GRAPHICAL V.

S.

DISPLAYS UDC 681.2.085.087.6

Govorov

In automated information systems not only various visual symbols determining the controlled parameters, but also various curves and specificaliy graphs must be displayed on visual indicators [1]. At the present time visual indicators employing a cathode-ray tube (CRT) are the most widely used and effective [2]. Since the data subject to visual reproduction in automated systems are first processed in digital several authors use a point (raster) method of forming curves on the CRT screens [3-5]. The disadvantages of method include the relatively long time required for forming the reproducing voltage of one point, the raster ture of the reproduced curve, and the necessity of calculating the coordinates of a large number of points [6]. sequently, several authors have maintained that a satisfactory solution to the problem has yet to be found [7,

from, this strucCon8].

The graph characterizing a process taking place in a certain object is a smooth curve, satisfying the Dirichlet boundary conditions in its mathematical description. Then according to the Weierstrass theorem, it can be represented with any previously specified error by a broken line, whose equation in Cartesian coordinates has the following form: n

(1)

f ( x ) ..~ P n ( x ) = ~ . , ci F i(x) ,

i=0 where F0(x), Ft(x) . . . . . Fn(x) are certain linear functional relationships: co, c 1. . . . . for the best fit of Pn(xi) to f(x i) in the section xi_ 1, x i.

c n are coefficients chosen

Relative to the speed with which Pn(X) tends to f(x), the most acceptable approximation will be that for which Eq. (1) contains the least number of terms. In the general ease, this is usually a piecewise linear approximation. The number and sizes of the linear portions xi+ 1-xi is this case are defined by the given approximationerror: [/(AX)lmax= [/(x)

Pn(X)]ma x

--

(2)

To reproduce the hodograph of a vector-function followed by the electron ray trace in the plane of the CRT screen, a voltage or current must be applied to its deflection screen proportional to the x and y components of the functional Pn(X), given in parametric form:

t %(t) =- 70 ( t - - fo) qol(t) = q%(h) -~ 71(t - - tl) x=

for to ~< t 4 tl, tl < t ~ t~,

~i(t)=~i_l(ti)@Ti(t__ti 9

,

,

.

.

.

.

.

.

.

.

.

.

.

) .

.

.

~n(t) = 'Pa--~ (t~) + 7~ (t - - t~)

ti
~

(3) ,

~

.

t n < t~< tn+ ~ .

Translated from Izmeritel'naya Tekhnika, No. 2, pp. 23-25, February, 1971. Original article submitted September 8, 1969.

0 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N, Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

211

U

a/~

sigo[~x~]

Fig. 1. Pl is a recorder of the length of the reproduced section; Rx and Ry are recorders of increments Ax i and Ayi; 11 and 12 are elements realizing the logical operation "AND;" T x and Ty are triggers for the sign of increment ZSXi and Ayi; V1-V m are tubes realizing the same logical operations as elements I; Pl ' Px' and Py are code-to-voltage or -current converters; T is a controlling trigger; D T 1 - D T 5 are diode-transformer elements; G is a sawtoothvoltage generator; CC is a coincidence circuit; 13 is an element realizing the logical operation "AND;" Fx and gy are voltage formers varying with a piecewise-linear approximation. p

to ~< t ~< t l ,

for * l ( t ) = % ( t d + ~l(t - - t,) %(0

= vo (t - - to)

ta < t ~< t~, t i < t < tz+ ~ ,

* i ( t ) = *t--1 (ti) "J- Vi (t - - ti) .

.

.

.

.

.

,

.

.

.

.

.

.

, n ( t ) = %,_ ~(tn) + vn (t - - t,,) tn < t < t,,+~

(4) where Yi and vi are linear coefficients. If a curve that does not have a point of inflection is given analytically, then the number of sections of the approximating curve is equal to [9]:

n ~ 0~25

(5)

i n C d ~ [ f ( x ) ] l d x . dx 2 [f(Ax)] max

Xo

If the maximum relative error is given [/(Ax)]

6 [[(AX)max -

max

const,

/(x)

(6)

then X n

(7)

n~

4 ] / 6 [[(Axl] max ~o

dx 2

[[--~x)~dx.

If the least distance between the approximating curve and the actual curve is given, which occurs quite often, then, designating it Ahma x, we find X~

n ~

"

4 V A h max

212

dx.

r

(x)]3 + 1

(8)

--#

I---

~ r D

f

v,

If in these expressions x 0 corresponds to the start of the curve, then assuming n= 1, 2, 3 . . . . . we can find the coordinates x I, x 2, x s . . . . of the end points of the sections that approximate the curve of the broken line. Knowing the coordinates of the boundary points of the approximating section, we can find the increments Ax, ky and the length of the section A l.

