2D, d, r, a, b are used and described in more detail in the following.

(33) According to the first embodiment of the present invention, three propagation time measurements are carried out for the determination of the three-dimensional position of the object 50. The three propagation time measurements are carried out by means of the transmitter 40 and the three receivers 10, 20, 30. The transmitter 40 emits a signal (the distance s denotes the way of the fastest signal from the transmitter 40 to the object 50, i.e. theoretically a wave or a particle stream is emitted. The distance s is the shortest propagation time and thus the first reflected signal at the object.) that propagates in the three-dimensional space until it hits an object 50 and is partly reflected and/or scattered at it. The reflected and/or scattered parts of the signal hit the three receivers 10, 20, 30 (along the distances e.sub.1, e.sub.2, e.sub.3). This yields the three propagation times that result from the propagation time from the transmitter 40 to the object 50 together with the propagation times from the object 50 to the individual receivers 10, 20, 30 (propagation time measurement).

(34) The consideration of the distance s that the signal travels from the transmitter 40 to the object 50 and of the individual distances e.sub.1, e.sub.2 and e.sub.3 that the signal travels from the object 50 to the respective receivers 10, 20, 30 leads to the three-dimensional geometric construct shown in FIG. 2 along with the auxiliary distances x.sub.2D, y.sub.2D, d, r, a, b.

(35) FIGS. 3a and 3b illustrate the geometric construct in a top view (FIG. 3a: top view of the z-axis) and in a side view (FIG. 3b: top view of the x-axis). FIG. 3b additionally depicts auxiliary distances q, y and the coordinate z.

(36) The two auxiliary distances y.sub.2D and d are always perpendicular to the x-axis and are connected with the object at the top. These auxiliary distances, together with the distance a, form a triangular structure (see FIGS. 3a and 3b).

(37) The geometrical construct provides the following equation:
a={square root over (d.sup.2z.sup.2)}+{square root over (y.sub.2D.sup.2z.sup.2)}(1)

(38) When the equation (1) for the distance a is solved for z, the following equation is obtained for the z-coordinate of the object 50 in the three-dimensional space:

(39) $\begin{array}{cc}z=\frac{1}{2}\sqrt{\frac{2y_{2D}^2{a}^2-y_{2D}^4-d^4+2d^2{y}_{2D}^2+2d^2{a}^2-a^4}{a}}& \left(2\right)\end{array}$
wherein the equation (2) has a positive sign with a>0 and a negative sign with a<0 (for the upper position of the object 50, i.e. above the plane in which the receivers 10, 20, 30 and the transmitter 40 are located). When the object 50 is below the plane in which the receivers 10, 20, 30 and the transmitter 40 are located, the sign is reversed.

(40) The auxiliary distance d in the equation (2) corresponds to the following equation:

(41) $\begin{array}{cc}d=\sqrt{{\left(U_{\mathrm{se}3}-\sqrt{x_{2D}^2+y_{2D}^2}\right)}^2-{\left(b-x_{2D}\right)}^2}& \left(3\right)\end{array}$

(42) The determination of the x-coordinate of the object in the three-dimensional space is performed by means of the following equation.

(43) $\begin{array}{cc}x=\frac{1}{4}\frac{-4f_2^2{U}_{\mathrm{se}1}+4f_1^2{U}_{\mathrm{se}2}-U_{\mathrm{se}2}{U}_{\mathrm{se}1}^2+U_{\mathrm{se}1}{U}_{\mathrm{se}2}^2}{f_2{U}_{\mathrm{se}1}+f_1{U}_{\mathrm{se}2}}& \left(4\right)\end{array}$

(44) The distances U.sub.se1, U.sub.se2 and U.sub.se3 are distances generated by the propagation time (sum of the distance s and the respective distance e.sub.n). These distances are determined via the propagation speed c of the signal and the propagation times t.sub.1, t.sub.2 and t.sub.3 that the signal requires from the transmitter 40 to the respective receiver 10, 20, 30 as follows:
U.sub.sen=c t.sub.n(5)

(45) The determination of the y-coordinate of the object in the three-dimensional space is performed by means of the following equation:
y={square root over (y.sub.2D.sup.2z.sup.2)}(6)
wherein the equation (6) for the y-coordinate in the three-dimensional space has a positive sign with y>q|a| and a>0, as well as with yq|a| and a<0. The equation (6) has a negative sign with all other value ranges.

(46) The auxiliary distance q mentioned in connection with the case distinction is calculated as follows:
q={square root over (d.sup.2z.sup.2)}(7)

(47) The auxiliary distances x.sub.2D and y.sub.2D mentioned in the above equations are the x-coordinate and y-coordinate of the object 50 in the case of the position determination of the object 50 in the two-dimensional space (where applicable in the second plane).

