846,871. Electric analog calculating. ELECTRIC & MUSICAL INDUSTRIES Ltd. Oct. 18, 1956 [Oct. 27, 1955], No. 30706/55. Class 37. An interpolating device comprises at least four input terminals, a greater number of output terminals representing a series of values of an argument, and a plurality of transformers one less in number than the input terminals; the latter being arranged so that their primaries may be energised by alternating signals applied to the input terminals and when so energised by co-phased signals the secondaries of the transformers cause alternating signals to be established at the output terminals having amplitudes related in accordance with a curvilinear interpolation function of the argument; which function is determined by the turns ratios of the respective transformers; the turns ratio being selected so that the interpolation function has for the first of the series of argument values a value determined by the amplitude of the alternating signal applied to the second input terminal and a slope determined by the amplitude difference between the signals applied to the first and third input terminals, and for another value of the argument a value determined by the amplitude of the alternating signal applied to the second last input terminal and a slope determined by the amplitude difference between the signals applied to the third last and last of the input terminals. Fig. 1 shows four discrete reference points P- 3 , P- 1 ,P 1 ,P 3 representing successive values of a control voltage, e.g. for an automatic machine tool, having ordinates x <SP>1</SP>-3, x <SP>1</SP> - 1, x<SP>1</SP>1 and x<SP>1</SP> 3 (representing e.g. displacement) and abscissaet (representing e.g. time) in unitary arithmetical progression. For a given set of reference points the gradient at each point is equivalent to that of a chord joining the adjacent reference points, and for variations of t a continuously varying ordinal value of x if obtained over the range of values from x- 1 to x 1 according to e.g. a cubic interpolation function x t = at <SP>3 </SP>+ bt<SP>2</SP> + ct + d (1) intersecting the reference points, whereby on successive interchange of such reference points a continuously varying value is obtained without discontinuities. It is shown that, for any set of four input variable signals. x<SP>1</SP> 3 = -11a + 9b - 3c + d (2) x<SP>1</SP>- 1 = -a + b - c + d (3) x<SP>1</SP> 1 = a + b + c + d (4) x<SP>1</SP> 3 = 11a + 9b + 3c + d (5) so that the interpolation device has four degrees of freedom subject to the four constraints defined by the above equations. In Fig. 2 four cophased alternating signals representing values x<SP>1</SP>- 3,x <SP>1-</SP>,x<SP>1</SP> 1 and x<SP>1</SP> 3 for successive abscissal values of t are applied to input terminals 4, 5, 6, 7, and there are provided multiple wound transformers 1, 2, 3 wherein first windings of each transformer are interconnected between inputs 4 and 5, a further six windings are series connected respectively between equally spaced contacts a 1 ,a 2 ,a 3 ,a 4 ,a 5 , a 6 and a 7 of a rotary make before break switch, and a final winding is series connected between inputs 6 and 7; the inputs 5 and 6 being connected respectively to contacts a 1 and a 7 and the switch arm 8 developing the required interpolated signal x t . The central switch contact a 4 floats at a voltage corresponding to d, the voltage per turn of each of the transformers is respectively: cb a - ,- and - and the number and sense of 39 27 the turns of each winding is given by the digits shown. The transformer arrangement energised by voltages representing equations 2 to 5 above on terminals 4, 5, 6, 7 respectively develops a cubic characteristic representing equation (1) passing through points P- 3 , P- 1 , P and P 3 above, so that rotation of arm 8 over contacts a 1 to a 7 develops voltage signals representing the interpolated value of x between X<SP>1</SP> 1 and x 1- 1 proportionately to the angular rotation of the arm; the groups of four input voltages being successively interchangeable. In Fig. 3 storage circuits 9, 10, 11, 12 13 of the kind set forth in Specification 796,995 deriving input signals from a record tape for the control of an automatic machine are respectively connected to contacts b 1 , c 3 , c 1 b 4 , b 2 c 4 , c 2 b 5 , and b 3 c 5 of two banks of switch contacts b 1 -b 5 and c 1 -c 5 each traversed by four movable contacts mutually separated by the width of one contact. Rectangles 14, 15 each symbolise an interpolating transformer arrangement as shown in Fig. 2; the inputs 4, 5, 6, 7 and 4<SP>1</SP>, 5<SP>1</SP>, 6<SP>1</SP> 7<SP>1</SP> of which are respectively derived from the four movable contacts of each switch bank and the outputs of which are respectively connected to circularly disposed contacts a 1 to a 7 and a 7 to a 1 of a rotary switch having an arm 8 and contacts a and a 7 each comprising two separate halves. In