Abstract:
Assuming that a given equation representing a quadratic curve is: 
     
       F(x, y)=ax.sup.2 +bxy+cy.sup.2 +dx+ey+f=0, 
     
     the method for generating quadratic curve signals repeatedly selects a point close to F (x, y)=0 in only one of either the region of F (x,y)≧0 or the region of F (x,y)&lt;0. This method allows to generate quadratic curve signals by using only a few parameters and without using complicated calculations. A hardware implementation is also disclosed.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates to a method for generating signals representing a quadratic curve such as a circle, an ellipse or a parabola, and more particularly to a method for generating quadratic curve signals best suited for use in a CRT display unit or a plotter. 
     2. Description of Prior Art 
     Known as a conventional method for generating signals representing a quadratic curve by repeating steps that select a new point from among eight points (x+1, y+1), (x+1, y), (x+1, y-1), (x, y-1), (x-1, y-1), (x-1, y), (x-1, y+1) and (x, y+1) adjacent to a current point (x, y) in a Cartesian coordinates system, is a method disclosed by a paper entitled &#34;Algorithm for drawing ellipses or hyperbolae with a digital plotter&#34; by M. L. V. Pitteway, Computer Journal, Vol. 10, November 1967, pp. 282-289. 
     This method first selects one octant from among the first octant in which point (x+1, y+1) or (x+1, y) can be selected, the second octant in which point (x+1, y) or (x+1, y-1) can be selected, the third octant in which point (x+1, y-1) or (x, y-1) can be selected, the fourth octant in which point (x, y-1) or (x-1, y-1) can be selected, the fifth octant in which point (x-1, y-1) or (x-1, y) can be selected, the sixth octant in which point (x-1, y) or (x-1, y+1) can be selected, the seventh octant in which point (x-1, y+1) or (x, y+1) can be selected, and the eighth octant in which point (x, y+1) or (x+1, y+1) can be selected. Then, by assuming that selectable points in the selected octant are (X 1 , Y 1 ) and (X 2 , Y 2 ) (e.g., X 1  =x+1, Y 1  =y+1, X 2  =x+1 and Y 2  =y in the first octant), that the equation of the quadratic curve is 
     
         F(x, y)=ax.sup.2 +bxy+cy.sup.2 +dx+ey+f=0, 
    
     and that X 3  =(X 1  +X 2 )/2 and Y 3  =(Y 1  +Y 2 )/2, either (X 1 , Y 1 ) or (X 2 , Y 2 ) is selected according to the sign of D(x,y)=F(X 3 , Y 3 ). Consequently, the next point is selected whether it be in the region of F (x,y)≧0 or in the region of F (x,y)&lt;0. 
     The method described in the above paper requires many parameters, complicated operations, and many operations for changing of parameters when changing the octant. And, it has a problem that it is difficult to be realized on hardware. 
     SUMMARY OF THE INVENTION 
     An object of this invention is to provide a method for generating quadratic curve signals which requires relatively few parameters, can generate signals representing a quadratic curve with only simple operations, and can be easily realized in hardware. 
     To attain the above objects, according to this invention, signals representing a line approximating a quadratic curve F (x, y)=0 are generated by repeatingly selecting a new point close to F (x,y)=0 from points in only one of either the region of F (x,y)≧0 or the region of F (x,y)&lt;0. 
     If the point to be selected is limited to only in the positive or only in the negative region of F (x,y), as described above, the next point is a point which does not change the sign of F (x,y) but if possible it reduces the absolute value of F (x,y). So the selection of a point is performed only by determining the sign. 
     For example, it is assumed that two candidate points (X 1 , Y 1 ) and (X 2 , Y 2 ) are selected in the octant selection step, from eight points around the current point. ((X 0 , Y 0 ) is the current point.) Then let 
     
