JP3118064B2 - Curve drawing device - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a curve drawing apparatus for drawing a curve connecting a start point and an end point based on approximate expression information connecting the start point, the end point, and the start end point.

[0002]

2. Description of the Related Art In a document creation processing system for displaying characters and figures, a set of characteristic points of outlines of characters and figures is stored, and at the time of display, the characteristic points are approximated by, for example, a Bezier curve. Then, the outline is developed in a bitmap memory, and thereafter, the inside of the outline is painted as necessary to display characters.

That is, for a bit map memory corresponding to two dimensions, an approximate function y = f (x) and a starting point A
Given [a, f (a)] and the end point B [b, f (b)], divide the space between AB into n in the x direction, for example,
The value of f (x) for each division point is sequentially obtained and developed on a memory, and a curve between AB is drawn by connecting two adjacent points with a straight line.

[0004]

When a curve between A and B is developed on a bit map memory as described above, if the number of divisions n is increased, the number of overlapping points increases, causing waste and a long processing time. I do. Conversely, if the degree of overlap is reduced by reducing the number of divisions, a discontinuous run may occur depending on the curve.

However, when drawing a discontinuous run, an unnatural character or figure is displayed. Therefore, conventionally, even if the processing time is long, the number of divisions n is set in advance so as not to form a discontinuous run. I took a lot. SUMMARY OF THE INVENTION It is an object of the present invention to provide an improved curve drawing apparatus in which the number of unnecessary waste points is small and the processing time is reduced by dividing the processing so as not to form a discontinuous run.

[0006]

Before describing the means for solving the above-mentioned problems, the principle of the present invention will be described first. Each bit of the bitmap memory is a two-dimensional address value (integer)
Therefore, the overlap on the bitmap memory means that the integer [x] and the integer [x + {x] ({
x is a division value), or an integer [f (x)] and an integer [f (x
+ △ x)] are the same. That is, any of [x + △ x] − [x] = 0 (1.1) | [f (x + △ x)] − [f (x)] | = 0 (1.2) It is time.

[0007] When discontinuity occurs, [x + △ x]-[x]> 1 (2.1) | [f (x + △ x)]-[f (x)] |> 1 ( 2.2) when any of

Therefore, if x is set to 1, no overlap or discontinuity occurs with respect to x. As for y, the amount of change △
y is Δy = f (x + Δx) −f (x) (3), and when f ′ (x) is a derivative of f (x), Δy = f ′ (x) · △ x .. (4), and in order to avoid discontinuity, the division value should be selected such that | f ′ (x) · △ x | ≦ 1 (5) always holds in the section [a, b]. I just need.

Since an approximate expression f (x) is given and its differential value can be easily obtained, it is easy to obtain a divided value satisfying the expression (5) in the section [a, b]. Next, means for solving the problem will be described. A curve drawing device for drawing a curve connecting a start point and an end point based on approximate expression information connecting a start point, an end point, and a start end point, (a) a curve drawing information memory for storing the start point, the end point, and the approximate expression information; (B) a division value calculation unit for obtaining a differential value of the approximate expression from the approximate expression information stored in the curve drawing information memory and calculating a division value based on the differential value; and (c) the start point and the end point. Sections and start and end points
Calculated by the division value calculation unit for the section obtained by dividing the point section
Whether to divide the drawing section based on the
A section division determination unit to a constant with respect to the drawing section is determined by; (d) interval division determination unit
Rendered based on division value calculated by the division value calculator Te
A curve drawing value calculation unit that calculates the curve drawing value by dividing the coordinate value of the section, and an approximation connecting the start point, end point, and start and end points
Draws a curve connecting the start and end points based on the formula information.
In curve drawing method of, (a) the start point, and stores the end point and the approximate expression information, a differential value of the approximate expression from the approximate expression information stored (b)
And calculating a division value based on the differential value, and (c) the start point and end point section and the start point and end point.
Based on the calculated division value for the section obtained by dividing the point section,
To determine whether to divide the drawing section, and (d) calculate the division value calculated for the determined drawing section.
Calculate curve drawing value by dividing the coordinate value of drawing section based on
I do. Is provided.

[0010]

The curve drawing information memory stores the start point, end point and approximate expression information between the start and end points of the curve to be drawn. The division value calculation unit reads the approximate expression information from the curve drawing information memory to calculate a differential value, and calculates a division value satisfying the expression (5) between the start point and the end point.

[0011] The section division judging section is configured to determine the start point and end point sections.
And the section obtained by dividing the start point and end point sections
The drawing area is calculated based on the division value calculated by the division value calculation unit.
It is determined whether to divide the interval. In the curve drawing value calculator,
Drawing section based on division value calculated by the division value calculating section against the determined drawn sections in the section division determination unit
The coordinate values are divided to calculate a curve drawing value. As described above, between the start point and the end point, the division value is obtained based on the differential value of the curve approximation equation between the start point and the end point, and the division is performed, and the curve drawing value corresponding to each divided point is calculated. In addition, it is possible to reduce the time required for the curve drawing process without causing overlapping and generating discontinuous runs.