(48) According to the first embodiment of the present invention, a transmitter 40 and two receivers 10, 20 of the three receivers 10, 20, 30 are required for the determination of the x-coordinate and y-coordinate (x.sub.2D and y.sub.2D) of the object 50 in the two-dimensional space. This is schematically illustrated in FIG. 4. The geometric construct depicted in FIG. 5a is generated in this arrangement. Two elliptical orbits (see FIG. 5b) are formed by the two propagation time distances e.sub.1 and e.sub.2. According to FIG. 5b, the position of the object 50 is in the intersection of the two elliptical orbits. The two first focal points of the two elliptical orbits coincide exactly at the position of the transmitter 40 (origin of the distance s).

(49) The following equations for the two propagation time distances (U.sub.se1 and U.sub.se2) result from the geometric constructs according to FIGS. 5a and 5b:
U.sub.se1={square root over (y.sub.2D.sup.2+x.sub.2D.sup.2)}+{square root over (y.sub.2D.sup.2+(2f.sub.1x.sub.2D).sup.2)}(8)
and
U.sub.se2={square root over (y.sub.2D.sup.2+x.sub.2D.sup.2)}+{square root over (y.sub.2D.sup.2+(2f.sub.2+x.sub.2D).sup.2)}(9)

(50) When the equation (8) for the propagation time distance U.sub.se1 is solved for y.sub.2D, the following equation is obtained:

(51) $\begin{array}{cc}{y}_{2D}=\frac{1}{2}\frac{\sqrt{-\left(-4f_1^2+U_{\mathrm{se}1}^2\right)\left(-U_{\mathrm{se}1}^2+4x_{2D}^2+4f_1^2-8f_1{x}_{2D}\right)}}{U_{\mathrm{se}1}}& \left(10\right)\end{array}$

(52) When the equation (9) for the propagation time distance U.sub.se2 is solved for y.sub.2D, the following equation is obtained:

(53) $\begin{array}{cc}{y}_{2D}=\frac{1}{2}\frac{\sqrt{-\left(-4f_2^2+U_{\mathrm{se}2}^2\right)\left(-U_{\mathrm{se}2}^2+4x_{2D}^2+4f_2^2-8f_2{x}_{2D}\right)}}{U_{\mathrm{se}2}}& \left(11\right)\end{array}$

(54) The equations, with a positive sign, for the semielliptical orbits (10) and (11) are equated and zeroed:

(55) $\begin{array}{cc}\frac{1}{2}\frac{\left(\begin{array}{c}{U}_{\mathrm{se}2}\sqrt{-\left(-4f_1^2+U_{\mathrm{se}1}^2\right)\left(-U_{\mathrm{se}1}^2+4x_{2D}^2+4f_1^2-8f_1{x}_{2D}\right)}-\\ U_{\mathrm{se}1}\sqrt{-\left(-4f_2^2+U_{\mathrm{se}2}^2\right)\left(-U_{\mathrm{se}2}^2+4x_{2D}^2+4f_2^2+8f_2{x}_{2D}\right)}\end{array}\right)}{U_{\mathrm{se}1}{U}_{\mathrm{se}2}}=0& \left(12\right)\end{array}$

(56) After solving the equation (12) for x.sub.2D, i.a., the following equation is obtained:

(57) $\begin{array}{cc}{x}_{2D}=\frac{1}{4}\frac{-4f_2^2{U}_{\mathrm{se}1}+4f_1^2{U}_{\mathrm{se}2}-U_{\mathrm{se}2}{U}_{\mathrm{se}1}+U_{\mathrm{se}1}{U}_{\mathrm{se}2}^2}{f_2{U}_{\mathrm{se}1}+f_1{U}_{\mathrm{se}2}}& \left(13\right)\end{array}$

(58) It should be noted that the equation (13) for the x-component in the two-dimensional space corresponds to the equation of the x-component (equation (4)) in the three-dimensional space. When the equations for x.sub.2D and y.sub.2D for the positive semielliptical orbit (+) are inserted into the equation (2) for the three-dimensional z-coordinate and into the equation (6) for the three-dimensional y-coordinate, the following equation (14) and equation (15) are obtained:

(59) $\begin{array}{cc}z=\frac{1}{32a}\sqrt{\begin{array}{c}-\frac{32\left(-4f_2^2+U_{\mathrm{se}2}^2\right)\left(-4f_1^2+U_{\mathrm{se}1}^2\right)\left(U_{\mathrm{se}2}^2-2U_{\mathrm{se}1}{U}_{\mathrm{se}2}-8f_1{f}_2+U_{\mathrm{se}1}^2-4f_1^2-4f_2^2\right)a^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}-\\ \frac{{\left(-4f_2^2+U_{\mathrm{se}2}^2\right)}^2{\left(-4f_1^2+U_{\mathrm{se}1}^2\right)}^2{\left(U_{\mathrm{se}2}^2-2U_{\mathrm{se}1}{U}_{\mathrm{se}2}-8f_1{f}_2+U_{\mathrm{se}1}^2-4f_1^2-4f_2^2\right)}^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^4}-\\ 256{\left(\begin{array}{c}{\left(U_{\mathrm{se}3}-\frac{1}{2}\sqrt{\frac{{\left(-4f_1^2{f}_2-4f_1{f}_2^2+f_1{U}_{\mathrm{se}2}^2+U_{\mathrm{se}1}^2{f}_2\right)}^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}}\right)}^2-\\ {\left(b+\frac{U_{\mathrm{se}2}{U}_{\mathrm{se}1}^2-U_{\mathrm{se}2}^2{U}_{\mathrm{se}1}-4U_{\mathrm{se}2}{f}_1^2+4U_{\mathrm{se}1}{f}_2^2}{4\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}\right)}^2\end{array}\right)}^2-\\ \frac{1}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}\left(\begin{array}{c}32\left(\begin{array}{c}{\left(U_{\mathrm{se}3}-\frac{1}{2}\sqrt{\frac{{\left(-4f_1^2{f}_2-4f_1{f}_2^2+f_1{U}_{\mathrm{se}2}^2+U_{\mathrm{se}1}^2{f}_2\right)}^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}}\right)}^2-\\ {\left(b+\frac{U_{\mathrm{se}2}{U}_{\mathrm{se}1}^2-U_{\mathrm{se}2}^2{U}_{\mathrm{se}1}-4U_{\mathrm{se}2}{f}_1^2+4U_{\mathrm{se}1}{f}_2^2}{4\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}\right)}^2\end{array}\right)\\ \left(-4f_2^2+U_{\mathrm{se}2}^2\right)\left(-4f_1^2+U_{\mathrm{se}1}^2\right)\left(U_{\mathrm{se}2}^2-2U_{\mathrm{se}1}{U}_{\mathrm{se}2}-8f_1{f}_2+U_{\mathrm{se}1}^2-4f_1^2-4f_2^2\right)\end{array}\right)+\\ 512\left(\begin{array}{c}{\left(U_{\mathrm{se}3}-\frac{1}{2}\sqrt{\frac{{\left(-4f_1^2{f}_2-4f_1{f}_2^2+f_1{U}_{\mathrm{se}2}^2+U_{\mathrm{se}1}^2{f}_2\right)}^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}}\right)}^2-\\ {\left(b+\frac{U_{\mathrm{se}2}{U}_{\mathrm{se}1}^2-U_{\mathrm{se}2}^2{U}_{\mathrm{se}1}-4U_{\mathrm{se}2}{f}_1^2+4U_{\mathrm{se}1}{f}_2^2}{4\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}\right)}^2\end{array}\right)a^2-256a^4\end{array}}& \left(14\right)\\ y=\frac{1}{32}\sqrt{\begin{array}{c}-\frac{64\left(-4f_2^2+U_{\mathrm{se}2}^2\right)\left(-4f_1^2+U_{\mathrm{se}1}^2\right)\left(U_{\mathrm{se}2}^2-2U_{\mathrm{se}1}{U}_{\mathrm{se}2}-8f_1{f}_2+U_{\mathrm{se}1}^2-4f_1^2-4f_2^2\right)}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}-\\ \frac{1}{a^2}\left(\begin{array}{c}-\frac{32\left(-4f_2^2+U_{\mathrm{se}2}^2\right)\left(-4f_1^2+U_{\mathrm{se}1}^2\right)\left(U_{\mathrm{se}2}^2-2U_{\mathrm{se}1}{U}_{\mathrm{se}2}-8f_1{f}_2+U_{\mathrm{se}1}^2-4f_1^2-4f_2^2\right)a^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}-\\ \frac{{\left(-4f_2^2+U_{\mathrm{se}2}^2\right)}^2{\left(-4f_1^2+U_{\mathrm{se}1}^2\right)}^2{\left(U_{\mathrm{se}2}^2-2U_{\mathrm{se}1}{U}_{\mathrm{se}2}-8f_1{f}_2+U_{\mathrm{se}1}^2-4f_1^2-4f_2^2\right)}^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^4}-\\ 256{\left(\begin{array}{c}{\left(U_{\mathrm{se}3}-\frac{1}{2}\sqrt{\frac{{\left(-4f_1^2{f}_2-4f_1{f}_2^2+f_1{U}_{\mathrm{se}2}^2+U_{\mathrm{se}1}^2{f}_2\right)}^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}}\right)}^2-\\ {\left(b+\frac{U_{\mathrm{se}2}{U}_{\mathrm{se}1}^2-U_{\mathrm{se}2}^2{U}_{\mathrm{se}1}-4U_{\mathrm{se}2}{f}_1^2+4U_{\mathrm{se}1}{f}_2^2}{4\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}\right)}^2\end{array}\right)}^2-\\ \frac{1}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}\left(\begin{array}{c}32\left(\begin{array}{c}{\left(U_{\mathrm{se}3}-\frac{1}{2}\sqrt{\frac{{\left(-4f_1^2{f}_2-4f_1{f}_2^2+f_1{U}_{\mathrm{se}2}^2+U_{\mathrm{se}1}^2{f}_2\right)}^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}}\right)}^2-\\ {\left(b+\frac{U_{\mathrm{se}2}{U}_{\mathrm{se}1}^2-U_{\mathrm{se}2}^2{U}_{\mathrm{se}1}-4U_{\mathrm{se}2}{f}_1^2+4U_{\mathrm{se}1}{f}_2^2}{4\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}\right)}^2\end{array}\right)\\ \left(-4f_2^2+U_{\mathrm{se}2}^2\right)\left(-4f_1^2+U_{\mathrm{se}1}^2\right)\left(U_{\mathrm{se}2}^2-2U_{\mathrm{se}1}{U}_{\mathrm{se}2}-8f_1{f}_2+U_{\mathrm{se}1}^2-4f_1^2-4f_2^2\right)\end{array}\right)+\\ 512\left(\begin{array}{c}{\left(U_{\mathrm{se}3}-\frac{1}{2}\sqrt{\frac{{\left(-4f_1^2{f}_2-4f_1{f}_2^2+f_1{U}_{\mathrm{se}2}^2+U_{\mathrm{se}1}^2{f}_2\right)}^2}{{\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}^2}}\right)}^2-\\ {\left(b+\frac{U_{\mathrm{se}2}{U}_{\mathrm{se}1}^2-U_{\mathrm{se}2}^2{U}_{\mathrm{se}1}-4U_{\mathrm{se}2}{f}_1^2+4U_{\mathrm{se}1}{f}_2^2}{4\left(U_{\mathrm{se}2}{f}_1+U_{\mathrm{se}1}{f}_2\right)}\right)}^2\end{array}\right)a^2-256a^4\end{array}\right)\end{array}}& \left(15\right)\end{array}$