         F(X.sub.1, Y.sub.1)-F(X.sub.0, Y.sub.0)=α 
    
     (the accrual of F when point (X 1 , Y 1 ) is selected), and 
     
         F(X.sub.2, Y.sub.2)-F(X.sub.0, Y.sub.0)=β 
    
     (the accrual of F when point (X 2 , Y 2 ) is selected). 
     Then, if points only in the region of F (x, y)≧0 are to be selected, the following steps are sufficient to decide the choice of the next point: 
     (1) Check the sign of α or β, 
     (2) Check the sign of F (X 2 , Y 2 ) if α≧0(β&lt;0), 
     (3) Check the sign of F (X 1 , Y 1 ) if α&lt;0(β≧0), 
     (4) Select (X 2 , Y 2 ) if F (X 2 , Y 2 )≧0 or F (X 1 , Y 1 )&lt;0, 
     (5) Select (X 1 , Y 1 ) if F (X 2 , Y 2 )&lt;0 or F (X 1 , Y 1 )≧0. 
     If points only in the region of F (x, y)&lt;0 are to be selected, the following steps are sufficient to decide the selection of the next point: 
     (1) Check the sign of α or β, 
     (2) Check the sign of F (X 1 , Y 1 ) if α≧0 (β&lt;0), 
     (3) Check the sign of F (X 2 , Y 2 ) if α&lt;0 (β≧0), 
     (4) Select (X 1 , Y 1 ) if F (X 2 , Y 2 )≧0 or F (X 1 , Y 1 )&lt;0, 
     (5) Select (X 2 , Y 2 ) if F (X 2 , Y 2 )&lt;0 or F (X 1 , Y 1 )≧0. 
     It should be noted that in the above steps only signs are checked. Thus, it is possible to provide symmetry to the flow of operations, which allows an easy realization with hardware. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a flowchart showing one embodiment of a method for generating quadratic signals according to the invention. 
     FIGS. 2(a)-(d) and 3(a)-(d) are diagrams illustrating the basic principle of the invention. 
     FIGS. 4(a)-(h), are diagrams illustrating eight octants. 
     FIG. 5 is a diagram illustrating α and β changes accompanying the octant changes. 
     FIG. 6 is a diagram showing a sequence of dots in drawing a circle of F=x 2  +y 2  -36=0 in the region of F≧0 according to the method of FIG. 1. 
     FIG. 7 is a diagram showing a sequence of dots in drawing a circle of F=x 2  +y 2  -36=0 in the region of F&lt;0 according to the method of FIG. 1. 
     FIGS. 8A, 8B, 8C, 8D, 8E, 8F, 8G and 8H show steps to draw a circle of F=x 2  +y 2  -72=0 in the region of F&lt;0 according to the method of FIG. 1. 
     FIGS. 9A, 9B, 9C, 9D, 9E and 9F show steps to draw an ellipse of F=x 2  +4y 2  -156=0 in the region of F&lt;0 according to the method of FIG. 1. 
     FIGS. 10A, 10B, 10C, 10D, 10E and 10F show steps to draw an ellipse of F=10x 2  -16xy+10y 2  -288=0 in the region of F&lt;0 according to the method of FIG. 1. 
     FIGS. 11A, 11B, 11C, 11D, 11E, 11F and 11G show steps to draw a parabola of F=4y-x 2  +2=0 in the region of F≧0 according to the method of FIG. 1. 
     FIG. 12 is a block diagram showing one exemplary configuration of an apparatus used for performing the method of FIG. 1. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     FIG. 1 is a flowchart showing an embodiment of the method for generating quadratic curve signals according to the invention. Prior to the description the embodiment of the invention shown in FIG. 1, basic principles of the invention will be described by referring to FIGS. 2 and 3. 
     FIG. 2 shows the method for selecting the next point in the region of F (x,y)≧0. In the figure, (X 0 , Y 0 ) indicates the current point, (X 1 , Y 1 ) and (X 2 , Y 2 ) the two candidates for the next point. In the case of FIG. 2(a), because both (X 1 , Y 1 ) and (X 2 , Y 2 ) are in the region of F (x, y)&lt;0,  (X 2 , Y 2 ) which is closer to F (x, y)=0 is selected. In the case of FIG. 2(b), although (X 2 , Y 2 ) is closer to F (x,y)=0 than (X 1 , Y 1 ), (X 1 , Y 1 ) is selected because (X 2 , Y 2 ) is in the region of F (x, y)&lt;0. In the case of FIG. 2(c), because both (X 1 , Y 1 ) and (X 2 , Y 2 ) are in the region of F (x, y)&lt;0, (X 1 , Y.sub. 1) being closer to F (x, y)=0 is selected. In the case of FIG. 2(d), although (X 1 , Y 1 ) is closer to F (x, y)=0 than (X 2 , Y 2 ), (X 2 , Y 2 ) is selected because (X 1 , Y 1 ) is in the region of F (x, y)&lt;0. 
     FIG. 3 shows the method for selecting the next point in the region of F (x, y)&lt;0. In the case of FIG. 3(a), because both (X 1 , Y 1 ) and (X 2 , Y 2 ) are in the region of F (x, y)&lt;0, (X 1 , Y 1 ) being closer to F (x, y)=0 is selected. In the case of FIG. 3(b), although (X 1 , Y 1 ) is closer to F (x, y)=0 than (X 2 , Y 2 ), (X 2 , Y 2 ) is selected because (X 1 , Y 1 ) is in the region of F (x, y)&lt;0. In the case of FIG. 3(c), because both (X 1 , Y 1 ) and (X 2 , Y 2 ) are in the region of F (x, y)&lt;0, (X 2 , Y 2 ) which is closer to F (x, y)=0 is selected. In the case of FIG. 3(d), although (X 2 , Y 2 ) is closer to F (x, y)=0 than (X 1 , Y 1 ), (X 1 , Y 1 ) is selected because (X 2 , Y 2 ) is in the region of F (x, y)&lt;0. 
     In the embodiment shown in FIG. 1, the following parameters are used: 
     Decision parameter: F (=ax 2  +bxy+cy 2  +dx+ey+f) 
     Direction parameters: α, β (dependent of x, y, a, b, c, d, e, octant) 
     Shape parameters: a, b, c (coefficients of x 2 , xy and y 2  in the quadratic equation) 
     Deviation parameters: T1, T2, T3 (dependent of a, b, c, octant) 
     α and β depend on the octant. There are eight octants. FIG. 4(a) shows the first octant in which a point (x+1, y+1) or (x+1, y) can be selected as the next point to the current point (x, y), FIG. 4(b) shows the second octant in which a point (x+1, y) or (x+1, y-1) can be selected as the next point, FIG. 4(c) shows the third octant in which a point (x+1, y-1) or (x, y-1) can be selected as the next point, FIG. 4(d) shows the fourth octant in which a point (x, y-1) or (x-1, y-1) can be selected as the next point, FIG. 4(e) shows the fifth octant in which a point (x-1, y-1) or (x-1, y) can be selected as the next point, FIG. 4(f) shows the sixth octant in which a point (x-1, y) or (x-1, y+1) can be selected as the next point, FIG. 4(g) shows the seventh octant in which a point (x-1, y+1) or (x, y+1) can be selected as the next point, FIG. 4(h) shows the eighth octant in which a point (x, y+1) or (x+1, y+1) can be selected as the next point. 
     In the first octant, α and β are: 
     
         α=F(x+1, y+1)-F(x, y) 
    
     
         β=F(x+1, y)-F(x, y) 
    
     In the second octant: 
     
         α=F(x+1, y-1)-F(x, y) 
    
     
         β=F(x+1, y)-F(x, y) 
    
     In the third octant: 
     
         α=F(x+1, y-1)-F(x, y) 
    
     
         β=F(x, y-1)-F(x, y) 
    
     In the fourth octant: 
     
         α=F(x-1, y-1)-F(x, y) 
    
     
         β=F(x, y-1)-F(x, y) 
    
     In the fifth octant: 
     
         α=F(x-1, y-1)-F(x, y) 
    
     
         β=F(x-1, y)-F(x, y) 
    
     In the sixth octant: 
     
         α=F(x-1, y+1)-F(x, y) 
    
     
         β=F(x-1, y)-F(x, y) 
    
     In the seventh octant: 
     
         α=F(x-1, y+1)-F(x, y) 
    
     
         β=F(x, y+1)-F(x, y) 
    
     In the eighth octant: 
     
         α=F(x+1, y+1)-F(x, y) 
    
     
         β=F(x, y+1)-F(x, y) 
    
