JP3502901B2 - Three-dimensional graphics image display apparatus and method - Google Patents

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a three- dimensional graphics image display device and method suitable for performing hidden surface processing and shadow display of a sphere in three-dimensional graphics .

[0002]

2. Description of the Related Art The hidden surface of a sphere in three-dimensional graphics
In order to perform processing and shading display, conventionally, a sphere is approximated by a polyhedron, and a sphere is approximated and displayed by a smooth shading method that smoothes shading display of polygons to be surfaces. However, this method has a problem that the contour of the sphere is inevitable to be polygonal and that if the quality is approximated to a circle and the quality is increased, the number of polygons is increased and thus the drawing time is long.

Meanwhile, although the arc so far different even algorithms as a drawing method have been reported, it hidden spheres
It is not used for surface processing and shading display. For example, Bresenham's algorithm uses only integer arithmetic to generate seamless point sequences. JP-A-7-935
In the No. 69 algorithm, the above-mentioned Presenham's algorithm is improved so that it can be performed at high speed for thick arcs and ellipses.

Dieter W. Fellner & Christoph Helmberg
Has published a new method for drawing general elliptic arcs using only integer and shift operations. (ACM Tran
sactions on Graphics, Vol.12, No.3, (1993) "Rubust Ren
dering of General Ellipsesand Elliptical Arc
s. ").

[0005]

The display method using these algorithms enables high-speed processing by being implemented by dedicated hardware. However,
Neither method is considered for application to hidden surface treatment and shading of spheres, and no actual application example has been reported.

Therefore, an object of the present invention is to provide a hidden surface treatment for spheres.
And shadow display can be applied to speed up arc drawing time 3
A three-dimensional graphics image display device and method are provided.

[0007]

In order to achieve such a problem, an image display device according to a first aspect of the present invention relates to a point sequence constituting an arc, of the (n + 1) th point of the point sequence on the display screen. Coordinates x (n + 1), y (n + 1)

[0008]

[Equation 9] x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・ sin (t) + y (n) ・ cos (t)

(Where t is the n + 1th point of the n + 1th point
The angle to rotate and move to a point, and the center of the circle as the coordinate origin)

Representation by the recurrence formula, indicating means for indicating the center of the arc, the start point and the end position of the arc, and the display of the arc point sequence by the recurrence formula based on the instructed information. Arithmetic processing means for acquiring each coordinate on the screen, and displaying the arc by displaying the point sequence at the position of each acquired coordinate, and along a scanning line of a circle representing a spherical surface.
The normal vector and depth coordinate of the point sequence of horizontal pixels
The sphere's hidden area is treated by filling it one after another while
And display control means for realizing the processing of shadow and shadow processing .

The invention of claim 2 is the third order of claim 1.
In the original graphics image display device, the arc is a circle, and sin (t), cos (t) in the recurrence formula
To

[0012]

Equation 10], characterized in that approximated by sin (t) = t cos ( t) = 1-t 2/2.

A three-dimensional graphics image according to claim 3
The image display device has a coordinate x (n + 1) of the (n + 1) th point of the point sequence on the display screen with respect to the point sequence forming the elliptical arc.
u (n + 1)

[0014]

[Equation 11] x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・ sin (t) + y (n) ・ cos (t) u (n + 1) = b / a ・ y (n + 1)

(Where a is the length of the x-axis of the ellipse, and b is y.
The length of the axis, the origin of the ellipse is used as the origin), and the instruction means for instructing the information indicating the center of the ellipse, the starting point and the ending point of the arc of the ellipse, and the information based on the indicated information. an arithmetic processing means for obtaining respective coordinates on the display screen of the point sequence of the ellipse by the recurrence formula, when you view the ellipse by displaying the point sequence to the positions of the coordinates of the acquired co
Of the horizontal pixel along the scan line of the circle representing the sphere
Paint one after another while obtaining the normal vector and depth coordinate of the point sequence
Performs hidden surface processing and shadow processing for the sphere by performing crushing.
And an image display control unit that appears.

