reference is now made to fig1 which illustrates a bend - responsive catheter 20 , in accordance with a preferred embodiment of the present invention . catheter 20 includes a distal end 22 , which is preferably inserted in the heart of a subject , and a proximal end 24 , which is coupled to a control console 26 . adjacent to distal end 22 , catheter 20 includes a first position - sensing element 28 and , proximal thereto , a second position - sensing element 30 , which serves to enable determination of a bending angle of catheter 20 , as will be described below . preferably , each of elements 28 and 30 comprises three substantially orthogonal , non - concentric coils , as described in the above - mentioned pct publication wo96 / 05768 , which generate signals responsive to magnetic fields applied by field generators 32 . these signals are conveyed via wires 34 to signal processing and computing circuitry 36 in console 26 , which preferably also provides driver and control signals to generators 32 . circuitry 36 analyzes the signals , as further described in the pct publication , in order to determine the six - dimensional translational and orientational coordinates of elements 28 and 30 in relation to a frame of reference established by generators 32 . alternatively , it is sufficient that one of elements 28 and 30 comprise three such coils , and that the other of the elements comprise a single coil , as described in the above - mentioned u . s . pat . no . 5 , 391 , 199 patent . as described in the patent , three - dimensional translational coordinates of the single - coil element are determined . further alternatively , sensors 28 and 30 may comprise other types and combinations of position sensors , known in the art . it is sufficient , for example , that element 28 be such as to enable determination of three - dimensional translational coordinates and two - dimensional angular elevation and azimuth coordinates with respect thereto , while three - dimensional coordinates are determined with respect to element 30 . if bending of catheter 20 is constrained to a plane , as shown in fig2 a and described below , it is sufficient to determine two - dimensional coordinates of element 30 . catheter 20 preferably includes a resilient longitudinal member 38 , for example , a coil spring element , which is fixed within the catheter along a longitudinal axis thereof . preferably , there is a sufficient distance between metal parts of member 38 and sensors 28 and 30 so that the metal parts do not significantly distort the magnetic fields at the sensors . such distortion may be caused , for example , by eddy currents induced in the metal parts or by bending of the magnetic field lines by ferromagnetic materials . on account of member 38 , catheter 20 has a generally constant elasticity over at least a portion 40 of its length , preferably extending at least from element 30 , or from another point proximal thereto , out to distal end 22 , or at least to element 28 . portion 40 of catheter 20 is preferably short enough , generally less that about 9 cm long , so that it is inserted entirely into a chamber of the heart with no more than a single bend in the portion . as a result , when portion 40 is bent , whereby element 30 is translationally displaced and orientationally rotated by a known angle relative to element 28 , portion 40 will assume an arcuate or helical shape having a known radius of curvature , determined by the known angle . fig2 a illustrates , for example , a case in which portion 40 of catheter 20 is bent in a plane , which we take to be the plane of the page without loss of generality . the length of portion 40 is taken to be l , as shown . respective first and second local coordinate axes 50 ( x o , y o , z o ) and 52 ( x 1 , y 1 , z 1 ) are defined at the positions of first and second elements 28 and 30 , wherein the local z - axis is taken in every case taken to be aligned with the longitudinal axis of catheter 20 , generally parallel to member 38 . the six - dimensional position coordinates of first element 28 are determined and used to define the element &# 39 ; s translational position and first local coordinate axes 50 . the orientation coordinates of second element 30 define second local axes 52 , which together with axes 50 determine a bend angle θ , as shown . an arc is thus defined having a radius of curvature given by r = l / θ , and a center of curvature 54 at a position y = r defined with respect to coordinate axes 50 or 52 . the elasticity of member 38 ensures that portion 40 will generally follow this arc , so that the position of any point within portion 40 of catheter 20 may be conveniently determined . fig2 b schematically illustrates the more general case , in which catheter 20 is free to twist in three dimensions . in the case shown here , portion 40 of catheter 20 has been twisted about its longitudinal axis by approximately 180 °, so that axes x 1 and y 1 of second local axes 52 are oriented in generally opposite respective directions to axes x 0 and y 0 of local axes 50 . the elasticity of member 38 causes portion 40 to assume a generally right - helical form , within the bounds of a cylinder 54 having a diameter r c and length d , as shown in the figure . the length d is defined by the translational displacement of element 30 relative to element 28 , but determining r c generally requires solving an integral equation . preferably , solutions to the equation are stored in the form of a look - up table , preferably within signal processing circuitry 36 , as is known in the art . r c and d then determine the pitch of the helical form , so that the position of any point within portion 40 of catheter 20 may again be conveniently determined . preferably , portion 40 of catheter 20 will not be allowed to twist by more than 180 ° in either the clockwise or counterclockwise direction , so that the relative rotational coordinates of elements 28 and 30 will be unambiguous . if necessary , however , the twist of portion 40 may be continuously monitored , by analyzing the signals received from the elements , as catheter 20 is being inserted into and manipulated inside the body , so that rotations of greater than 180 ° will be detected . these greater twist angles are then used in appropriately determining r c , as described above . in the preferred embodiments described above , it is assumed that portion 40 of catheter 20 is free