     It should be noted that, by these definitions, α changes while β does not, in a transition between the first and second octants, or between the third and fourth octants, or the fifth and sixth, or the seventh and eighth octants. Similarly, β changes but α does not, in any transition between the second and third, or the fourth and fifth, the sixth and seventh, or the eighth and first octants. Thus, in any transition between adjacent octants, only one of the parameters α and β will change in value and must be updated. 
     As illustrated later, T1 is a parameter which must be added to β after selecting a point that displaces by (+1) or (-1) along either X or Y direction from the current point (x, y). T1 has the following values: 
     In the first octant, 2a(=β(x+1, y)-β(x, y)), 
     In the second octant, 2a(=β(x+1, y)-β(x, y)), 
     In the third octant, 2c(=β(x, y-1)-β(x, y)), 
     In the fourth octant, 2c(=β(x, y-1)-β(x, y)), 
     In the fifth octant, 2a(=β(x, y-1y)-β(x, y)), 
     In the sixth octant, 2a(=β(x-1, y)-β(x, y)), 
     In the seventh octant, 2c(=βx, y+1)-β(x, y)), 
     In the eighth octant, 2c(=β(x, y+1)-β(x, y)). 
     Thus, T1 is 2a in the first, second, fifth and sixth octant, and is 2c in the third, fourth, seventh and eighth octants. In other words, T1 has only two values for all octants. Therefore, in the following, T1 is referred as T1 (=2a) for the first, second, fifth and sixth octant, and T1&#39; (=2c) in the third, fourth, seventh and eighth octants. 
     As illustrated later, T2 is a parameter which must be added to α after selecting a point that displaces by (+1) or (-1) along either X or Y direction from the current point (x, y), and must be added to β after selecting a point that displaces by (+1) or (-1) in X direction and by (+1) or (-1) in Y direction, from the current point (x, y). T2 has the following values: 
     In the first octant, 
     2a+b(=α(x+1, y)-α(x, y)=β(x+1, y+1)-β(x, y)), 
     In the second octant, 
     2a-b(=α(x+1, y)-α(x, y)=β(x+1, y-1)-β(x, y)), 
     In the third octant, 
     2c=b(=α(x, y-1)-α(x, y)=β(x+1, y-1)-β(x, y)), 
     In the fourth octant, 
     2c+b(=α(x, y-1)-α(x, y)=β(x-1, y-1)-β(x, y)), 
     In the fifth octant, 
     2a+b(=α(x-1, y)-α(x, y)=β(x-1, y-1)-β(x, y)), 
     In the sixth octant, 
     2a-b(=α(x-1, y)-α(x, y)=β(x-1, y+1)-β(x, y)), 
     In the seventh octant, 
     2c-b(=α(x, y+1)-α(x, y)=β(x-1, y+1)-β(x, y)), 
     In the eighth octant, 
     2c+b(=α(x, y+1)-α(x, y)=β(x+1, y+1)-β(x, y)). 
     As illustrated later, T3 is a parameter which must be added to α after selecting a point that displaces by (+1) or (-1) in X direction and by (30 1) or (-1) in Y direction, from the current point (x, y). T3 has the following values: 
     In the first octant, 
     2a+2c+2b(=α(x+1, y+1)-α(x, y)) 
     In the second octant, 
     2a+2c-2b(=α(x+1, y-1)-α(x, y)) 
     In the third octant, 
     2a+2c-2b(=α(x+1, y-1)-α(x, y)) 
     In the fourth octant, 
     2a+2c-2b(=α(x+1, y-1)-α(x, y)) 
     In the fifth octant, 
     2a+2c+2b(=α(x-1, y-1)-α(x, y)) 
     In the sixth octant, 
     2a+2c-2b (=α(x-1, y+1)-(x, y)) 
     In the seventh octant, 
     2a+2c-2b (=α(x-1, y+1)-α(x, y)) 
     In the eighth octant, 
     2a+2c+2b(=α(x+1, y+1)-α(x, y)) 
     Thus, T3 is 2a+2c+2b in the first, fourth, fifth and eighth octants, and is 2a+2c-2b in the second, third, sixth and seventh octants. In other words, T3 has only two values for all octants. Therefore, in the following, T3 is referred to as T3 (=2a+2c+2b) for the first, fourth, fifth and eighth octants, and T3&#39; (=2a+2c-2b) in the second, third, sixth and seventh octants. 
     Table 1 below shows the values of α, β, T1 (T1&#39;), T2 and T3 (T3&#39;) in the eight octants. 
     In Table 1, the equations in the change column (either the α or β column) are: 
     
         α=2β-α+2c 
    
     
         α=2β-α+2a 
    
     
         β=α-β+b 
    
     
         β=α-β-b 
    
     These are equations for finding α and β for the next octant by using α and β for the current octant, when changing the octant. Three digits in parentheses in the octant column are codes indicating each octant. 
     It should be noted that the above equations, for finding α and β for the next octant, apply for transitions between two adjacent octants in either direction. This is because these equations express a symmetrical function, the sum, of the old and new values of the changing parameter (α or β) in terms of other parameters that do not change in the subject transition, as is easily seen. 
     
                                           TABLE 1__________________________________________________________________________Octant         α              β     T1 T2  T3__________________________________________________________________________First2ax + bx + by + 2cy +              2ax + by + a + d                         2a 2a + b                                2a + 2c + 2b(111)a + b + c + d + eChangeα 32  2 β - α + 2cSecond2ax - bx + by - 2cy +              2ax + by + a + d                         2a 2a - b                                2a + 2c - 2b(110)a - b + c + d + 3               (T3&#39;)Change             β = α + bThird2ax - bx + by - 2cy +              -bx - 2cy + c - e                         2c 2c - b                                2a + 2c - 2b(010)a - b + c + d + e        (T1&#39;)  (T3&#39;)Changeα = 2 β - α + 2aFourth-2ax - bx - by - 2cy +              -bx - 2cy + c - e                         2c 2c + b                                2a + 2c + 2b(000)a + b + c - d - e        (T1&#39;)Change             β = α - bFifth-2ax - bx - by - 2cy +              -2ax - by + a - d                         2a 2a + b                                2a + 2c + 2b(100)a + b + c - d - eChangeα = 2 β - α + 2cSixth-2ax + by - by + 2cy +              -2ax - by + a - d                         2a 2a - b                                2a + 2c - 2b(101)a - b + c - d + e           (t3&#39;)Change             β = α - β + bSeventh-2ax + bx - by + 2cy +              bx + 2cy + c + e                         2c 2c - b                                2a + 2c - 2b(001)a - b + c - d + e        (T1&#39;)  (T3&#39;)Changeα = 2 β = α + 2aEighth2ax + bx + by + 2cy +              bx + 2cy + c + e                         2c 2c + b                                2a + 2c + 2b(011)a + b + c + d + e        (T1&#39;)Change             β = α - β - bFirst2ax + bx 30 by + 2cy +              2ax + by + a + d                         2a 2a + b                                2a + 2c + 2b(111)a + b + c + d + e__________________________________________________________________________ 
    
     Now referring to FIG. 1, the preferred embodiment of the invention is described. First, the start point (X s , Y s ) is to be given. Then, as shown in the block 2, values for F, α, β, T1, T1&#39; and b are obtained at the start point and an octant is selected. For example, when drawing a circle 
     
         F=x.sup.2 +y.sup.2 -36=0, 
    
     if it is assumed that the start point is (-5, 5) and the initial octant is the first octant, then (by Table 1) 
     
         F=(-5).sup.2 +5.sup.2 -36=14 
    
     
         α=2x(-5)+2x5+2=2 
    
     
         β=2x(-5)+1=-9 
    
     
         T1=T1&#39;=2 
    
     
         b=0 
    
     are set. And, as shown in the block 4, values for T3, T3&#39; and T2 are found from the following equations (by Table 1): 
     
         T3=T1+T1&#39;+2b 
    
     
         T3&#39;=T1+T1&#39;-2b 
    
     
         T2=T1(T1&#39;)±b (-sign for octants 2, 3, 6 and 7) 
    
     For the above example, 
     
         T3=T3&#39;=4 
    
     
         T2=2. 
    
     Table 2 below shows α, β, T1 (T1&#39;), T2 and T3 (T3&#39;) in each octant for F=x 2  +y 2  -36. 
     