Ut Fig. 2

V

i I

'

'

"

C,Cbl

The boundary points of the approximation sections are determined from these expressions if there are no points of i n f l e c tion in the sections, t f such points exist, the curve is partitioned into a number of sections (according to the sign of the second derivative), and the nodal points of the approximation are d e t e r mined for each of them.

But if the coordinates of the points are determined over the course of time, then the digital device is confronted with AXi and Ayi for the same interval of t i m e At i. Here in the general case Ax i . Ay i, and At i . Atj, where i . j. To reproduce a curve varying as a broken line, we must know not only the coordinate increments but also the length Ali:

a,, = VA' -

+

~ -2( c

The values of Ax i and Ayi d e t e r m i n e the a m p l i t u d e of the voltages or currents applied to the deflection system of the CRT, and A l i determines their length. In this case the rate at Fig. 3 which the electron ray is shifted along the section A / i in the plane of the screen will be equal to Vi= A l l / A t i. But A / i = a A t i, where a = c o n s t . Accordingly, Vi=const; i.e., the scan rate of the electron ray is made independent of the length of the reproduced sections. Figure 1 shows the d i a g r a m of a device that generates voltages Ux(t) and Uy(t), proportional to x and y, which can subsequently be converted to proportional currents according to the given Ax i, Ay~ and A li, arriving in digital code. Initially the coordinates of the starting point of the curve x 0 and Y0 are transcribed from storage S to the coordinate register (these elements are not shown on the diagram). After their conversion to points and a m p l i f i c a tion, the electron ray is applied to the starting point on the CRT screen. Since no intensifier pulses are generated in this case, the trace of the electron ray on the screen is not noticeable. The quantities P l , Pm and Py are transcribed from the storage device to registers Rv Rx, and Ry. After the electron ray reaches point Ix 0, Y0], a triggering pulse Ut is applied from the local control device (not shown on the diagram) to trigger T and elements I 1 and 12, flipping trigger T. Triggers T x and Ty are tripped only if Ax i and Ayi are negative, since

sign(Axt) =

1 for 0 for

AxeO

sign(Ayi)=

1 for 0 for

Ag i < 0 , Ag~>I 0.

(9)

The m o m e n t that controlling trigger T is tripped indicates the start of data arrival from registers R i, Rx, and through robes V 1 - V M to converters Pl, Px, and P.. The voltages they generate are proportional to A l p Axi,and Y a~Yi respectively. These voltages are applied to diode-transformer elements D T I - D T s . The circuit of DT 1 with the controlling trigger T is shown in Fig. 2. Negative voltage U l is taken from the Pl output t h r o u g h d i o d e D to the primary winding of the transformer Tr. When the trigger is tripped by pulse U t, transistor T 2 saturates. In this mode its e m i t t e r - c o l l e c t o r resistance is low (on the order of 5 a), and the T 2 collector is almost at ground potential. A current proportional in amplitude to U~ flows through the primary winding of the trans former. 213

The pulse from the controlling trigger T (see Fig. 1) also triggers the sawtooth voltage generator (G), which generates a voltage that varies linearly with time. The sawtooth pulse and a voltage pulse from the DT 1 output arrive at the c o i n c i d e n c e circuit (CC). At the m o m e n t when the sawtooth pulse is equal in a m p l i t u d e to the pulse from the DT r the coincidence circuit generates a pulse that trips controlling trigger T into the initial position and sets registers R1, Rx, and Ry to zero. As a result, the length of the pulse generated by controlling trigger T is proportional to A l i. Elements D T z - D T 5 operate analogously. Depending on the sign of Ax i, current flows through the primary winding of the transformer in either DT z or DT 3. The voltage pulse on the DT z output, just as on the Px output (analogously to Py), is equal in length to the pulse generated by the controlling trigger. Current pulses from the secondary winding of DT 2 or DT 3 are applied to former Fx, whose schematic is shown in Fig. 3 and described in [10, 11]. When the transistors are connected in this way, the collector current does not depend on the collector voltage but is only determined by e m i t t e r current; therefore, the voltage on the capacitor varies as: n--1

Uc(t ) :

Ux(t)