[0012]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS One embodiment of the present invention will be described with reference to FIGS. FIG. 1 is a block diagram of an embodiment of the present invention, and FIG. 2 is an operation flowchart of the embodiment. In FIG. 1, reference numeral 1 denotes a curve drawing information memory, which stores start and end points of a curve to be drawn and approximate expression information.

Reference numeral 2 denotes a division value calculation unit that calculates a division value that satisfies Expression (5) in a given section. Reference numeral 3 denotes a section division determination unit, which will be described later. Reference numeral 4 denotes a curve drawing value calculation unit that divides the curve based on the division value calculated by the division value calculation unit and calculates a curve drawing value.

5 is a bit map memory, 6 is a display unit, 7
Is a data input / output interface (I / O), and 8 is a processor (CPU) for performing processing. Before describing the operation of the embodiment, a method of calculating a division value that satisfies Expression (5) in section [a, b] in division value calculation section 2 and a section division determination method in section division determination section 3 in the embodiment. The principle of will be described.

The division value needs to satisfy the equation (5).
Therefore, the condition that the overlap is small and the discontinuity does not occur in the y direction is | f ′ (x) · △ x | = 1 (6).

Further, f ′ (x) that satisfies Expression (6) in the section [a, b] and minimizes △ x (maximizes the number of divisions) is f ′ (x) = f ′ (x) _{max a, b} (7) where f '(x) _{max a, b} is given by the maximum value of f' (x) in the section [a, b].

Therefore, the division value Δx can be obtained by substituting the equation (7) into the equation (6), Δx = 1 / f ′ (x) _{max a, b} (8) Therefore, in the section [a, b], f ′
A method for _obtaining f ′ (x) _{max a, b} that maximizes (x) will be described below.

Assuming that the approximate expression f (x) of the curve is given by f (x) = Ax ³ + Bx ² + Cx + D (9), the derivative of the expression (9) is obtained to obtain f ′
(X) is expressed by ^{f '(x) = 3Ax 2} + 2Bx + C ≡A 0 x 2 + B 0 x + C 0 ... (10).

Equation (10) is a quadratic equation, and the root discriminant D
If D = B ₀ ² -4 A ₀ C ₀ ≧ 0 (11), there is a real root, and the value x _m of the curve point in equation (10) is x _m = −B ₀ / 2A ₀ (12) ), and, x = f in x _m 'value f of (x)' (m) is f '(m) = (- given by ^{_{B 0 2 + 4A 0 C 0}} ) / 4A 0 ... (13).

Therefore, if D ≧ 0: (a) If x _m is in the section [a, b], f ′ (x) _{max a, b} = max ｛| f ′ (a) |, | f ′ (m ) |, | f '(b ) |} ... (14) (b) x m is the interval [a, b]. if the outer _{f' (x) max a,} b = max {| f '(a) |, | F ′ (b) |｝ (15) In the case of D <0 (c) In this case, there is no real root, and therefore f ′ (x) _{max a, b} = max ｛| f ′ (a) |, | f ′ (b) |｝ (16), and the equation (14), (15) or (16) is substituted into the equation (8) to obtain the division value △ x.

When the section [a, b] is divided by the divided value △ x obtained by substituting the equations (14), (15) or (16) into the equation (8), the f (x) direction Does not generate a discontinuous run, but if △ x> 1, a discontinuous run occurs in the x direction. Therefore, the divided value in this case △
x is △ x = 1. When the division value △ x is obtained in this manner, many overlaps occur in the f (x) direction. That is, the division value △ x shown in Expression (8) is f ′ (x)
This is a division value in which discontinuous runs are not generated at a point indicating _{max a, b} . At other points, overlapping occurs and wasteful processing is performed.

Therefore, if the section [a, b] is divided and a new division value is obtained corresponding to the divided section, the overlap can be reduced. The section division determination unit 3 determines whether to divide this section. When the section [a, b] is divided by the division value △ x given by the equation (8), the division number N ₀ becomes N ₀ = | (ba) | / △ x (17).

Therefore, the section [a, b] is changed to the section [a, (a
+ B) / 2] and the section [(a + b) / 2, b],
Interval [a, (a + b) / 2] with respect to calculate the division number N ₁₁ by equation (7) to (17). Based on the number of divisions N ₀ calculated from the section [a, b] and the number of divisions N ₁₁ calculated from the section [a, (a + b) / 2], the section division determination unit 3 calculates 2N ₁₁ / N ₀ <α (18) However, if α <1.0, the subdivision is stopped, and the interval [a, (a + b) /
2] is divided into N ₁₁ pieces to calculate a drawing value.

If the section division judging section 3 judges that the section needs to be divided, the section [a, (a + b) / 2] is added to the section [a, (a +
b) / 4] and the section [(a + b) / 4, (a + b) / 2]
Then, a division value N ₂₁ for the section [a, (a + b) / 4] is calculated, and re-determination is performed based on the condition of 4N ₂₁ / N ₀ <α (19).