According to a fourth aspect of the present invention , a three-dimensional graphic for calculating and displaying coordinates of a sequence of points forming an arc for display.
In the cursor image display method, the coordinate x (n) at the (n + 1) th point of the point sequence on the display screen with respect to the point sequence forming the arc.
+1), y (n + 1)

[0017]

[Equation 12] x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・ sin (t) + y (n) ・ cos (t)

(Where t is the nth point and the (n + 1) th point
The angle to rotate and move , the center of the circle is the coordinate origin)

The information indicating the center of the arc, the starting point and the end point of the arc is indicated by the recurrence formula of, and based on the instructed information, the point sequence of the arc is displayed on the display screen by the recurrence formula. Acquire each coordinate and display the above-mentioned point sequence at the position of each acquired coordinate to display an arc and
The normal of the point array of horizontal pixels along the scanline of the representing circle.
Fill one after another while obtaining the vector and depth coordinates.
It is characterized in that the hidden surface processing and the shadow processing of the sphere are realized by carrying out.

The invention of claim 5 is the third order of claim 4.
In the original graphics image display method, the arc is a circle, and sin (t), cos (t) in the recurrence formula
To

[0021]

Equation 13], characterized in that approximated by sin (t) = t cos ( t) = 1-t 2/2.

The invention of claim 6 is a three-dimensional graph for calculating and displaying the coordinates of a sequence of points forming an arc of an ellipse for display.
In the fixed image display method, the coordinates x (n + 1), u (n + 1) of the (n + 1) th point of the point sequence on the display screen with respect to the point sequence forming the elliptic arc are calculated.

[0023]

[Equation 14] x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・ sin (t) + y (n) ・ cos (t) u (n + 1) = b / a ・ y (n + 1)

(Where a is the length of the x-axis of the ellipse, b is y
The length of the axis and the center of the ellipse are used as the origin) and the information indicating the center of the ellipse, the starting point and the end point of the arc of the ellipse is designated, and based on the designated information, the Acquire each coordinate on the display screen of the point sequence of the ellipse by the chemical formula, and display the ellipse by displaying the point sequence at the position of each obtained coordinate, and display the ellipse along the scanning line of the circle representing the spherical surface.
The normal vector and depth coordinate of the point sequence of horizontal pixels
The sphere's hidden area is treated by filling it one after another while
It is characterized in that it realizes processing and shadow processing .

[0025]

BEST MODE FOR CARRYING OUT THE INVENTION Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

FIG. 1 shows a graphic L to which the present invention is applied.
The circuit configuration of SI is shown.

The graphic LSI is mounted in an image processing device (host system) such as a personal computer. In FIG. 1, a graphic LSI is connected to a system bus 1 of a host system such as a personal computer. The graphic LSI is similar to the conventional LSI in that it generates a point sequence of a figure to be drawn by a DDA (digital differential analyzer), and can be created within the range of existing technology.

Drawing commands, ie circles, ellipses, spheres, etc.
A drawing command including parameter information necessary for drawing such as the type of command to be drawn, center position, start point, end point position, etc. is sent from the host system to the graphic command control unit 2 via the system bus 1. The parameter information included in the drawing command is temporarily stored in the RAM 3.

DDA (digital differential analyzer) unit 4
Calculates the pixel position of the figure to be drawn according to the image processing program installed in advance, and draws the figure in the video frame buffer 7 via the pixel control unit 5.

The ALU (logical operation unit) 10 is an RA
Using the parameter information stored in M3, each pixel position of the figure designated by the drawing command is calculated. This calculation method will be described in detail later.

For this calculation, the register group 11 holds the value of the variable and the flag register 12 holds the flag information. The register 13 holds an address for reading / writing data in the RAM 3.

The RAM 3 is provided with an area 21 for storing the values of the variables used in the pixel position calculation and the calculation result, and a storage area (pixel light FIFO) 22 for sequentially transferring the calculation result to the pixel control unit 5. Has been. In the present embodiment, as the calculation result, the coordinates (X, Y, Z) of the drawn point sequence and the brightness B obtained from the normal vector are obtained.

The pixel light control unit 5 reads the coordinate data and the luminance data from the pixel light FIFO 22 and draws them in the video frame buffer 7 using the Z buffer method. As a result, only the point whose Z value indicated by the drawn image data of the point sequence is in front of the Z value of the Z buffer 8 is drawn with the brightness B. The image data in the video frame buffer 7 is digital-to-analog converted in the video control unit 6 and sent to the monitor 9 as a video signal.

As a result, the monitor 9 displays the figure designated by the drawing command.