(60) The preferred embodiment discussed in detail here is, of course, only an exemplary way of calculating the coordinates. However, the invention is not at all restricted to this example. It is rather also possible to derive other equations from the geometric situation illustrated in the Figures by means of which the person skilled in the art can calculate the individual coordinates of the object position. It is mentioned only exemplarily that in a polar coordinate system all equations look completely different. However, the invention is based, i.a., on the general basic idea that two of the determined three propagation times define two intersecting elliptical orbits, as illustrated in FIG. 5b. Accordingly, the processor is preferably configured to determine a first elliptical orbit or a first semielliptical orbit on the basis of a first propagation time which the signal requires from the transmitter via the object to the first receiver within a second plane and to determine a second elliptical orbit or a second semielliptical orbit on the basis of a second propagation time which the signal requires from the transmitter via the object to the second receiver within the second plane, wherein the position of the object in the three-dimensional space is calculated using the position of the intersection of the two elliptical orbits or semielliptical orbits in the second plane without the invention being restricted to a specific kind of calculation.

(61) The above-discussed preferred embodiment is based on the assumption that the transmitter 40 emits a signal essentially isotropically. However, other transmitters, such as, for example, the semicircular transmitter 60 illustrated in FIG. 6, can also be used instead. The semicircular transmitter 60 according to a second embodiment of the present invention comprises a main body 62 through which a slit 61 (also referred to a opening or gap) extends on a peripheral side of the main body 62. A signal is emitted originating from the slit 61 so as to propagate from there in the three-dimensional space. It can thus be achieved that a signal has a defined starting point on a semicircle, i.e., the signal escapes through the slit 61 from the semicircular transmitter 60 and thereupon propagates in the three-dimensional space above the semicircular transmitter 60.

(62) In other words, a semicircular transmitter 60 is a transmitter having a signal source in the form of a semicircular gap 61. The signal is emitted at the same time from the signal source into the three-dimensional half space. The semicircular shape of the signal source enables an object to be detected above the semicircular transmitter 60, i.e., for z>0 (cf. FIG. 7). Variations of the semicircular transmitter 60 in which the signal source is smaller or greater than 180 degrees are certainly also possible.

(63) FIG. 7 depicts a three-dimensional illustration of the situation according to the second embodiment of the present invention. The signal emitted by the semicircular transmitter 60 is reflected by the object 50 to be detected into the three receivers 10, 20, 30. In addition to the distances s, e.sub.1, e.sub.2, e.sub.3, d, a, b, r, y.sub.2D, x.sub.2D known from FIG. 2, the radius h of the semicircular transmitter 60 is indicated in FIG. 7.

(64) The arrangement of the receivers 10, 20, 30 and the transmitter, here the semicircular transmitter 60, is preferably analogous to the arrangement as shown in FIG. 1. It is to be noted that when a semicircular transmitter 60 is used the center of the semicircle corresponds to the position of the transmitter 40 according to FIG. 1, i.e., the center of the semicircle preferably is, on the one hand, within the same plane as the receivers 10, 20, 30 and, on the other hand, on a straight line with the receivers 10, 20. Thus, the origin of the signal is at a predetermined distance h (radius of the semicircular transmitter 60) from the origin of the coordinate system (cf. FIG. 7).

