JPS6232519B2 - - Google Patents

Description

[Detailed description of the invention]

[Industrial Field of Application] The present invention provides a graphic signal for outputting a graphic to a graphic display device or a graphic recording device that displays or records a graphic using a large number of discretely arranged pixels. This relates to a device that generates electricity. [Prior Art and Problems to be Solved by the Invention] Conventionally, this type of device has relied heavily on software, as seen in general graphic display devices, and has used linear approximations to approximate circular arcs or sine approximations. It was either done by combining waves or using Lissage shapes. However, in either case, it has been difficult to draw an arbitrary arc with high precision while reducing the burden on the software. In addition, digital graphic display devices such as dot display elements consisting of pixels that can essentially take only two states, bright and dark, raster scanning displays that use digital IC memory as a video buffer, and XY plotters, , it is not possible to use the method using the Lissage figure. [Objective and Summary of the Invention] The present invention has been made in view of these drawbacks, and its purpose is to reduce the burden on software for display devices and recording devices configured with discretely arranged pixels. It is an object of the present invention to provide a device that generates a graphic signal that can be displayed with fairly high precision or in black and white binary display without any multiplication. In order to display a figure on a digital figure display device in which a large number of discrete pixels are arranged, first, according to the present invention, the x and y coordinate values of a sequence of points that approximate the shape of the figure as accurately as possible are determined. Then, by increasing the brightness of all points on the display screen that correspond to the dot sequence obtained by the present invention, or by moving the pen one after another to the position of the dot sequence, , just output shapes that appear to be smoothly connected. [Example] The content of the present invention will be explained below by taking as an example the operating principle and device configuration sequence for generating circular arcs, but the present invention can also be used to generate straight lines and other figures. Constructing the generating device is as easy as for circular arcs. First, the operating principle used in the present invention will be explained. In this invention, the start and end points of the arc to be output are given as relative coordinates from the center of the circle containing the arc, and the direction in which the arc is drawn is given, so that pixels along the arc can be traced one after another. That is, for example, in a display unit, an X-Y plotter, etc. that is composed of grid-like pixels, the origin of the XY coordinates (0, 0,
0) in the clockwise or counterclockwise direction, the direction of the increment to be taken differs for each of the eight octants from the first octant to the eighth octant, as shown by the arrows. The basic operation of the present invention is to appropriately select the directions of these two pixels one after another to generate an arc signal. First, place the current point P ₀ on the starting point. Select one of P ₁ or P ₂ adjacent to the current point P ₀ as the next point. Next, the current point is moved to the selected point, and the next point is selected in the same manner.
By repeating this process until reaching near the end point, a sequence of points (ie, a graphic signal) that approximates a circular arc is generated. The principle of selecting adjacent pixels is as follows. The equation of a circle centered at the origin is x ² + y ² = r ² ... (1) where r is the radius. Note that a general _{circle (} x
For -x ₀ ) ² + (y-y ₀ ) ² = r ² , x ₀ and y ₀ may be added to the x and y coordinate values obtained as the following results, respectively. Now, change equation (1) to the following equation (2), substitute the coordinates of an appropriate point to the right-hand side, and calculate the left-hand side, that is, f
If (x, y) is 0, the point is on the circle shown by equation (1), and if it is not 0, it is not on the circle. The larger the absolute value of the left side, the further away the point is from the circle shown by equation (1). f (x, y) = x ² + y ² − r ² ... (2) Therefore, the left side of equation (2) determined by the grid points adjacent to the starting point, that is, the absolute value of f (x, y) By selecting the smallest one, the optimal grid point, that is, the direction of the increment can be determined. Now, as shown in Figure 2, consider the generation of an arc within the first octant as an example (the same argument can be applied to other octants by simply changing the selection direction of the points as shown in Figure 1). ). Point P ₁ to be selected next to current point P (x, y)
The value of f in equation (2) at (x, y+1) or point P ₂ (x-1, y+1) is given by the following equations. f ₁ = f (x, y + 1) = f (x, y) + 2y + 1 ... (3) f ₂ = f (x - 1, y + 1) = f (x, y) + 2y - 2x + 2 = f ₁ - 2x + 1 ... (4) By the way, at this time, the arc signal generator invented by Kamae et al.
According to the displacement comparison method shown in the literature (Kamae et al.: “Dot representation of figures”, Institute of Electronics and Communication Engineers Paper Examination, Vol. 56-A, No. 7, 1973), the following rule A is given.

[Table] A technique has been known for generating an arc by selecting points according to the table and repeating this process. On the other hand, in the apparatus of the present invention, the above-mentioned conventional technique is further improved as follows. That is, a predetermined constant (threshold value) T is set corresponding to the figure to be generated, and the following rule B is applied.