(1) Method for generating a point sequence of circular arcs When a circular arc of a circle as shown in FIG. 8 is represented by a discrete sequence of points, the sequence of points is expressed by the following equation X (n), Y ( n) is converted into an integer to obtain the coordinate on the corresponding grid (see FIG. 9). For simplicity, the center of the circle is taken as the origin.

[0036]

X (n) = r ・ cos (t ・ n) y (n) = r ・ sin (t ・ n) t is the angle formed by the points, and if t = 1 / r, then on the grid A seamless point sequence is obtained. This sequence of points can be expressed by the following recurrence formula using a rotation matrix of two-dimensional coordinates.

[0037]

(18) x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・ sin (t) + y (n ) ・ Cos (t) sin () and cos () are approximated to execute the recurrence formula of Formula 2 for one rotation of the arc, that is, to execute the circumference length of about 2πR times. , Formula 2 obtains Formula 19 by Taylor expansion. Note that terms higher than t ³ can be ignored.

[0038]

By Equation 19] sin (t) ≒ t cos ( t) ≒ 1-t 2/2 This is calculated the difference of the point sequence,

[0039]

[Expression 20] Δx = x (n + 1) -x (n) =-y / r-1 / 2r ² x Δy = y (n + 1) -y (n) = + x / r-1 / 2r ^When performing 1/8 rotation of ² y arc, the term of t ² is x =
It can be ignored except when 0 and y = 0,

[0040]

[Expression 21] Δx = -y / r (however, -1 / 2r ² x when y = 0) Δy = x / r x and y are real numbers, but the points to draw when the integer part changes Since the integer part and the decimal part are separate integer variables, the value of the integer part is increased and decreased when the decimal part overflows and underflows, and at the same time You can change the position. When this fractional part is expressed by a numerator of a fraction with r as a denominator, X (n) = [integer part of X (n)] + fX (n) / r
Is. Assuming that the variables of this minority part are fx and fy for x and y, respectively, from equation (21), Δx and Δy
Since it can be seen that the absolute value does not exceed 1, only the minority parts fx and fy need to be calculated in the calculation of the recurrence formula in Formula 20. When r is applied to both sides of the equation (20), division by r is lost and the following equation is obtained.

[0041]

[Mathematical formula-see original document] fx (n + 1) = fx (n) -y (n) fy (n + 1) = fy (n) + x (n) This fractional part variable is 0- (r-1). Takes a value and changes the integer part when it exceeds that range.

A flowchart of this process is shown in FIG. The process of FIG. 2 will be described. In step 101, the initial value of the coordinate of the arc whose center coordinate of the circle is (x0, y0) is set. In step 102, the coordinates (rX, rY) of the circle centered on the origin and the minority part variables are initialized, and in step 103, the calculation of the minority part is repeated. In step 104, the coordinate is incremented when r or more, and in step 107, the coordinate is decremented when 0 or less. RX in step 111
By repeating until rY> 1/8 arc,
That is, a point sequence of t = 0 to 45 degrees is obtained. Since the circular arc is symmetric with respect to the x-axis and the y-axis, the entire 1/8 circle can be drawn using the obtained 1/8 circular arc.

Depending on the size of the arc to be drawn, the following measures are necessary to obtain an arc having no unevenness.
If the diameter of the circular arc is even, the center of the circle is x, y from the grid.
Both are shifted by 0.5, and the point sequence generated by setting the initial value of the angle t to 0 and cutting off the decimal point or less is used. Since the centers are offset, the point sequences X (n) and Y (n) have a decimal part from the beginning, and correspondingly, the decimal part variables fx and f
A value (r / 2) corresponding to 0.5 is set as the initial value of y. If the diameter is odd, the radius will have a fractional part, so correspondingly, set a value (r / 2) corresponding to 0.5 to the initial value of the fractional part fx of X (n). , The center of the circle is matched with the lattice, and the point sequence generated by setting the initial value of the angle t to ½r is truncated.

In Fellner et al.'S report, the calculation by (Equation 21) has a large error, but when considering a 1/8 circular arc, the error is in a negligible range. Calculations based on (20) are as accurate as their method. Further, even if it is used for drawing a sphere, the error is sufficiently small and practical.

(2) Method for generating a sequence of points of an elliptical arc A sequence of points of an elliptical arc as shown in FIG. 10 uses a new variable U (n) for the y coordinate. Let the lengths of the axes in the x and y directions be a and b

[0046]

X (n) = a · cos (n · t) U (n) = b · sin (n · t) where Y (n) = a / b · U (n) Column (X
(N) and Y (n) are arcs having a radius a, and U (n) is obtained from the point sequence of the arcs, but division and integration can be omitted as follows.