(65) According to the second embodiment of the present invention, three propagation time measurements are performed for the determination of the three-dimensional position of an object 50. The three propagation time measurements are performed by means of the semicircular transmitter 60 and the three receivers 10, 20, 30. A signal that propagates in the three-dimensional space until it hits an object 50 (distance s) and is partly reflected is emitted by the semicircular transmitter 60 as described above. The reflected parts of the signal hit the three receivers 10, 20, 30 (distances e.sub.1, e.sub.2, e.sub.3). This yields the three propagation times that result from the propagation time from the semicircular transmitter 60 to the object 50 (time that the signal requires for the distance s) together with the propagation times from the object 50 to the individual receivers 10, 20, 30 (time that the signal requires for the respective distances e.sub.1, e.sub.2, e.sub.3) (propagation time measurement).

(66) The three receivers 10, 20, 30 and the center of the semicircle (radius origin) are preferably arranged or fixed in one plane (at the same height). The receiver 10 and the receiver 20 are preferably positioned on a straight line along with the semicircular transmitter 60 (preferably on the dashed line), i.e., preferably directly on the x-axis. In the x-direction the receiver 30 can have the distance b and in the y-direction the distance a from the coordinate origin O. The distances a and b can have any value. Preferably, a should be unequal to zero.

(67) The consideration of the distance s that the signal travels from the semicircular transmitter 60 to the object 50 and of the individual distances e.sub.1, e.sub.2 and e.sub.3 that the signal travels from the object 50 to the respective receivers 10, 20, 30 leads to the three-dimensional geometric construct shown in FIG. 7 along with the depicted auxiliary distances.

(68) FIG. 8a illustrates the geometric construct of FIG. 7 in a top view (top view of the z-axis) and FIG. 8b illustrates the geometric construct of FIG. 7 in a side view (top view of the x-axis).

(69) The two auxiliary distances y.sub.2D and d are always perpendicular to the x-axis and are connected with the object 50 at the top. These auxiliary distances, together with the distance a, form a triangular structure according to FIGS. 8a and 8b.

(70) The geometrical construct provides the following equation:
a={square root over (d.sup.2z.sup.2)}+{square root over (y.sub.2D.sup.2z.sup.2)}(16)

(71) When the equation (16) for the distance a is solved for z, the following equation is obtained for the z-coordinate of the object 50 in the three-dimensional space:

(72) $\begin{array}{cc}z=\frac{1}{2}\frac{\sqrt{2y_{2D}^2{a}^2-y_{2D}^4-d^4+2d^2{y}_{2D}^2+2d^2{a}^2-a^4}}{a}& \left(17\right)\end{array}$
wherein the equation (17) has a positive sign with a>0 and a negative sign with a<0 (for the upper position of the object 50). For the lower position, the sign is reversed. Upper position means that the object 50 is above the plane defined by the three receivers 10, 20, 30. Lower position accordingly means that the object 50 is below the plane defined by the three receivers 10, 20, 30.

(73) The auxiliary distance d in the equation (17) corresponds to the following equation:

(74) 0$\begin{array}{cc}d=\sqrt{{\left(U_{\mathrm{se}3}-\sqrt{x_{2D}^2+{\left(y_{2D}-h\right)}^2}\right)}^2-{\left(b-x_{2D}\right)}^2}& \left(18\right)\end{array}$

(75) The determination of the x-coordinate of the object 50 in the three-dimensional space is performed by means of the equation (19) and the equation (20). The equation (19) indicates the x-coordinate of the object 50 for a propagation time distance with U.sub.se1U.sub.se2 and the equation (20) the x-coordinate for a propagation time distance with U.sub.se1<U.sub.se2. For the sake of completeness, it is mentioned that it can likewise read U.sub.se1>U.sub.se2 and U.sub.se1U.sub.se2.