                       TABLE 2______________________________________                         T1         T3Octant α      β    (T1&#39;) T2   (T3&#39;)______________________________________First  2x + 2y + 2  2x + 1    2     2    4(111)Second 2x - 2y + 2  2x + 1    2     2    4(110)Third  2x - 2y + 2  -2y + 1   2     2    4(010)Fourth -2x - 2y + 2 -2y + 1   2     2    4(000)Fifth  -2x - 2y + 2 -2x + 1   2     2    4(100)Sixth  -2x + 2y + 2 -2x + 1   2     2    4(101)Seventh  -2x + 2y + 2 2y + 1    2     2    4(001)Eighth 2x + 2y + 2  2y + 1    2     2    4(011)______________________________________ 
    
     Then, as shown in the block 6, the signs for α and β are checked. If α and β have different signs, the octant first selected is a correct octant. In the above example, since α=2, β=-9 and the signs for α and β are different, the octant is the correct one. 
     If α and β have equal signs, the octant change process shown in the block 8 is performed. As clearly seen from Table 1, changing the value of α according to the equations in Table 1 while maintaining β is sufficient to change from the first octant to the second octant, from the third to the fourth, from the fifth to the sixth, or the seventh to the eighth. Also, changing the value of β according to the equations in Table 1 while maintaining α is sufficient to change from the second octant to the third octant, from the fourth to the fifth, from the sixth to the seventh, or the eighth to the first. In particular, when the octant is continuously changed, changes of α and β are caused alternately (see FIG. 5). Then, by checking whether α was changed in the last octant change or not, in the block 10, it is found which one of α and β should now be changed in this octant change. For example, if the current first octant is now to be changed for the second octant, it is found that change of α is now required because β was (or would have been) changed in the last octant change. 
     If the necessity of change of α is detected, it is decided whether the current octant is the first or fifth octant, or not, in block 12. If so, as shown in the block 14, an operation 
     
         α=2β-α+2c 
    
     is performed to change the value of α. This means that the current octant is changed to the second or the sixth octants, respectively. In the above example, this changes the first octant to the second octant. If in the block 12 it is decided that the current octant is not the first or the fifth octant, it is the third or the seventh octant, so that an operation 
     
         α=2β-α+2a 
    
     is performed in the block 16 to change the value of α. This means that the current octant is changed to the fourth or the eighth octant. 
     However, when the block 10 provides an affirmative result in judgment, the necessity of change of β is detected, and then, as shown in the block 18, it is judged whether the current octant is the second or sixth octant, or not. If so, as shown in the block 20, an operation 
     
         β=α-β+b 
    
     is performed to change β. This means that the current octant is changed to the third or the seventh octant. If the block 18 provides a negative decision, the current octant is the fourth or the eighth octant, so that an operation 
     
         β=α-β-b 
    
     is performed to change β, as shown in block 22. This means that the current octant is changed to the fifth or the first octant. 
     Along with the change of octant as described above, the value of T1 (T1&#39;), T2 and T3 (T3&#39;) are also changed according to Table 1, as briefly indicated in block 24 of FIG. 1. It is clear from Table 1 that new values for all of them corresponding to the new octant can be determined using the values set in the block 2 or 4. 
     Then, the signs of the new α and β are checked, again in the decision block 6. If α and β have different signs, the point selection process in block 39 is performed. If they still have the same sign, the octant change process in block 8 is again performed. This process continues until α and β have different signs. 
     When α and β have different signs, it is first judged in the block 32 whether F and α have the same or different signs. It is equivalent to the checking of signs of F and β because, when it is intended to draw a curve in the region of F≧0, F is positive (including zero), so the fact that F and α have the same sign means that α is positive (or zero) and β is negative. When it is intended to draw a curve in the region of F&lt;0, F is negative, so the fact that F and α have the same sign means that α is negative and β is positive (or zero). 
     If it is judged in block 32 that they have the same sign, the signs of F and F+β are compared, as shown in block 34. If the same sign, the point that displaces by (+1) or (-1) along either X or Y direction is selected, as shown in the block 36. Thus, if it is assumed to be the first octant, (X+1, Y) is selected. If F and F+β are judged in block 34 to have different signs, the point that displaces by (+1) or (-1) in the X direction and (+1) or (-1) in the Y direction is selected, as shown in the block 42. Now, if it is assumed to be the first octant, (X+1, Y+1) is selected. 
     If F and α are judged in block 32 to have different signs, the signs of F and F+α are compared in the block 40. If the same sign, the point that displaces by (+1) or (-1) in the X direction and (+1) or (-1) in the Y direction is selected as shown in the block 42. If F and F+α are judged to have different signs, the point that displaces by (+1) or (-1) along either X or Y direction is selected, as shown in the block 36. 
     After the process of block 36 is executed, the values of parameters are updated, as shown in the block 38, according to the equations: 
     
         F=F+β 
    
     
         α=α+T2 
    
     
         β=β+T1 (T1&#39;). 
    
     After the process of the block 42 is executed, the values of parameters are updated, as shown in the block 44, according to the equations: 
     
         F=F+α 
    
     
         α=α+T3 (T3&#39;) 
    
     
         β=β+T2. 
    
     Then, returning to the block 6, the signs of α and β are checked. If they are different, the point selection process of block 30 is again performed. If, however, the signs are the same, the octant change process of block 8 is performed next, as described above. 
     FIG. 6 shows a circle of F=x 2  +y 2  -36=0 that is drawn in the region of F≧0 according to the method of FIG. 1 by assuming the start point of (-5, 5). Tables 3 and 4 below, taken together as one table, show F, α, β and the octant change when drawing the curve of FIG. 6, also recalling Table 2 above. 
     
                                           TABLE 3__________________________________________________________________________                     Point   NextF          α   β                     selection                             (x, y)__________________________________________________________________________P1    14   2         -9   (x + 1, y)                             (-4, 5)P2    5    4         -7   (x + 1, y + 1)                             (-3, 6) (F + β)      (α + T2)                (β + T1)P3    9    8         -5   (x + 1, y)                             (-2, 6) (F + α)      (α + T3)                (β + T2)P4    4    10        -3   (x + 1, y)                             (-1, 6) (F + β)      (α + T2)                (β + T1)P5    1    12        -1   (x + 1, y)                             (0, 6) (F + β)      (α + T2)                (β + T1) 0    14        1 (F + β)      (α + T2)                (β +  T1)P6    0    -10       1    (x + 1, y)                             (l, 6)(Change of (α = 2β - α + 2c)octant)P7    1    -8        3    (x + 1, y)                             (2, 6)P8    4    -6        5    (x + 1, y)                             (3, 6)P9    9    -4        7    (x + 1, y - 1)                             (4, 5) 5    0         9P10   5    0         -9   (x + 1, y - 1)                             (5, 4)(Change ofoctantP11   5    4         -7   (x + 1, y - 1)                             (6, 2)P12   9    8         -5   (x, y - 1)                             (6, 2)P13   4    10        -3   (x, y - 1)                             (6, 1)P14   1    12        -1   (x, y - 1)                             (6, 0) 0    -10       1    (x, y - 1)                             (6, -1)P15   0    -10       1    (x, y - 1)                             (6, -1)octant__________________________________________________________________________ 
    
     
                       TABLE 4______________________________________                      Point     NextF         α  β  selection (x, y)______________________________________P16     1     -8       3     (x, y - 1)                                  (6, -2)P17     4     -6       5     (x, y - 1)                                  (6, -3)P18     9     -4       7     (x - 1, y - 1)                                  (5, -4)   5     0        9P19     5     0        -9    (x - 1, y - 1)                                  (4, -5)Change ofoctantP20     5     4        -7    (x - 1, y - 1)                                  (3, -6)P21     9     8        -5    (x - 1, y)                                  (2, -6)P22     4     10       -3    (x - 1, y)                                  (1, -6)P23     1     12       -1    (x - 1, y)                                  (0, -6)   0     14       1P24     0     -10      1     (x - 1, y)                                  (-1, -6)(Change ofoctantP25     1     -8       3     (x - 1, y)                                  (-2, -6)P26     4     - 6      5     (x - 1, y)                                  (-3, -6)______________________________________ 
    
     FIG. 7 shows a circle of F=x 2  +y 2  -36=0, which is drawn in the region of F&lt;0 according to the method of FIG. 1 by assuming the start point of (-4, 4). Table 5 below shows F, α, β and the octant change when drawing the curve of FIG. 7, while also recalling Table 2 above. 
     