By selecting points according to [Table] and repeating this process, a graphic signal for outputting an arc is generated. In the case of a circular arc, it will be proven later that it is sufficient to set T=r (radius of the circle). Next, FIG. 3 shows an example of the configuration of an apparatus according to the present invention for generating arcs in the first octant. In Fig. 3, 1 is the X counter, 2 is the Y counter, and 3 and 4 are the X counter 1 and Y counter, respectively.
A 1-bit shifter for immediately doubling the contents of counter 2 and outputting it; 5 is an F register for storing the value of f in equation (2); 6 and 7 are arithmetic units that combine adders; 8 is a rule B threshold T
9 is a comparator, 10 is a buffer register that stores the intermediate calculation result of the value of f,
11 is an X increment generation circuit, 12 is a Y increment generation circuit,
13 is an end point detection circuit. To operate this device, first set the X and Y coordinate values (X _s = r, Y _s = 0) of the starting point P of the circular arc to
Set the value of f(X _s , Y _s ) (in this case, the point X _s , Y _s ) is on the circle, so this value is 0) to the F register 5. Furthermore, the value of the radius r of the circle is set in the T register 8. Next, the operation is started by applying the clock signal C to the arithmetic unit 6. First, the contents of F register 5 (f) and 1-bit shifter 4
The arithmetic unit 6 calculates the value f ₁ =f+2y+1 using the output (2y), and sends the arithmetic result to the comparator 9 and buffer register 10. continue
The content _T (=
r), and if | _{f 1} | ) is stored in the F register 5 as the new value of f, and then sent to the arithmetic unit 6 as follows.
Add next clock signal C. Note that at this time, nothing is sent to the arithmetic unit 7, and the X counter 1,
The 1-bit shifter 3 and the X increment generation circuit 11 remain retained. On the other hand, if |f ₁ |≧r, buffer register 10
and without sending a signal to the Y increment generation circuit 12,
The value f ₁ is sent as it is to the arithmetic unit 7, and by combining it with the output (2x) of the 1-bit shifter 3, the arithmetic unit 7 calculates the value f ₂ = f ₁ - 2x + 1. This is stored as is in the F register 5 and a new value is generated. f, and sends a signal to the X increment generating circuit 11 to reduce the contents of the X counter 1 by 1, and then applies the next clock signal C to the arithmetic unit 6. By repeating the above operation, a counterclockwise circular arc in the first octant is generated, and the signal is sent to the X counter 1, Y counter 2, or X increment at each timing as the X, Y coordinate value or the X, Y increment. Generation circuit 11, Y increment generation circuit 12
taken out from The end point may be detected by the end point detection circuit 13 when the contents of the X counter 1 and the Y counter 2 become equal (x=y). By doing the same thing for each of the first to eighth octants, a complete circle can be generated. Furthermore, by devising the end point detection conditions as shown in Figure 4, the starting point (X _s , Y _s )
Any counterclockwise arc can be drawn from to the end point (X _E , Y _E ) (the same applies to clockwise arcs). In FIG. 4, for example, the condition for terminating the arc generation in the first octant is x=y or x= _xE . For other octants, conditions for terminating arc generation are shown in FIG. The present invention has been described above using the generation of circular arcs as an example, but Rule B may be similarly applied when generating straight lines or other figures. For example, in the case of a straight line in the first octant passing through the origin, equations (2), (3), and (4) are f(x, y) = ay - bx = 0 (where a≧b>0), respectively.
...(2)' f ₁ = f (x, y + 1) = f (x, y) + a ... (3)' f ₂ = f (x-1, y + 1) = f (x, y) + a + b
...(4)' is replaced, and it is proved as described later that the threshold T of rule B should be set to a/2. Similarly, the general straight line f (x, y) = ay + bx + c
After converting the coordinate system so that =0 passes through the origin, a similar method can be applied to each octant, and the threshold T of rule B is |a|/2 when |a|≧|b| When |a|<|b|, it is sufficient to set |b|/2. Although the generation of circular arcs and straight lines has been described above as an example, it will be explained below that correct circular arcs and straight lines can be generated by setting the threshold value T described above. () In the case of circular arc occurrence It will be explained that T (threshold value) should be set to r (radius of circle). First, consider a paraboloid of revolution expressed by equation (2), f(x, y)=x ² +y ² -r ² . . . (2) as shown in FIG. Points r _{1 and} r ₂ that satisfy equations (5) and (6) on both sides of x=r on the x-axis (however, r ₁ > r, r > r ₂ > 0,
r≧1). r ² ₁ −r ² = r (= f (r ₁ , 0)) ……(5) r ² ₂ − r ² = −r (= f (r ₂ , 0)) ……(6) (5) , transform equation (6) to obtain equations (7) and (8). r ₁ =√ ² + ...(7) r ₂ =√ ² − ...(8) From equations (7) and (8), equation (9) is obtained. (√ ² +−√ ² −) ² = 2r ² −2√ ⁴ − ²
...(9) By the way, (2r ² -1) ² - (2√ ⁴ - ² ) ² =1>0
...(10) Therefore, the relationship of equation (11) is derived. 2r ² −1>2√ ⁴ − ² ...(11) Equation (12) is further derived from equation (11). 2r ² −2√ ⁴ − ² > 1 ...(12) Obtain (13) from equation (12). (√ ² +−√ ² −) ² > 1 (= 1 ² )
...(13) From equations (7), (8), and (13), we obtain the result expressed by equation (14). r ₁ − _{r 2} > 1 ...(14) From equation (14), f(x,
It can be seen that the width in the radial direction of the ring-shaped region R such that the absolute value of the displacement of y) |f(x,y)| is equal to or less than r is greater than 1. In the generation of a counterclockwise circular arc in the first octant, if the current point is P ₀ = (X, Y), then P ₀ is selected according to the rules of the present invention, so it is already |f(X, Y) The condition |<r is satisfied. That is, P ₀ must be inside region R. Now, the next point to be selected is P ₁ = (X,
Either Y+1) or P ₂ = (X-1, Y+1), but the width of X and Y-1 is 1, X is within region R, and the width of region R is greater than or equal to 1. Therefore, it is impossible for both X and X-1 to exist outside region R. That is, |f ₁ (X, Y+1)|, |f ₂ (X-
1, Y+1) | cannot be greater than or equal to r (it is possible for both to be <r in rare cases. In that case, first select adjacent points that are determined to satisfy <r. (This is the next point to be selected.) In this way, the ring-shaped region R is always
Since points within the circle are selected one after another, a sequence of points that approximate a circle, that is, a circle, can be generated. Furthermore, when the radius of the circle is r, the width of the region R is w
From equations (5) and (6), w=r ₁ −r ₂ =√ ² +−√ ² −. Table 1 is obtained by calculating this w for various values of r.