(76) $\begin{array}{cc}{x}_{\mathrm{SP}3}=\frac{1}{2}\frac{\left(\begin{array}{c}8h^2{f}_1^2{f}_2+8f_2^3{U}_{\mathrm{se}1}^2-8U_{\mathrm{se}2}^2{f}_1^3-2U_{\mathrm{se}1}{U}_{\mathrm{se}2}^3{f}_1+8h^2{f}_1^3-8h^2{f}_2^3+2f_1{U}_{\mathrm{se}1}^2{U}_{\mathrm{se}2}^2-2h^2{f}_1{U}_{\mathrm{se}1}^2-8f_2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}{f}_1^2-\\ 8h^2{f}_1{f}_2^2+2U_{\mathrm{se}1}^3{f}_2{U}_{\mathrm{se}2}-2h^2{f}_2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}+2h^2{f}_2{U}_{\mathrm{se}2}^2+2f_1{h}^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}+8f_2^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}{f}_1-2f_2{U}_{\mathrm{se}1}^2{U}_{\mathrm{se}2}^2+\\ \sqrt{{\left(U_{\mathrm{se}1}-U_{\mathrm{se}2}\right)}^2{h}^2\left(-U_{\mathrm{se}2}^2+h^2+4f_2^2\right)\left(\begin{array}{c}{U}_{\mathrm{se}1}^2{U}_{\mathrm{se}2}^2-2U_{\mathrm{se}1}^3{U}_{\mathrm{se}2}+U_{\mathrm{se}1}^4-h^2{U}_{\mathrm{se}2}^2+2h^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}-h^2{U}_{\mathrm{se}1}^2-\\ 4f_2^2{U}_{\mathrm{se}1}^2+4f_2^2{h}^2-8f_1{f}_2{U}_{\mathrm{se}1}^2+8f_1{f}_2{h}^2-4f_1^2{U}_{\mathrm{se}2}^2+8f_1^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}-\\ 8f_1^2{U}_{\mathrm{se}1}^2+4f_1^2{h}^2+16f_1^2{f}_2^2+32f_1^3{f}_2+16f_1^4\end{array}\right)}\end{array}\right)}{-4f_1^2{U}_{\mathrm{se}2}^2-4f_2^2{U}_{\mathrm{se}1}^2+4f_2^2{h}^2+4f_1^2{h}^2-h^2{U}_{\mathrm{se}2}^2-h^2{U}_{\mathrm{se}1}^2-8f_1{f}_2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}+8f_1{f}_2{h}^2+2h^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}}& \left(19\right)\\ x_{\mathrm{SP}4}=-\frac{1}{2}\frac{\left(\begin{array}{c}-8h^2{f}_1^2{f}_2-8f_2^3{U}_{\mathrm{se}1}^2+8U_{\mathrm{se}2}^2{f}_1^3+2U_{\mathrm{se}1}{U}_{\mathrm{se}2}^3{f}_1-8h^2{f}_1^3+8h^2{f}_2^3-2f_1{U}_{\mathrm{se}1}^2{U}_{\mathrm{se}2}^2+2h^2{f}_1{U}_{\mathrm{se}1}^2+8f_2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}{f}_1^2+\\ 8h^2{f}_1{f}_2^2-2U_{\mathrm{se}1}^3{f}_2{U}_{\mathrm{se}2}+2h^2{f}_2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}-2h^2{f}_2{U}_{\mathrm{se}2}^2-2f_1{h}^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}-8f_2^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}{f}_1+2f_2{U}_{\mathrm{se}1}^2{U}_{\mathrm{se}2}^2+\\ \sqrt{{\left(U_{\mathrm{se}1}-U_{\mathrm{se}2}\right)}^2{h}^2\left(-U_{\mathrm{se}2}^2+h^2+4f_2^2\right)\left(\begin{array}{c}{U}_{\mathrm{se}1}^2{U}_{\mathrm{se}2}^2-2U_{\mathrm{se}1}^3{U}_{\mathrm{se}2}+U_{\mathrm{se}1}^4-h^2{U}_{\mathrm{se}2}^2+2h^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}-h^2{U}_{\mathrm{se}1}^2-\\ 4f_2^2{U}_{\mathrm{se}1}^2+4f_2^2{h}^2-8f_1{f}_2{U}_{\mathrm{se}1}^2+8f_1{f}_2{h}^2-4f_1^2{U}_{\mathrm{se}2}^2+8f_1^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}-\\ 8f_1^2{U}_{\mathrm{se}1}^2+4f_1^2{h}^2+16f_1^2{f}_2^2+32f_1^3{f}_2+16f_1^4\end{array}\right)}\end{array}\right)}{-4f_1^2{U}_{\mathrm{se}2}^2-4f_2^2{U}_{\mathrm{se}1}^2+4f_2^2{h}^2+4f_1^2{h}^2-h^2{U}_{\mathrm{se}2}^2-h^2{U}_{\mathrm{se}1}^2-8f_1{f}_2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}+8f_1{f}_2{h}^2+2h^2{U}_{\mathrm{se}1}{U}_{\mathrm{se}2}}& \left(20\right)\end{array}$

(77) The distances U.sub.se1, U.sub.se2 and U.sub.se3 are distances generated by the propagation time (sum of the distance s and the respective distance e.sub.n). These distances are determined via the propagation speed c of the signal and the propagation times t.sub.1, t.sub.2 and t.sub.3 of the signal as follows:
U.sub.sen=c t.sub.n(21)

(78) The determination of the y-coordinate of the object 50 in the three-dimensional space is performed by means of the following equation:
y={square root over (y.sub.2D.sup.2z.sup.2)}(22)
wherein the equation (22) for the y-coordinate in the three-dimensional space has a positive sign with y>q|a| and a>0, as well as with yq|a| and a<0. The equation (22) has a negative sign with all other value ranges.

(79) The auxiliary distance q mentioned in connection with the case distinction is calculated as follows:
q={square root over (d.sup.2z.sup.2)}(23)

(80) The auxiliary distances x.sub.2D and y.sub.2D mentioned in the above equations are the x-coordinate and y-coordinate of the object 50 in the case of the position determination of the object 50 in the two-dimensional space.

(81) FIG. 9 shows a semicircular transmitter 60 and two receivers 10, 20 for the determination of the x-coordinate and y-coordinate (x.sub.2D and y.sub.2D) of the object 50 in the two-dimensional space.

(82) On the basis of FIG. 9, the geometric construct according to FIG. 10a can be generated. Furthermore, four intersecting semielliptical orbits according to FIG. 10b are generated by this arrangement.