The variable U (n) is divided into an integer part and a decimal part as in the case of X and Y, and the calculation of the point sequence is repeated with the decimal part as the fraction of b. That is, (Equation 22
At the same time, the following equation is calculated for the numerator fU of the minority part of U.

[0048]

[Equation 24] fU (n) = fU (n) + x (n) Since the value x (n) added to fU is the size in which a is loaded below the decimal point, divide this by a ² / b. Thus, the desired value U (n) = y · b / a can be obtained. That is, the integer part of U (n) is obtained by regarding the fU (n) calculated by (Equation 24) as the numerator of the fraction with a ² / b as the denominator and applying overflow and underflow.

After all, although division and integration are required to obtain a ² / b only once at the beginning, the repetitive calculation can be performed only by addition and subtraction and comparison processing. Further, when a <b, if the denominator of the decimal part when calculating the arcs of X (n) and Y (n) is b instead of a, the denominator of the decimal part of U (n) is a. You can, and you will not be able to divide and integrate.

In the case of an ellipse, since there is no symmetry in the exchange of X and Y, it is necessary to find a point sequence of 1/4 of the circumference, and calculation based on (Equation 20) is necessary. To it in addition to the calculation of the equation (21 formula) 1 / 2r ^2-consuming 2
We also calculate for one term. These are also treated as fractions as before. The denominator is expressed as a fraction with 2r ² and its numerator is an integer variable in gX and gY. Similarly, overflow and underflow are the higher terms fX.
And fY, or borrow. From the equation (4), the following equations are obtained for gX and gY.

[0051]

GX (n + 1) = gx (n) -2 * fy-rX gY (n + 1) = gy (n) + 2 * fx-rY The formula for the variable U is the same as the formula for Y. Therefore, it can be omitted and the overflow and underflow of gY can be applied to fU.

For the sake of versatility, it is regarded as a generator using the method of generating the point sequence of this elliptic arc, and it can be used as a subroutine for obtaining the next point by calling it. This elliptic arc generation subroutine and the ellipse drawing flow chart using it are shown in FIGS. For the sake of simplicity, the part that generates a point sequence of 1/4 of the full ellipse is shown. In step 201 of FIG. 3, the main axis a of the ellipse, the radius length b of the other axis, the integer part of the initial value of the point sequence of the elliptic arc, and the initial values of the respective minor parts are a and b as denominators. Set the initial value of the elliptic arc generator by the value of the numerator of the fraction. Details are shown in FIG. Step 202
Call the elliptic arc generation subroutine in to get the next point representing the circle and elliptical arc. As data of the point sequence, a total of three integer values of two coordinate axes X and Y of the point sequence forming an arc, and a variable U forming an elliptic arc by combining X, and vectors rX, rY, rU from the center of the ellipse are obtained. To be

The elliptic arc generation subroutine is step 220 and subsequent steps in FIG. 5, but in comparison with the arc generation processing in FIG. 2, the calculation of (Equation 25) is added in step 221.
The overflow and underflow of gX and gY are added to steps 222-229. Then, even when the diameter of the elliptic axis is an odd number, it is effective to consider the initial value of each parameter as in the case of the circle.

(3) Method of displaying shadow of sphere In displaying the shadow of a sphere, when the inside of a circle representing the sphere is filled with a specified color, the brightness of the pixel is changed corresponding to the normal vector of the sphere. Is a method of expressing a three-dimensional sphere. At the same time, in order to apply the hidden surface processing using the z buffer, it is also necessary to calculate the change in the depth direction of the spherical surface and set the Z value in each pixel to be filled.

In this method, filling is performed one after another while obtaining a normal vector and a depth coordinate Z of a point sequence of horizontal pixels along a scanning line of a circle representing a spherical surface. At this time, the X coordinate and the depth coordinate Z along the scanning line draw a circular arc, and the normal vector also changes in a circular arc. Therefore, use the above-mentioned high-speed elliptic arc drawing method. You can Then, by performing this for all the scanning lines inside the circle representing the sphere, all the pixels that fill the entire sphere are drawn. It is a feature of this method that most of the calculations required for this purpose can be executed at high speed because only integer addition and subtraction and comparison operations are performed.