to move within a body cavity , and that the shape and configuration of portion 40 are determined substantially by its own elasticity . portion 40 is caused to bend by a combination of a compressive axial force , generally exerted from proximal end 24 of catheter 20 by a user , such as a physician , and a lateral deflecting force exerted on distal end 22 by body tissue with which the distal end is in contact . fig3 schematically illustrates an alternative preferred embodiment of the present invention , in which catheter 20 bends controllably , not necessarily in an arcuate or helical form , by means of a steering mechanism 56 . preferably , mechanism 56 comprises an electronically - or mechanically - controlled deflection element , operating under the control of console 26 , as described in the above - mentioned pct patent application no . pct / il97 / 00159 . alternatively , mechanism 56 may comprise any suitable catheter steering or deflection device known in the art . catheter 20 is sufficiently rigid , except in an immediate vicinity of mechanism 56 , so as to bend only in the immediate vicinity of the mechanism . the position coordinates of elements 28 and 30 are used to measure the deflection angle θ , whereby the location of any point along portion 40 of catheter 20 may be determined . preferably , the measured deflection angle is also used to provide feedback for closed - loop control of mechanism 56 . fig4 schematically illustrates another preferred embodiment of the present invention , which is similar to the embodiments described above except that in place of second position - sensing element 30 , catheter 20 as shown here includes a bend sensor 80 , responsive to the angle of bending of the catheter . bend sensor 80 preferably comprises at least one piezoelectric element , or more preferably , three such elements 82 , 84 and 86 as shown in the figure . the piezoelectric elements are mechanically coupled to resilient member 38 , so that when member 38 is bent , as described above , the bending force is conveyed to and acts upon the elements . as is known in the art , the piezoelectric crystals generate voltage signals that are generally proportional to this bending force , which signals are conveyed by wires 34 to signal processing circuitry 36 in console 26 . each of elements 82 , 84 and 86 includes a piezoelectric crystal having a crystal axis aligned orthogonally to the axes of the other two elements , so that each crystal generates signals responsive to bending of catheter 20 about a different axis . thus , as shown in fig4 element 82 generates signals responsive to twisting of catheter 20 about its longitudinal axis , and elements 84 and 86 generate signals responsive to left - right and up - down bending , respectively . due to the generally constant elasticity of member 38 , the signals generated by elements 82 , 84 and 86 can be used to derive the bend and twist angles of portion 40 of catheter 20 . these angles are taken together with the translational and orientational coordinates determined with respect to position - sensing element 28 , in order to determine the positions of the plurality of points of interest along the length of catheter 20 . other types of bend sensors may be used in place of sensor 80 shown in fig4 . for example , strain gauges may be substituted - for piezoelectric elements 82 , 84 and 86 . such strain gauges have an electrical resistance that varies as a function of mechanical strain applied thereto , as is known in the art . alternatively , fiberoptic sensors , as are known in the art , may be used to determine the bend angle of catheter 20 , by measuring the loss and back - reflection of light conveyed through an optical fiber embedded in the catheter . furthermore , additional bend sensors of other types may be positioned at different locations along the length of catheter 20 , so that multiple bends or bends of non - constant radius of curvature can be detected . more generally speaking , while the preferred embodiments of the present invention have been described above with reference to one or two position - sensing elements 28 and 30 and a single bend sensor 80 , it will be appreciated that for some applications , catheter 20 may preferably comprise a greater number of position sensors and / or of bend sensors . such additional sensors may be particularly useful when a portion of the length of the catheter must be tracked within a convoluted passage , or when the catheter is brought to bear against and is desired to conform to a convoluted surface within a body cavity . preferably , however , the number of such sensors is held to the minimum needed to achieve the desired accuracy of determination of the plurality of points along the length of the catheter . although for simplicity of illustration , catheter 20 has been shown and described above as comprising only the sensors and other elements necessary for the operation of the present invention , in preferred embodiments of the present invention , the catheter preferably includes other sensing and / or therapeutic devices , as are known in the art . the principles of the present invention may then be applied , for example , to map physiological activity or apply local therapeutic treatment within a body cavity , such as a chamber of the heart , with greater ease and accuracy than methods and devices known in the art . it will be appreciated that the principles of the present invention may be applied , as well , to other flexible medical probes , such as endoscopes . it will further be appreciated that the preferred embodiments described above are cited by way of example , and the full scope of the invention is limited only by the claims .
this invention is an invasive probe apparatus including flexible elongate probe having a distal portion adjacent to a distal end thereof for insertion into the body of a subject , which portion assumes a predetermined curve form when a force is applied thereto . first and second sensors are fixed to the distal portion of the probe in known positions relative to the distal end , which sensors generate signals responsive to bending of the probe . signal processing circuitry receives the bend responsive signals and processes them to find position and orientation coordinates of at least the first sensor , and to determine the locations of a plurality of points along the length of the distal portion of the probe .