                                           TABLE 5__________________________________________________________________________                     Point   NextF          α β selection                             (x, y)__________________________________________________________________________Q1    -4   2       -7     (x  + 1, y  + 1)                             (-3, 5)Q2    -2   6       -5     (x + 1, y)                             (-2, 5) (F  + α)      (α + T3)              (β + T2)Q3    -7   8       -3     (x  + 1, y)                             (-1, 5) (F  + β)      (α + T2)              (β + T1)Q4    - 31 100     - 1    (x + 1, y)                             (0, 5) (F  + β)      (α + T2)              (β + T1) -11  12      1 (F  + β)      (α + T2)              (β + T1)Q5    -11  -8      1      (x  + 1, y)                             (1, 5)(Change of (2 β - α + 2c)octant)Q6    -10  -6      3      (x + 1, y)                             (2, 5) (F  + β)      (α + T2)              (β + T1)Q7    -7   -4      5      (x  + 1, y)                             (3, 5) (F  + β)      (α + T2)              (β + T1)Q8    -2   -2      7      (x  + 1, y  - 1)                             (4, 4) (F + β)      (α + T2)              (β + T1) -4   2       9 (F + α)      (α + T3)              (β + T2)Q9    -4   2       -7     (x + 1, y - 1)                             (5, 3)(Change of         (α - β + b)octant)Q10   -2   6       -5     (x, y 31  1)                             (5, 2) (F + α)      (α + T3)              (β + T2)Q11   -7   8       -3     (x, y - 1)                             (5, 1) (F + β)      (α + T2)              (β + T1)Q12   - 10 10      -1     (x, y - 1)                             (5, 0) (F + β)      (α + T2)              (β + T1)__________________________________________________________________________ 
    
     FIGS. 8A, 8B, 8C, 8D, 8E, 8F, 8G and 8H show steps to draw a circle of F=x 2  +y 2  -72=0 in the region of F&lt;0 according to the method of FIG. 1 by assuming the start point of (0, 8). Table 6A, 6B, 6C, 6D, 6E, 6F, 6G and 6H show F, α, β, the octant, T1, T1&#39;, T2, T3 and T3&#39; corresponding to FIGS. 8A to 8H, respectively. 
     
                                           TABLE 6A__________________________________________________________________________NO F    α         β             Octant                 T1  T1&#39;                        T2  T3 T3&#39;__________________________________________________________________________0  FFFF8   FFFF2 00001             2   002 002                        002 004                               0041  FFFF9   FFFF4 00003             2   002 002                        002 004                               0042  FFFFC   FFFF6 00005             2   002 002                        002 004                               0043  FFFF2   FFFFA 00007             2   002 002                        002 004                               0044  FFFF9   FFFFC 00009             2   002 002                        002 004                               0045  FFFF5   00000 FFFF5             3   002 002                        002 004                               004__________________________________________________________________________ 
    
     
                                           TABLE 6B__________________________________________________________________________NO F    α        β             Octant                 T1  T1&#39;                        T2  T3 T3&#39;__________________________________________________________________________6  FFFF5   00004        FFFF7             3   002 002                        002 004                               0047  FFFF9   00008        FFFF9             3   002 002                        002 004                               0048  FFFF2   0000A        FFFFB             3   002 002                        002 004                               0049  FFFFC   0000E        FFFFD             3   002 002                        002 004                               00410 FFFF9   00010        FFFFF             3   002 002                        002 004                               00411 FFFF8   FFFF2        00001             4   002 002                        002 004                               004__________________________________________________________________________ 
    
     
                                           TABLE 6C__________________________________________________________________________NO F     α         β             Octant                 T1  T1&#39;                        T2  T3 T3&#39;__________________________________________________________________________12 FFFF9 FFFF4         00003             4   002 002                        002 004                               00413 FFFFC FFFF6         00005             4   002 002                        002 004                               00414 FFFF2 FFFFA         00007             4   002 002                        002 004                               00415 FFFF9 FFFFC         00009             4   002 002                        002 004                               00416 FFFF5 00000         FFFF5             5   002 002                        002 004                               00417 FFFF5 00004         FFFF7             5   002 002                        002 004                               004__________________________________________________________________________ 
    
     
                                           TABLE 6D__________________________________________________________________________NO F     α        β              Octant                  T1 T1&#39; T2 T3 T3&#39;__________________________________________________________________________18 FFFF9 00008        FFFF9 5   002                     002 002                            004                               00419 FFFF2 0000A        FFFFB 5   002                     002 002                            004                               00420 FFFFC 0000E        FFFFD 5   002                     002 002                            004                               00421 FFFF9 00010        FFFFF 5   002                     002 002                            004                               00422 FFFF8 FFFF2        00001 6   002                     002 002                            004                               00423 FFFF9 FFFF4        00003 6   002                     002 002                            004                               004__________________________________________________________________________ 
    
     
                                           TABLE 6E__________________________________________________________________________NO F    α         β             Octant                 T1  T1&#39;                        T2 T3  T3&#39;__________________________________________________________________________24 FFFFC   FFFF6 00005             6   002 002                        002                           004 00425 FFFF2   FFFFA 00007             6   002 002                        002                           004 00426 FFFF9   FFFFC 00009             6   002 002                        002                           004 00427 FFFF5   00000 FFFF5             7   002 002                        002                           004 00428 FFFF5   00004 FFFF7             7   002 002                        002                           004 00429 FFFF9   00008 FFFF9             7   002 002                        002                           004 004__________________________________________________________________________ 
    
     
                                           TABLE 6F__________________________________________________________________________NO F    α        β             Octant                 T1  T1&#39;                        T2 T3  T3&#39;__________________________________________________________________________30 FFFF2   0000A        FFFFB             7   002 002                        002                           004 00431 FFFFC   0000E        FFFFD             7   002 002                        002                           004 00432 FFFF9   00010        FFFFF             7   002 002                        002                           004 00433 FFFF8   FFFF2        00001             8   002 002                        002                           004 00434 FFFF9   FFFF4        00003             8   002 002                        002                           004 00435 FFFFC   FFFF6        00005             8   002 002                        002                           004 004__________________________________________________________________________ 
    