The flow of this processing is roughly divided into two. Two flowcharts: an arc that represents the outer circumference of the sphere, a part that finds a horizontal pixel row along one scanning line that fills the sphere, and a part that fills the pixel by calculating the Z coordinate component and shadow component in the horizontal pixel row. Is shown in FIGS. 6 and 7. In addition, the brightness of the pixel drawn by this processing is schematically represented as a numerical value of 1 to 9. In both of these two, the elliptic arc generation subroutine described above is utilized.

Therefore, a subroutine for saving all the parameters required for generating the point sequence in the memory and a subroutine for restoring the parameters so that the calculation of the point sequence can be restarted from the middle later are prepared. .

First, in order to obtain a point sequence of a circle representing a sphere, an elliptic arc generation subroutine is initialized in step 301, and the elliptical arc generation subroutine is called in step 302 to start the horizontal pixel array starting point along the scanning line. And ask for the length. Then, in step 303, the parameters of this elliptic arc generation subroutine are once saved so that the next point of the arc can be immediately obtained later.

The horizontal pixel string drawing subroutine in and after step 320 performs drawing in consideration of hidden surface processing and shadow processing in three-dimensional graphics. In order to calculate the Z value and the normal vector at each point of the obtained horizontal pixel row, the elliptic arc generation subroutine is initialized again in step 321, and the point value of the elliptic arc is obtained in step 322 to obtain the Z value in the horizontal pixel row. And obtain the normal vector. In the Z-buffer method for hidden surface processing, it is effective and common to use a Z value in a unit smaller than the pixel resolution, so the Z value for X is a point sequence of an elliptic arc. A three-dimensional pixel waiting for a Z value is obtained by obtaining a point sequence of an elliptic arc represented by X and U.

In the shadow processing, the brightness to be set is determined from the pixel value in proportion to the inner product of the normal vector of the object and the light source vector, and therefore the normal vector for each pixel is obtained here. The normal vector of the surface of the sphere is simply equal to the vector from the center of the sphere to the sphere, and the point sequence (X, Y) of the arc obtained when obtaining the point sequence of the elliptic arc can be used. Drawing is performed using a subroutine that can draw a pixel with the brightness to be set based on the obtained Z value and normal vector value of the pixel. The image data obtained by this drawing is shown in FIG. Since the horizontal pixel column, which is the segment of the sphere, is bilaterally symmetric, the left and right points are drawn in steps 323 and 324. The surface normal vector of the sphere has symmetry of the sphere, but since the brightness of each pixel is calculated and displayed according to the obtained normal vector and the direction of the light source vector, the symmetry is finally displayed in the shaded display. It is also possible to draw an empty sphere.

Since four horizontal pixel rows are obtained by utilizing the symmetry of the upper and lower sides of the sphere and the symmetry with respect to the X coordinate and the Y coordinate, the drawing subroutines are called in steps 305, 306, 308 and 309, respectively. , The entire sphere can be drawn.

The starting point and length of the horizontal pixel row in the next scan line is obtained by returning the parameters of the elliptic arc generation subroutine previously saved in step 310 and continuing. Then, by calculating the arc of 1/8 of the circumference, all the areas can be filled.

In addition to the embodiments described above, the following modes can be implemented.

1) In the above embodiment, an example in which image processing for drawing a circle, an ellipse, and a sphere is performed by an LSI has been described. However, when the drawing processing is performed by a general-purpose personal computer, the CPU shown in FIG. ~ The processing procedure of FIG. 7 may be executed. 2 to 7 are recorded in a recording medium such as a hard disk storage device.

2) In the above embodiment, a circle, an ellipse,
Although information such as the radius required for drawing the sphere, the starting point and the ending point of the drawing, etc. is specified in the form of a command, these information can be specified from the keyboard input or from the graphic creation program.

[0066]

As described above, in the present invention, cos (t) and sin (t) in the recurrence formula are constants,
The coordinates of x (n + 1), y (n + 1) are x (n), y
It can be easily obtained by four arithmetic operations using the value of (n) and the above constant. This shortens the coordinate calculation time.

Further , the present invention further applies s when the arc is a circle.
in (t), since performed the calculation of cos (t) approximation of ^{t, 1-t 2/2} , further coordinate calculation time is shortened.

Further, according to the present invention , the sphere display can be obtained by performing the hidden surface processing or the shadow processing on the circle figure, and the sphere display processing can be shortened.