Therefore, the general expression of the judgment condition of the section division judgment unit 3 is given by 2 ^m N _mn / N ₀ <α (20). Next, the operation will be described with reference to FIG.

Process S1 In process S1, the CPU 8 stores the start point, the end point, and the approximate expression information in the curve drawing information memory 1 via the I / O 7. Process S2 In process S2, the division value calculation unit 2 stores the curve drawing information memory 1
Is read, an inflection point value x _m shown in Expression (12) is calculated, and it is determined whether or not x _m is shown in the section [r, s]. In the first judgment, r of section [r, s]
And s are a and b input to the curve drawing information memory 1.

Processing S3 and S4 If YES in processing S2, the processing shown in equation (14) is performed in processing S3, and if NO, processing shown in equation (15) or (16) is performed in processing S4. The obtained result is substituted into equation (8) to calculate a division value △ x, and the process proceeds to step S5.

Process S5 In process S5, it is determined whether or not the process of calculating the division number N ₀ for the input section [a, b] has been completed.
If NO, the process proceeds to step S7, and if YES, the process S6.
Move on to

Step S6 If the decision result in the step S5 is YES, in the step S6, the section division judging section 3 judges whether or not 2 ^m N _mn / N ₀ shown in the equation (20) is smaller than α. If YES, the process proceeds to step S8, and if NO, the process proceeds to step S7.

Step S7 If the decision result in the step S5 is NO, in a step S7 the section used in the step S2 is divided into two and stored in the curve drawing information memory 1, and the divided section is processed again in the step S2. ~ S
Step 5 is executed.

Step S8 In step S8, the curve drawing value calculation unit 4 determines the interval [r, r] based on the division value △ x calculated in step S3 or step S4.
s] is calculated, and a curve drawing value corresponding to each of the divided values is calculated from Expression (9) and recorded in the bitmap memory 5.
The data recorded in the bitmap memory 5 is displayed on the display unit 6.

Process S9 In the process S9, the determination result in the process S6 becomes YES,
In step S7, the section is divided, and it is determined whether or not there is a section in which the curve drawing value has not been calculated in step S8 in the divided section. If there is an unprocessed section, step S2 is performed. The process is performed again, and in the case of NO, the process ends.

In the embodiment, the case where the approximate expression is a function of y = f (x) has been described.
Even when given by the parameters of (t) and y = g (t), a division value is calculated for both x and y, and the smaller value of the calculated division values is used as the division value. Has the same effect.

Although one embodiment of the present invention has been described above, the present invention is not limited to this embodiment, and various modifications can be made.

[0035]

As described above, according to the present invention, the following effects can be obtained. (A) Since the division between the start point and the end point is determined based on the differential value of the curve approximation equation between the start point and the end point, the curve drawing value corresponding to each of the divided points is calculated. Curve generation processing time can be reduced without generating discontinuous runs. (B) When the division value is small, the designated section is divided and the division value is calculated again to calculate the curve drawing value, so that the overlap is reduced and the processing time can be shortened.

[Brief description of the drawings]

FIG. 1 is a configuration diagram of an embodiment of the present invention.

FIG. 2 is an operation flowchart of the embodiment.

FIG. 3 is an explanatory diagram of calculation of a division value.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 Curve drawing information memory 2 Division value calculation part 3 Section division determination part 4 Curve drawing value calculation part 5 Bitmap memory 6 Display part 7 Interface (I / O) 8 Processor (CPU)

──────────────────────────────────────────────────続 き Continuation of the front page (56) References JP-A-2-81281 (JP, A) JP-A-3-269686 (JP, A) JP-A-3-78888 (JP, A) JP-A-2-78 170287 (JP, A) JP-A-64-55678 (JP, A) JP-A-64-82282 (JP, A) JP-A-3-176783 (JP, A) JP-A-3-105579 (JP, A) JP-A-3-105578 (JP, A) (58) Fields investigated (Int. Cl. ⁷ , DB name) G06T 11/20 110 G09G 5/20

Claims (2)

(57) [Claims] 1. A curve drawing apparatus for drawing a curve connecting a start point and an end point based on approximate expression information connecting a start point, an end point, and a start end point, (a) a curve storing the start point, the end point, and the approximate expression information A drawing information memory; (b) a division value calculation unit that obtains a differential value of the approximate expression from the approximate expression information stored in the curve drawing information memory and calculates a division value based on the differential value; An interval division determining unit that determines whether to divide a drawing interval based on the division value calculated by the division value calculation unit with respect to the start point and the end point section and the section obtained by dividing the start point and the end point section; A) a curve drawing value calculation unit that divides a coordinate value of the drawing section based on the division value calculated by the division value calculation unit with respect to the drawing section determined by the section division determination unit to calculate a curve drawing value; With, Curve drawing apparatus, characterized in that. 2. When the approximate expression information is an approximate expression given as a parameter for x and y, the division value calculation unit calculates a division value for both x and y, and calculates a smaller division value for the calculated division value. 2. The curve drawing device according to claim 1, wherein the value is a division value.