     
                                           TABLE 6G__________________________________________________________________________NO F    α        β              Octant                  T1  T1&#39;                         T2 T3 T3&#39;__________________________________________________________________________36 FFFF2   FFFFA        00007 8   002 002                         002                            004                               00437 FFFF9   FFFFC        00009 8   002 002                         002                            004                               00438 FFFF5   00000        FFFF5 1   002 002                         002                            004                               00439 FFFF5   00004        FFFF7 1   002 002                         002                            004                               00440 FFFF9   00008        FFFF9 1   002 002                         002                            004                               00441 FFFF2   0000A        FFFFB 1   002 002                         002                            004                               004__________________________________________________________________________ 
    
     
                                           TABLE 6H__________________________________________________________________________NO F    α       β             Octant                 T1  T1&#39;                        T2  T3 T3&#39;__________________________________________________________________________42 FFFFC   0000E       FFFFD 1   002 002                        002 004                               00443 FFFF9   00010       FFFFF 1   002 002                        002 004                               00444 FFFF8   FFFF2       00001 2   002 002                        002 004                               004__________________________________________________________________________ 
    
     FIGS. 9A, 9B, 9C, 9D, 9E and 9F show steps to draw an ellipse of F=x 2  +4y 2  -156=0 in the region of F&lt;0 according to the method of FIG. 1, by assuming the start point of (0, 6). Table 7A, 7B, 7C, 7D, 7E and 7F show F, α, β, the octant, T1, T1&#39;, T2. T3 and T3&#39; corresponding to FIGS. 9A to 9F, respectively. 
     
                                           TABLE 7A__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________0  FFFF4   FFFD3        00001             2   002                    008                       002                          00A                             00A1  FFFF5   FFFD5        00003             2   002                    008                       002                          00A                             00A2  FFFF8   FFFD7        00005             2   002                    008                       002                          00A                             00A3  FFFFD   FFFD9        00007             2   002                    008                       002                          00A                             00A4  FFFD6   FFFE3        00009             2   002                    008                       002                          00A                             00A5  FFFDF   FFFE5        0000B             2   002                    008                       002                          00A                             00A__________________________________________________________________________ 
    
     
                                           TABLE 7B__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________6  FFFEA   FFFE7        0000D             2   002                    008                       002                          00A                             00A7  FFFF7   FFFE9        0000F             2   002                    008                       002                          00A                             00A8  FFFF0   FFFF3        00011             2   002                    008                       002                          00A                             00A9  FFFF1   FFFF5        00013             2   002                    008                       002                          00A                             00A10 FFFF6   FFFFF        00015             2   002                    008                       002                          00A                             00A11 FFFFB   00001        FFFEA             3   002                    008                       008                          00A                             00A__________________________________________________________________________ 
    
     
                                           TABLE 7C__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________12 FFFFC   0000B        FFFF2             3   002                    008                       008                          00A                             00A13 FFFEE   00013        FFFFA             3   002                    008                       008                          00A                             00A14 FFFE8   FFFFB        00002             4   002                    008                       008                          00A                             00A15 FFFEA   FFFF3        0000A             4   002                    008                       008                          00A                             00A16 FFFF4   FFFFB        00012             4   002                    008                       008                          00A                             00A17 FFFEF   00005        FFFFB             5   002                    008                       002                          00A                             00A__________________________________________________________________________ 
    
     
                                           TABLE 7D__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________18 FFFF4   0000F        FFFED             5   002                    008                       002                          00A                             00A19 FFFE1   00011        FFFEF             5   002                    008                       002                          00A                             00A20 FFFF2   0001B        FFFF1             5   002                    008                       002                          00A                             00A21 FFFF3   0001D        FFFF3             5   002                    008                       002                          00A                             00A22 FFFF6   0001F        FFFF5             5   002                    008                       002                          00A                             00A23 FFFF5   00029        FFFF7             5   002                    008                       002                          00A                             00A__________________________________________________________________________ 
    
     
                                           TABLE 7E__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________24 FFFEC   0002B        FFFF9             5   002                    008                       002                          00A                             00A25 FFFE5   0002D        FFFFB             5   002                    008                       002                          00A                             00A26 FFFE0   0002F        FFFFD             5   002                    008                       002                          00A                             00A27 FFFDD   00031        FFFFF             5   002                    008                       002                          00A                             00A28 FFFDC   FFFD7        00001             6   002                    008                       002                          00A                             00A29 FFFDD   FFFD9        00003             6   002                    008                       002                          00A                             00A__________________________________________________________________________ 
    
     
                                           TABLE 7F__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________30 FFFE0   FFFDB        00005             6   002                    008                       002                          00A                             00A31 FFFE5   FFFDD        00007             6   002                    008                       002                          00A                             00A32 FFFEC   FFFDF        00009             6   002                    008                       002                          00A                             00A33 FFFF5   FFFE1        0000B             6   002                    008                       002                          00A                             00A34 FFFD6   FFFEB        0000D             6   002                    008                       002                          00A                             00A35 FFFE3   FFFED        0000F             6   002                    008                       002                          00A                             00A__________________________________________________________________________ 
    
     FIGS. 10A, 10B, 10C, 10D, 10E and 10F show steps to draw an ellipse of F=10x 2  -16xy+10y 2  -288=0 in the region of F&lt;0 according to the method of FIG. 1, by assuming the start print of (6, 8). Table 8A, 8B, 8C, 8D, 8E and 8F show F, α, β, the octant, T1, T1&#39;, T2, T3 and T3&#39; corresponding to FIGS. 10A to 10F, respectively. 
     
                                           TABLE 8A__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________0  FFFC8   FFFDC        00002             2   014                    014                       024                          008                             0481  FFFCA   00000        FFFDA             3   014                    014                       024                          008                             0482  FFFCA   00048        FFFFE             3   014                    014                       024                          008                             0483  FFFC8   FFFCC        00012             4   014                    014                       004                          008                             0484  FFFDA   FFFD0        00026             4   014                    014                       004                          008                             0485  FFFAA   FFFD8        0002A             4   014                    014                       004                          008                             048__________________________________________________________________________ 
    
     
                                           TABLE 8B__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________6  FFFD4   FFFDC        0003E             4   014                    014                       004                          008                             0487  FFFB0   FFFE4        00042             4   014                    014                       004                          008                             0488  FFFF2   FFFE8        00056             4   014                    014                       004                          008                             0489  FFFDA   FFFF0        0005A             4   014                    014                       004                          008                             04810 FFFCA   FFFF8        0005E             4   014                    014                       004                          008                             04811 FFFC2   00000        FFFAE             5   014                    014                       004                          008                             048__________________________________________________________________________ 
    