=

t

c i=0

n--1 ,

0

-

-

C

I i n , i (t - - t i ) , i=0

where t i + l - t i = r i is the length of the input pulse, proportional to A l it Ik, i is the a m p l i t u d e of the current pulse in the collector circuit,. Iin, i is the a m p l i t u d e of the current pulse in the input circuit of the former; and a i = const. Then from this expression voltage Ux(t) (see Fig. 1) varies as a p i e c e w i s e - l i n e a r approximation. The Uy(t) former circuit operates in the same way as the Ux(t) circuit. A field established by voltages Ux(t) and uy(t) (oi after their transformation, by currents) causes the CRT deflection system to deflect the electron ray, and a broken line is reproduced on the screen. To m a k e it visible, pulses Uz from the controlling trigger output are used as intensifier pulses. If the curve is extended, i.e., the sum of the incremental coordinates Ax i and Ay i corresponds to a voltage that exceeds Ec (Fig. 3), then the coordinates of an i n t e r m e d i a t e point are c a l c u l a t e d in the d i g i t a l device and subsequently sent to the coordinate register. Since the electron ray is first applied to this point, formers Fx and Fy should be set to zero. Accordingly, when the coordinates of the i n t e r m e d i a t e point appear at the output of the + digital device, the local controller generates a pulse Uin t, which is applied to e l e m e n t I z, and a pulse from the c o i n c i d e n c e circuit, characterizing the moment the reproduction of the preceding part of the curve ceased, triggers formers Fx and Fy, i.e., discharges the capacitors by applying a discharge pulse to the bases of T 3 and T 4. Depending on the polarity of the residual voltage, capacitor C discharges through one of them. Since the length of the pulses characterizing the reproduction t i m e of separate parts of the curve are not equal, the pulse generated by the coincidence circuit U!n~ is applied to the l o c a l controller, informing it that the next part of the curve has been reproduced, and succeeding data can be transcribed into registers Ri, Rx, and Ry. The speed of this circuit is determined not so much by the frequency responses of the elements comprising it, as by the luminous efficiency of the CRT luminophor. In fact, if the converters and DT elements use tunnel diodes, the t i m e increments along the curve can be chosen approximately equal to 20 nsec. When standard paraphase amplifiers are used, and i f the curve is reproduced on the screen of a CRT that has an electrostatic deflection system, or if two-stage amplifiers are used with the windings of a CRT with magnetic deflection system connected into the collector circuits of their transistors, the t r a c e rate of the electron ray in the plane of the CRT screen will be so high that the luminophor will scarcely be excited. To test the primary elements of this circuit, the converters and DT units were supplied with high-frequency transistors, and the voltage formers Fx and Fy, with low-frequency. In this case the t i m e increments along the curve were 1 gsec. When the trace rate of the electron ray was equal to 3 m m / g s e c and the critical flicker rate was 30 Hz, the curve displayed on the screen of an SI-1 oscilloscope was easily visible under normal external illumination. The reproduction t i m e of the curve is determined by the number of node points of the approximation. When a section of a straight l i n e is reproduced, its mapping t i m e is directly proportional to the trace rate of the electron ray in the plane of the screen.

214

A feature of this circuit is that it can reproduce the g r a p h i c - d i g i t a l data describing the curve. it is included in the combined indicator described in [12]. LITERATURE 1. 2. 8. 4. 5. 6. 7. 8. 9. 10. 11. 12.

In this case

CITED

Electronic Data Display Systems [in Russian], Voenizdat, Moscow (1966). V . S . Govorov, I z m e r i t e l ' . Tekh., No. 8 (1968). Yu. R. Novikov and D. P. Rodzevich., Proc. of the Arctic and Antarctic NII, No. 277 (1966). Electronics Weekly, No. 243 (1965). V . I . Rybak and L. N. Shishonok, A v t o m a t i k a i Priborostroenie, No. 1 (1963). V, S. Govorov, Priborostroenie, No. 7, Kiev (1969). Ivan E. Sutherland, "Computer graphics," Datamation, 12, No. 5 (1966). V . D . Kurganov, in. Computer Technology [in Russian], Mashinostroenie, No. 4, Moscow (1964). L . P . Afinogenov, Izv. VUZ Priborostroenie, No. 2 (1963). V . S . Govorov, Priborostroenie, No. 2, Kiev (1966). V . S . Govorov, I z m e r i t e l ' . Tekh., No. 6 (1970). V . S . Oovorov, A v t o m a t . i Telemekhan., No. 4 (1968).

215