(83) According to the present invention, in particular the second embodiment, the position of the object 50 is at the intersection of the two elliptical orbits (semielliptical orbits). On the basis of the condition that the signal of the semicircular transmitter propagates above the plane, the intersection determining the position of the object 50 is likewise located above the two-dimensional construct (above the x.sub.2D axis) according to FIG. 9. It is clear that the semicircular transmitter 60 can also transmit in any direction other than the one in this example. It is in any case clear that due to the restricted transmission area of the semicircular transmitter 60 the intersection relevant to the position of the object is defined.

(84) The following equations for the two propagation time distances (U.sub.se1 and U.sub.se2) result from the geometric construct:
U.sub.se1={square root over ((y.sub.2Dh).sup.2+x.sub.2D.sup.2)}+{square root over (y.sub.2D.sup.2+(2f.sub.1x.sub.2D).sup.2)}(24)
and
U.sub.se2={square root over ((y.sub.2Dh).sup.2+x.sub.2D.sup.2)}+{square root over (y.sub.2D.sup.2+(2f.sub.2+x.sub.2D.sup.2).sup.2)}(25)

(85) When the equation (24) for the propagation time distance U.sub.se1 is solved for y.sub.2D, the following equation is obtained:

(86) $\begin{array}{cc}{y}_{2D}=\frac{1}{2}\frac{\left(\begin{array}{c}{h}^2{U}_{\mathrm{se}1}^2-h^4+\\ 4f_1^2{h}^2-\\ 4f_1{x}_{2D}{h}^2+U_{\mathrm{se}1}\end{array}\right)\sqrt{h^2\left(\begin{array}{c}16f_1^4-8f_1{x}_{2D}{h}^2+\\ 16f_1^2{x}_{2D}^2-32f_1^3{x}_{2D}+\\ 8f_1^2{h}^2+4x_{2D}^2{h}^2+\\ h^4+8f_1{x}_{2D}{U}_{\mathrm{se}1}^2-\\ 8f_1^2{U}_{\mathrm{se}1}^2-4U_{\mathrm{se}1}^2{x}_{2D}^2-\\ 2h^2{U}_{\mathrm{se}1}^2+U_{\mathrm{se}1}^4\end{array}\right)}}{\left(U_{\mathrm{se}1}^2-h^2\right)h}& \left(26\right)\end{array}$

(87) When the equation (25) for the propagation time distance U.sub.se2 is solved for y.sub.2D, the following equation is obtained:

(88) $\begin{array}{cc}{y}_{2D}=\frac{1}{2}\frac{\left(\begin{array}{c}{h}^2{U}_{\mathrm{se}2}^2-h^4+\\ 4f_2^2{h}^2-\\ 4f_2{x}_{2D}{h}^2+U_{\mathrm{se}2}\end{array}\right)\sqrt{h^2\left(\begin{array}{c}16f_2^4-8f_2{x}_{2D}{h}^2+\\ 16f_2^2{x}_{2D}^2-32f_2^3{x}_{2D}+\\ 8f_2^2{h}^2+4x_{2D}^2{h}^2+\\ h^4+8f_2{x}_{2D}{U}_{\mathrm{se}2}^2-\\ 8f_2^2{U}_{\mathrm{se}2}^2-4U_{\mathrm{se}2}^2{x}_{2D}^2-\\ 2h^2{U}_{\mathrm{se}2}^2+U_{\mathrm{se}2}^4\end{array}\right)}}{\left(U_{\mathrm{se}2}^2-h^2\right)h}& \left(27\right)\end{array}$

(89) When the equation (26) and the equation (27), with a positive sign, are equated, zeroed and solved for x.sub.2D, the equation for the x.sub.2D coordinate is obtained. It is the same as the one for the x-coordinate and thus corresponds to the equation (19) and the equation (20).

(90) It is refrained from the insertion of the individual results for the second embodiment due to the complexity of the expressions. It is clear to the person skilled in the art that this can be made essentially analogously to the way described on the basis of the first embodiment in order to obtain a terminal equation for the coordinates of the object 50 in the three-dimensional space. However, it is also explicitly noted that it is not absolutely necessary to set up the terminal equations since a successive calculation of the above described calculation steps is sufficient to determine the position of the object 50 in the three-dimensional space.

(91) FIG. 11 depicts a third embodiment of the present invention comprising three sensors 100, 200, 300. Each of the sensors 100, 200, 300 can be a component that has both transmitting functionality and receiving functionality. According to the present invention, at least three sensors 100, 200, 300 having receiving functionality are to be provided and at least one of the at least three sensors 100, 200, 300 can additionally be provided with transmitting functionality. However, it is also possible that all of the at least three sensors 100, 200, 300 are provided with transmitting and receiving functionalities, for switching through the system, e.g., to transmit from different transmitting positions (either at the same time or in a time-shifted way).

(92) According to the third embodiment, the determination of the three-dimensional position of an object 50 can also be performed on the basis of three propagation time measurements. The three propagation time measurements are performed according to the third embodiment by means of three sensors 100, 200, 300, wherein at least one sensor (e.g., sensor 100) can transmit and receive. It is likewise possible that separate transmitting and receiving devices are installed closely to each other for sending and receiving from more or less the same position. In other words, each of the sensors 100, 200, 300 can also consist of two separate transmitting and receiving devices arranged in direct proximity to each other.