     
                                           TABLE 8C__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________12 FFFC2   00008        FFFB2             5   014                    014                       004                          008                             04813 FFFCA   00010        FFFB6             5   014                    014                       004                          008                             04814 FFFDA   00018        FFFBA             5   014                    014                       004                          008                             04815 FFFF2   00020        FFFBE             5   014                    014                       004                          008                             04816 FFFB0   00024        FFFD2             5   014                    014                       004                          008                             04817 FFFD4   0002C        FFFD6             5   014                    014                       004                          008                             048__________________________________________________________________________ 
    
     
                                           TABLE 8D__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________18 FFFAA   00030        FFFEA             5   014                    014                       004                          008                             04819 FFFDA   00038        FFFEE             5   014                    014                       004                          008                             04820 FFFC8   FFFDC        00002             6   014                    014                       024                          008                             04821 FFFCA   00000        FFFDA             7   014                    014                       024                          008                             04822 FFFCA   00048        FFFFE             7   014                    014                       024                          008                             04823 FFFCB   FFFCC        00012             8   014                    014                       004                          008                             048__________________________________________________________________________ 
    
     
                                           TABLE 8E__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________24 FFFDA   FFFD0        00026             8   014                    014                       004                          008                             04825 FFFAA   FFFD8        0003A             8   014                    014                       004                          008                             04826 FFFD4   FFFDC        0003E             8   014                    014                       004                          008                             04827 FFFB0   FFFE4        00042             8   014                    014                       004                          008                             04828 FFFF2   FFFE8        00056             8   014                    014                       004                          008                             04829 FFFDA   FFFF0        0005A             8   014                    014                       004                          008                             048__________________________________________________________________________ 
    
     
                                           TABLE 8F__________________________________________________________________________NO F    α        β             Octant                 T1 T1&#39;                       T2 T3 T3&#39;__________________________________________________________________________30 FFFCA   FFFF8        0005E             8   014                    014                       004                          008                             04831 FFFC2   00000        FFFAE             1   014                    014                       004                          008                             04832 FFFC2   00008        FFFB2             1   014                    014                       004                          008                             04833 FFFCA   00010        FFFB6             1   014                    014                       004                          008                             04834 FFFDA   00018        FFFBA             1   014                    014                       004                          008                             04835 FFFF2   00020        FFFBE             1   014                    014                       004                          008                             048__________________________________________________________________________ 
    
     FIGS. 11A, 11B, 11C, 11D, 11E, 11F and 11G show steps to draw a parabola of F=4y-x 2  +2=0 in the region of F≧0 according to the method of FIG. 1, by assuming the start point of (-8, 18). Table 9A, 9B, 9C, 9D, 9E, 9F and 9G show F, α, β, the octant, T1, T1&#39;, T2, T3 and T3&#39; corresponding to FIGS. 11A to 11G, respectively. 
     
                                           TABLE 9A__________________________________________________________________________NO  F   α        β             Octant                 T1 T1&#39; T2 T3  T3&#39;__________________________________________________________________________0   0000A   0000B        FFFFC             3   FFE                    000 000                           FFE FFE1   00006   0000B        FFFFC             3   FFE                    000 000                           FFE FFE2   00002   0000B        FFFFC             3   FFE                    000 000                           FFE FFE3   0000D   00009        FFFFC             3   FFE                    000 000                           FFE FFE4   00009   00009        FFFFC             3   FFE                    000 000                           FFE FFE5   00005   00009        FFFFC             3   FFE                    000 000                           FFE FFE__________________________________________________________________________ 
    
     
                                           TABLE 9B__________________________________________________________________________NO  F   α        β             Octant                 T1 T1&#39; T2 T3  T3&#39;__________________________________________________________________________6   00001   00009        FFFFC             3   FFE                    000 000                           FFE FFE7   0000A   00007        FFFFC             3   FFE                    000 000                           FFE FFE8   00006   00007        FFFFC             3   FFE                    000 000                           FFE FFE9   00002   00007        FFFFC             3   FFE                    000 000                           FFE FFE10  00009   00005        FFFFC             3   FFE                    000 000                           FFE FFE11  00005   00005        FFFFC             3   FFE                    000 000                           FFE FFE__________________________________________________________________________ 
    
     
                                           TABLE 9C__________________________________________________________________________NO  F   α        β             Octant                 T1 T1&#39; T2 T3  T3&#39;__________________________________________________________________________12  00001   00005        FFFFC             3   FFE                    000 000                           FFE FFE13  00006   00003        FFFFC             3   FFE                    000 000                           FFE FFE14  00002   00003        FFFFC             3   FFE                    000 000                           FFE FFE15  00005   00001        FFFFC             3   FFE                    000 000                           FFE FFE16  00001   00001        FFFFC             3   FFE                    000 000                           FFE FFE17  00002   FFFFF        00003             2   FFE                    000 FFE                           FFE FFE__________________________________________________________________________ 
    
     
                                           TABLE 9D__________________________________________________________________________NO  F   α        β             Octant                 T1 T1&#39; T2 T3  T3&#39;__________________________________________________________________________18  00001   FFFFD        00001             2   FFE                    000 FFE                           FFE FFE19  00002   00003        FFFFF             1   FFE                    000 FFE                           FFE FFE20  00001   00001        FFFFD             1   FFE                    000 FFE                           FFE FFE21  00002   FFFFF        00004             8   FFE                    000 000                           FFE FFE22  00001   FFFFD        00004             8   FFE                    000 000                           FFE FFE23  00005   FFFFD        00004             8   FFE                    000 000                           FFE FFE__________________________________________________________________________ 
    
     
                                           TABLE 9E__________________________________________________________________________NO  F   α        β             Octant                 T1 T1&#39; T2 T3  T3&#39;__________________________________________________________________________24  00002   FFFFB        00004             8   FFE                    000 000                           FFE FFE25  00006   FFFFB        00004             8   FFE                    000 000                           FFE FFE26  00001   FFFF9        00004             8   FFE                    000 000                           FFE FFE27  00005   FFFF9        00004             8   FFE                    000 000                           FFE FFE28  00009   FFFF9        00004             8   FFE                    000 000                           FFE FFE29  00002   FFFF7        00004             8   FFE                    000 000                           FFE FFE__________________________________________________________________________ 
    
     
                                           TABLE 9F__________________________________________________________________________NO  F   α        β             Octant                 T1 T1&#39; T2 T3  T3&#39;__________________________________________________________________________30  00006   FFFF7        00004             8   FFE                    000 000                           FFE FFE31  0000A   FFFF7        00004             8   FFE                    000 000                           FFE FFE32  00001   FFFF5        00004             8   FFE                    000 000                           FFE FFE33  00005   FFFF5        00004             8   FFE                    000 000                           FFE FFE34  00009   FFFF5        00004             8   FFE                    000 000                           FFE FFE35  0000D   FFFF5        00004             8   FFE                    000 000                           FFE FFE__________________________________________________________________________ 
    