operation, arm 8 rotates continuously at constant speed and is geared to the moving contact groups of the store switch (which may be a continuously rotating uniselector) so that for a single arm rotation the store switch contacts move over one contact width and when arm 8 lies on contact a 4 or a 1 0 the next successive set of four discrete input signals is applied to each alternate transformer arrangement in turn. As shown, arm 8 is about to transfer from the outputs of transformer arrangement 15 to those of 14, and stores 11, 12, 13, 9, 10 carry signals representing five successive reference points in order. Since inputs 6<SP>1</SP> and 5<SP>1</SP> are connected to the same store, there is no discontinuity in signal output from contact a at the transfer, and during traversal of contacts a 1 to a 7 of transformer arrangement 14, input 4 receives the signal previously applied to input 5 and input 6 receives the signal previously applied to input 7<SP>1</SP> so that there is no discontinuity in slope at the transfer. As arm 8 traverses contact a 4 during interpolation between the third and fourth reference points, contact b 2 is disconnected and store 11 is cleared of the signal representing the first reference point and a signal representing the sixth reference point is stored therein As arm 8 traverses contact a 7 the output signal is again derived from transformer arrangement 15 without discontinuity, and interpolation is again effected between the fourth and fifth reference points. At contact a 1 o , contact C 2 of the store switch is disengaged and store 12 is cleared and a signal representing a seventh reference point is stored therein. The cycle proceeds sequentially over a succession of groups of reference point signals to produce a smoothly varying output signal representing a continuous interpolation therebetween. Fig. 4 shows a modified transformer circuit which may be used to replace 14 and 15 in the arrangement of Fig. 3 wherein ganged pairs of relay switches RA1, RA2 and RB1, RB2 in their normal condition connect inputs 4 and 7 to transformer 3 as in Fig. 2. In its off normal condition, RA1 opencircuits input 4 and shortcircuits the corresponding winding of transformer 3, RB1 opencircuits input 7 and shortcircuits its corresponding winding of transformer 3, and RA2 and RB2 in their off normal condition shortcircuit the winding of transformer 2 corresponding to input 7. A discontinuity is assumed to occur at a reference point corresponding to interchange of transformer arrangements 14, 15; i.e. when arm 8 lies on contact a 1 . During the range of derivation of the output from arrangement 15, RB1 <SP>1</SP> is off normal while during the range of derivation from 14, RA1 is off normal; the prime denoting a switch associated with transformer circuit 15. Input points 4, 7 <SP>1</SP> are then disconnected so as to suppress the constraint on the interpolated curve gradient on either side of the discontinuity reference point and by shortcircuiting the cubic transformers removes the corresponding degree of freedom so that the interpolation equation (11) is reduced to a quadratic of the form. X t = bt<SP>2</SP> +ct+d (6) and the gradient at any point t is x<SP>c</SP>t=2 bt+c (7) so that interpolation is controlled by constraints represented by x<SP>1-</SP> 1 =b-c+d d (8) x<SP>1</SP> 1 =b+c+d (9) x 1 3 =9b+3c+d (10) For discontinuities at two successive reference points, RA1 and RB1<SP>1</SP> may be set to off normal for negotiation of the first discontinuity e.g. when arm 8 lies on contact a 7 ; while RB1 and RA1<SP>1</SP> are set off normal for the interpolation range corresponding to passage of arm 8 over the range of contacts of transformer arrangement 14; both pairs of switches RA1 RA2, RB1 RB2 in 14 being off normal. Transformer 2 is thus shortcircuited over. RA2, RB2 whereby two degrees of freedom are removed. If a discontinuity occurs at each end of the range of contacts a 1 to a 7 of the interpolating switch, RA2 and RB2 are held off normal throughout the range. A transcendental interpolation function may be employed since X t +a o +a 1 cos t + a 2 cos 2t + --- +a n cos nt (11) +b 1 sin t+b 2 sin 2t + - - - + bn sin nt may satisfy 2n + 1 constraints by proper choice of coefficients, and given five inputs for corresponding sets of constraint signals the interpolation function may be x t =a+b cos t+c sin t+d cos 2t+ e sin 2t (12) with reference points at abscissal values of - #, -#/2, O, #/2 and # and with constraints defined by X<SP>1</SP> #/2 = a-c-d (13) 2 X<SP>1</SP>O= a+b+d (14) # X <SP>1</SP>- = a+c-d (15) 12 X <SP>1</SP> O -X- # X<SP>1</SP>O-X-#/# = b-2e (16) X<SP>1</SP> #-X<SP>1</SP>O = -b-2e (17) where X<SP>1#</SP>- #, X<SP>1</SP>- #/2, X O , X #/2 and X<SP>1</SP> # are the corresponding ordinal inputs. In such an interpolating device (Fig. 5), rectangles 16, 17, 18, 19 respectively represent four multiwinding toroidal transformers whe