(93) According to FIG. 12, a signal that propagates in the three-dimensional space until it hits an object 50 (distance s) and is partly reflected is emitted by the sensor 100. The reflected parts of the signal hit the three sensors 100, 200, 300 (distances e.sub.1, e.sub.2, e.sub.3). This yields the three propagation times that result from the propagation time from the sensor 100 to the object 50 together with the propagation times from the object 50 to the individual sensors 100, 200, 300 (propagation time measurement).

(94) The three sensors 100, 200, 300 are preferably arranged in one plane (i.e., at the same height). In the x-direction the sensor 300 can have the distance b and in the y-direction the distance a from the coordinate origin O. The distances a and b can have any value wherein the distance a preferably is unequal to zero. The coordinate origin O is preferably on a straight line that preferably extends between the sensors 100 and 200. However, it is clear that the position of the coordinate origin O has no influence on the workability of the present invention.

(95) The consideration of the distance s that the signal travels from the sensor 100 to the object 50 and of the individual distances e.sub.1, e.sub.2 and e.sub.3 that the signal travels from the object 50 to the respective sensors 100, 200, 300 leads to the three-dimensional geometric construct shown in FIG. 12 along with the depicted auxiliary distances (x.sub.2D, y.sub.2D, a, b, r, d).

(96) A top view of the geometric construct according to FIG. 12 (top view of the z-axis) yields together with the auxiliary distances the geometric construct according to FIG. 13a. A side view of the geometric construct according to FIG. 12 (top view of the x-axis) yields together with the auxiliary distances the geometric construct according to FIG. 13b.

(97) The two auxiliary distances y.sub.2D and d are always perpendicular to the x-axis and are connected with the object at the top. These form together with the distance a a triangular structure according to FIGS. 13a and 13b.

(98) The geometrical construct provides the following equation:
a={square root over (d.sup.2z.sup.2)}+{square root over (y.sub.2D.sup.2z.sup.2)}(28)

(99) When the equation (28) for the distance a is solved for z, the following equation is obtained for the z-coordinate of the object 50 in the three-dimensional space:

(100) $\begin{array}{cc}z=\frac{1}{2}\frac{\sqrt{2y_{2D}^2{a}^2-y_{2D}^4-d^4+2d^2{y}_{2D}^2+2d^2{a}^2-a^4}}{a}& \left(29\right)\end{array}$
wherein the equation (29) has a positive sign with a>0 and a negative sign with a<0 (for the upper position of the object 50). For the lower position, the sign is reversed.

(101) The auxiliary distance d in the equation (29) corresponds to the following equation:

(102) $\begin{array}{cc}d=\sqrt{{\left(U_{\mathrm{se}3}-\frac{U_{\mathrm{se}1}}{2}\right)}^2-{\left(b-x_{2D}\right)}^2}& \left(30\right)\end{array}$

(103) The determination of the x-coordinate of the object 50 in the three-dimensional space is performed by means of the following equation:

(104) $\begin{array}{cc}x=-\frac{1}{2}\frac{-f_1^2+f_2^2+U_{\mathrm{se}1}{U}_{\mathrm{se}2}-U_{\mathrm{se}2}^2}{f_1+f_2}& \left(31\right)\end{array}$

(105) The distances U.sub.se1, U.sub.se2 and U.sub.se3 are distances generated by the propagation time (sum of the distance s and the respective distance e.sub.n). These distances are determined via the propagation speed c of the signal and the propagation times t.sub.1, t.sub.2 and t.sub.3 of the signal as follows:
U.sub.sen=c t.sub.n(32)

(106) The determination of the y-coordinate of the object 50 in the three-dimensional space is performed by means of the following equation:
y={square root over (y.sub.2D.sup.2z.sup.2)}(33)
wherein the equation (33) for the y-coordinate in the three-dimensional space has a positive sign with y>q|a| and a>0, as well as with yq|a| and a<0. The equation (33) has a negative sign with all other value ranges.

(107) The auxiliary distance q mentioned in connection with the case distinction is calculated as follows:
q={square root over (d.sup.2z.sup.2)}(34)

(108) The auxiliary distances x.sub.2D and y.sub.2D mentioned in the above equations are the x-coordinate and y-coordinate of the object 50 in the case of the position determination of the object 50 in the two-dimensional space (where applicable in the second plane).

(109) A transmitter 100 and two receivers 100, 200 are required for the determination of the x-coordinate and y-coordinate (x.sub.2D and y.sub.2D) of the object 50 in the two-dimensional space (see FIG. 14). The geometric construct of FIG. 15a is derived on the basis of FIG. 14. According to FIG. 15b, two circular orbits or semicircular orbits are formed by the two propagation time distances. The positive result of the equation (37) and the equation (38) for the circular orbits describes the upper semicircular orbit and the negative the lower.