【table】

[Proved to T1]

In Fig. 7, when Q ₀ exists at any position on a line segment of length 1 in area B (this is the point selected by the present rule just before Q ₀ , so naturally it is in area B). ), Q ₁ is on a line segment of length 1,
It should also be on the line segment of length 1 of _Q2 .
On the other hand, according to the assumption |a|≧|b|, the slope of the straight line L is less than 1, so the upper boundary line L ^t of the region B passes through a point G on the line segment, and the lower boundary line L ^b of the region B passes through point F on the line segment. At this time, the line segment has a length of 1, and the line segment between Q ₁ and Q ₂ also has a length of 1, so at least one of Q ₁ or Q ₂ is always on the line segment, that is, in area B. exists within. Note that point Q ₁ and point Q ₂ exist in area B at the same time because point Q ₁ is on point F (i.e., straight line L ^b ) and point Q ₂ is on point G (i.e., on straight line L ^t ). Limited to cases where each exists on top of the other. Similarly, when |a|<|b|, it is clear that the threshold T should be set to T=|b|/2. [Effects of the Invention] As explained above, the graphic signal generation device of the present invention can generate high-precision graphic signals at high speed by combining inexpensive and simple circuits such as adders, comparators, registers, etc. There are advantages.

[Brief explanation of the drawing]

Fig. 1 is a diagram showing the selection direction of points in each octant in arc generation according to the present invention, Fig. 2 is a diagram showing the generation of a counterclockwise circular arc in the first octant, and Fig. 3 is a diagram showing the arc of Fig. 2. FIG. 4 is a diagram illustrating an example of the configuration of an apparatus of the present invention configured to generate a circular arc, and is a diagram showing end point detection conditions in each octant that are necessary when drawing an arbitrary arc by combining the apparatuses shown in FIG. 3. FIG. 5 is a diagram for explaining that in the case of arc generation according to the present invention, the threshold value T may be set to the radius r of the circle. FIGS. 6 and 7 are diagrams for explaining that in the case of straight line generation according to the present invention, the threshold value T may be set to |a|/2. 1...X register, 2...Y register, 3, 4
...1 bit shifter, 5...F register, 6,7
... Arithmetic unit, 8 ... T register, 9 ... Comparator, 10 ... Buffer register, 11 ... X increment generation circuit, 12 ... Y increment generation circuit, 13 ...
End point detection circuit.

Claims (1)

[Claims] 1. A signal for causing a device that outputs a figure by specifying discrete x and y coordinates to output a figure expressed in the form of a function f(x, y) = 0. In a figure signal generator that generates ₁ = f
An F register that stores the value of (X, Y), a T register that stores a predetermined threshold T corresponding to the figure to be generated, and the absolute value of the contents of the F register, that is, |f
(X, Y)|, and further the absolute value |f
a comparator that performs a comparison by taking the difference between (X, Y)| and the content T of the T register;
X increment generation circuit that generates Y increments and Y
an increment generating circuit; an arithmetic unit for updating the value of f ₁ =f(X, Y) in the F register; and checking the contents of the X register and the Y register, and determining the contents of the X register and/or An end point detection circuit that determines the end point when the contents of the Y register satisfy a preset condition and outputs a signal instructing to end the figure generation operation. Graphic signal generator. 2 When generating a graphic signal to output a circle or arc of radius r expressed by f(x, y) = x ² + y ² - r ² = 0, the threshold value T stored in the T register is , T=r. 3. When generating a graphic signal for outputting a straight line expressed by f(x, y) = ay - bx = 0, the threshold value T stored in the T register is set as |a|>
In the case of |b|, T=|a|/2 is set, and in the case of |a|<|b|, T=|b|/2 is set. The graphic signal generator described in Section 1.