Patent GB-1225164-A (GB)
IPC: G06F17/17
Filed: 1968-04-19
Abstract: 1,225,164. Analogue, digital interpolation systems. HITACHI Ltd., and HITACHI ELECTRONICS CO. Ltd. 19 April, 1968 [19 April, 1967], No. 18690/68. Headings G4A and G4G. In a two-dimensional curve interpolation system, the successive x, y co-ordinates of specific points of a curve are obtained from a drawing and digitally stored by a potentiometric coordinate curve follower incorporating analogue to digital conversion and memory (Fig. 1, not shown), and the digital information is read out from 32 (Fig. 2, not shown) and stored in a memory programmed to read out in predetermined time sequence to a D/A converter and a curve plotter. The plural specific curve points are defined by x co-ordinates, X 1 , X 2  . . - x i-1 , xi, x i+1  and the intervals are interpolated by function where (#t) is a function of time varying from 0 # 1, and a, b, . . . are coefficients derived from the co-ordinate information of a point x i  and a predetermined number of adjacent points. When F is of order n> 1, the values of a, b, &c. are considered as coefficients of #(t), so that two analogue computing circuits calculate #(t) and coefficients a, b,..., from which F[#(t), a, b, . . .] is derived by combination. Representing x, y co-ordinates of a curve by where # is a function of time, #(t) = 1 - e<SP>-t</SP>, representing a first order delay; a x , b x , c x , a y , b y , c y  are coefficients of #; and k x , k y  are constants, it is shown that satisfied by the circuit of Fig. 3 comprising integrators 71, 72, 73, adders 74, 75 and inverter 76 receiving signals representing a x , b x , c x  and k x . On application of initial signal values a 0 , b 0 , c 0 , k 0  with zero integrator output; at t=0, x 0  = k 0 , and at t = 00, x 00  = a 0  + b 0  + c 0  + k 0 . Addition of incremental values # a , # b , # c , # k , to the respective circuits in steady state gives x (t ) = # a #<SP>3</SP> + # b #<SP>2</SP> + # c # + # k  + (a 0 +b 0 +c 0 +k 0 ) representing successive x-axis value variations of the desired curve, by repeatedly inserting coefficients a x , b x , C x , k x  at predetermined values at certain time intervals. The required interpolated co-ordinates of points on the curve are digitally stored in the memory of the DP system. Fig. 4 shows a circuit utilizing second order equations in for evaluating both X and Y interpolated co-ordinates. Holding circuits 93 x , 94 x  respond to initial and final specific signals x i , x i-1  for an interpolation step and are controlled by circuit 100 to energize multipliers 95 x , 96 x  cross-connected to adders 97 x , 98 x  feeding 1st order delay circuits 91 x , 92 x  having transfer functions where T is time constant and S is an operand. It is shown by mathematics that, at a steady state, the output x from adder 99 x  is given by x = x i-1 +¢(-x i+1 +4x i -3x i-1 )#+ ¢(x i+1 -2x i +x i-1 )#<SP>2</SP> wherein x = x i-1  for t = 0, # = 0 x = x i  for t=00, # = 1 so that the interval x i-1 to x; is interpolated by a second order equation using the final value x i+1  of the next succeeding step as reference value. Similarly, for interpolation over y intervals, utilizing elements 93y, 94y, 95y, 96y, 97y, 98y, 91y, 92y, 99y, it is shown that, The resultant interpolated analogue signals may be applied to an X-Y recorder or C.R.O. and are shown by mathematics to define a partial parabola extending from (x i-1 Y i-1 ) to (x i , Y i ) and continuous interpolation is obtained by successive co-ordinate settings. In a modification (Fig. 5) an additional signal holding circuit 101 x  is provided for setting the final value in the last interpolation step, an additional ¢ multiplier 102 x , and an additional first order delay circuit 103 x  of transfer function with additional adders 104 x , 105 x  interconnected as shown. The holding circuits receive signals x i , x i-1 , x i-2  respectively and it is shown by mathematics that the interval between points x i-1  and x i  is third order interpolated using the final value X i+1  in the next step as reference value. Utilizing a similar arrangement for y interpolation, it is shown that the inclination of the curve at the commencement of each interpolation step coincides with that of the preceding step so that a continuous curve can be represented (Fig. 6, not shown) by successively supplying co-ordinate information of the points thereon under control of circuit 100. In a further modification (Fig. 7, not shown), a second order interpolater incorporates series transfer delay circuits.