     
                                           TABLE 9G__________________________________________________________________________NO  F   α        β             Octant                 T1 T1&#39; T2 T3  T3&#39;__________________________________________________________________________36  00002   FFFF3        00004             8   FFE                    000 000                           FFE FFE37  00006   FFFF3        00004             8   FFE                    000 000                           FFE FFE38  0000A   FFFF3        00004             8   FFE                    000 000                           FFE FFE39  0000E   FFFF3        00004             8   FFE                    000 000                           FFE FFE40  00001   FFFF3        00004             8   FFE                    000 000                           FFE FFE41  00005   FFFF3        00004             8   FFE                    000 000                           FFE FFE__________________________________________________________________________ 
    
     FIG. 12 shows a configuration of an apparatus used for implementing the method of FIG. 1. First, the parameters F, α, β, T1, T1&#39; and b representing a curve to be drawn as well as the octant are given through a data bus 50 and a multiplexer 52. The parameters F, α, β, T1, T1&#39; and b are stored in an F register 60, α register 54, β register 56, T1 register 62 , T1&#39; register 64 and b register 58, respectively. The octant is provided to an octant section 74. A pair of start coordinates (X s , Y s ) is set in an X counter 84 and a Y counter 86, respectively. 
     Then, an adder control circuit 78 receives an instruction to perform operation according to the following equations through the data bus 50 and the multiplexer 52: 
     
         T3=T1+T1&#39;+2b 
    
     
         T3&#39;=T1+T1&#39;-2b 
    
     
         T2=T1(T1&#39;)±b 
    
     According to the instruction, an adder 80 performs the above operations using output from the T1, T1&#39; and b registers 62, 64 and 58, respectively, and supplies the results to T3, T3&#39; and T2 registers 68, 70 and 66, respectively. 
     Then, a first sign judging section 72 receives outputs from the α and β  registers 54 and 56 and compares the signs of α and β. The first sign judging section 72 supplies an octant change request signal to the octant section 74 through a line 73 if the signs of α and β are the same The octant section 74 also receives through a line 75 a signal indicating whether change of α was performed in the last octant change or not. However, it is unknown whether α was changed in the last octant change when the octant is first provided. So a signal indicating whether change of α should be assumed in the last octant change or not is supplied at the same time when an octant is provided from outside. 
     When the octant section 74 receives a signal indicating that a change of α was (or would have been) performed in an octant preceding to the given octant, it causes the adder 80 to perform an operation 
     
         β=α-β+b 
    
     through the adder control circuit 78 if the given octant is the second, third, sixth or seventh octant, and supplies the result to the β register 56. The octant section 74 causes the adder 80 to perform an operation 
     
         β=α-β-b 
    
     through the adder control circuit 78 if the given octant is the first fourth, fifth or eighth octant, and supplies the result of the β register 56. 
     If the section 74 receives a signal indicating that the change of α was not performed in an octant preceding to the given octant, it causes the adder 80 to perform an operation 
     
         α=2β-α+2c 
    
     through the adder control circuit 78 if the given octant is the first, second, fifth or sixth octant, and supplies the result to the α register 54. if the given octant is the third, fourth, seventh or eighth octant, it causes the adder 80 to perform an operation 
     
         α=2β-α+2a, 
    
     and supplies the result to the α register 54. Also, it causes the adder 80 to perform an operation of T2=T1(T1&#39;)±b. The octant section 74 generates a code representing the new octant which becomes the current octant after the change. 
     If the signs of α and β become different after the octant change, the first sign judging section 72 does not issue the octant change request signal any more. Then, the second sign judging section 76 receives the outputs of the α register 54 and the F register 60 and checks the signs of F and α. If they are the same, the section 76 instructs the adder control circuit 78 to perform an operation to generate F+β. According to this, the adder 80 receives the outputs of the F and β registers 60 and 56, performs the operation (F+β), and supplies the result to a step control circuit 82, through the multiplexer 52. 
     The step control circuit 82 is also supplied with the output of the F register 60, and a signal representing the current octant from the octant section 74. The step control circuit 82 generates output as listed in Table 10 below. 
     
                       TABLE 10______________________________________   Signs forOctant  F and F + β              X up    X down Y up  Y down______________________________________First   Same       on      off    off   off   Different  on      off    on    offSecond  Same       on      off    off   off   Different  on      off    off   onThird   Same       off     off    off   on   Different  on      off    off   onFourth  Same       off     off    off   on   Different  off     on     off   onFifth   Same       off     on     off   off   Different  off     on     off   onSixth   Same       off     on     off   off   Different  off     on     on    offSeventh Same       off     off    on    off   Different  off     on     on    offEighth  Same       off     off    on    off   Different  on      off    on    off______________________________________ 
    
     If the second sign judging circuit 76 detects that the signs of F and α are different, it instructs the adder circuit 78 to perform an operation to generate F+α. The adder 80 receives the outputs of the F and α registers 60 and 54, performs the operation (F+α), and supplies the result to the step control circuit 82. In this case, the step control circuit 82 generates as listed in Table 11. 
     
                       TABLE 11______________________________________   Signs forOctant  F and F + α              X up    X down Y up  Y down______________________________________First   Same       on      off    on    off   Different  on      off    off   offSecond  Same       on      off    off   on   Different  on      off    off   offThird   Same       on      off    off   on   Different  off     off    off   onFourth  Same       off     on     off   on   Different  off     off    off   onFifth   Same       off     on     off   off   Different  off     on     off   offSixth   Same       off     on     on    off   Different  off     on     off   offSeventh Same       off     on     on    off   Different  off     off    on    offEighth  Same       on      off    on    off   Different  off     off    on    off______________________________________ 
    
     The X and Y counters 84 and 86, respectively, increase or decrease the values of X and Y by one according to output supplied from the step control circuit 82. The output of the step control circuit 82 is also supplied to the adder control circuit 78. When the step control circuit 82 outputs a signal to increment only one of either X or Y by ±1, the adder control circuit 78 causes the adder 80 to perform the following operations to update the values of F, α and β. 
     
         F=F+β 
    
     
         α=α+T2 
    
     
         β=β+T1 (T1&#39;) 
    
     When the step control circuit 82 outputs signals to increment both X and Y by ±1, the adder control circuit 78 causes the adder 80 to perform the following operations to update the values of F, α and β. 
     
         F=F+α 
    
     
         α=α+T3(T3&#39;) 
    
     
         β=β+T2 
    
     Thereafter, the next point will be obtained using the new parameters. When the values of the X and Y counters 84 and 86 reach the end point coordinates set in X and Y end point registers 88 and 90, respectively, drawing of the curve is terminated by signals from a stop check circuit 92. 
     Since the above embodiment changes the octant by noticing the signs of α and β, the change of octant can be continuously performed until the signs of α and β become different, and, therefore, a sharp curve in which a plurality of octant changes are continuously occurring can easily be drawn. 
     In addition, double lines that never cross with each other can easily be drawn by first drawing a line approximate to F (x, y)=0 in a region of F≧0, and then drawing a line approximate to F=0 in the region of F&lt;0. 
     As seen from the foregoing description, the invention reduces the number of parameters, simplifies the operation, and makes realization in hardware easy by selecting a new point close to F (x, y)=0 in only one of either region of F (x, y)≧0 or F (x, y)&lt;0 for generating signals representing F (x, y)=0.