[Brief description of drawings]

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

FIG. 2 is a flowchart showing a processing procedure for generating a point sequence of arcs.

FIG. 3 is a flowchart showing a processing procedure for generating a point sequence of an elliptic arc.

FIG. 4 is a flowchart showing a processing procedure for generating a point sequence of an elliptic arc.

FIG. 5 is a flowchart showing a processing procedure for generating a point sequence of an elliptic arc.

FIG. 6 is a flowchart showing a processing procedure for generating a point sequence of an elliptic arc.

FIG. 7 is a flowchart showing a processing procedure for generating a point sequence of an elliptic arc.

FIG. 8 is an explanatory diagram illustrating a procedure for drawing a circle.

FIG. 9 is an explanatory diagram showing a dot sequence of circles.

FIG. 10 is an explanatory diagram for explaining a procedure for drawing an ellipse.

FIG. 11 is an explanatory diagram showing image data for displaying a sphere.

[Explanation of symbols]

1 system bus 2 Graphic command control unit 3 RAM 4 DDA unit 5 pixel light control unit 6 Video control unit 7 video frame buffer 8 Z buffer 9 monitors

   ─────────────────────────────────────────────────── ─── Continued front page       (56) References JP-A-7-93569 (JP, A)                 JP-A-7-29033 (JP, A)                 JP-A-2-69882 (JP, A)                 JP 60-114969 (JP, A)                 Fujio Yamaguchi "Computer Graph               Six "(May 30, 1979), daily publication               Kogyo Shimbun, pp. 112-113                 Hosaka Mamoru "Computer Graphics"               X "(January 28, 1974), industrial map               Sho Co., pp. 81-82                 ACM Transactions               on Graphics, Vol. 12,               No. 3, (1993), "Robust               Rendering of Generator               al Ellipse sand Ell               optical Arcs. "

Claims (6)

(57) [Claims] 1. The coordinates x (n + 1), y of the (n + 1) th point of the point sequence on the display screen with respect to the point sequence forming an arc.
(N + 1) is given by [Formula 1] x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・ sin (t) + y (n) ・ cos (t) (where t is the angle to rotate the nth point to the n + 1th point, the center of the circle is the coordinate origin), and the center of the arc , Instruction means for instructing information indicating the starting point and the end point position of the arc, and based on the instructed information, each coordinate on the display screen of the point sequence of the arc is calculated using only integer addition and subtraction by the recurrence formula. Obtained arithmetic processing means and a point sequence of horizontal pixels along a scanning line of a circle representing a spherical surface while displaying an arc by displaying the point sequence at the position of each acquired coordinate. Three-dimensional graphics image display device comprising display control means for realizing hidden surface processing and shading processing of a sphere by successively filling the same while obtaining the normal vector and depth coordinates of . 2. The three-dimensional graphics image display device according to claim 1, wherein the arc is a circle, and sin (t) and cos (t) in the recurrence formula are expressed by the following formula: sin (t) t) = t cos (t) = 3 -dimensional graphic image display apparatus characterized by approximated by 1-t ^2/2. 3. The coordinates x (n + 1) of the (n + 1) th point of the point sequence on the display screen with respect to the point sequence forming the elliptical arc.
u (n + 1) can be expressed by [Formula 3] x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・ sin (t ) + y (n) ・ cos (t) u (n + 1) = b / a ・ y (n + 1) (where a is the length of the x-axis of the ellipse, b is the length of the y-axis, The center of the ellipse is used as the origin) and the instruction means for instructing the information indicating the center of the ellipse, the start point and the end point of the arc of the ellipse, and the recurrence equation based on the instructed information. The arithmetic processing means for obtaining each coordinate on the display screen of the point sequence of the ellipse using only addition and subtraction of integers, and displaying the ellipse by displaying the point sequence at the position of each obtained coordinate. An image that realizes hidden surface processing and shading processing of the sphere by successively filling the circle representing the spherical surface while obtaining the normal vector and depth coordinate of the point sequence of horizontal pixels along the scanning line. A three-dimensional graphics image including display control means Display devices. 4. A three-dimensional graphics image display method for calculating and displaying the coordinates of a sequence of points forming an arc for display, wherein n of said sequence of points on the display screen is related to the sequence of points forming the arc.
The coordinates x (n + 1) and y (n + 1) at the + 1st point are expressed by the following equation: x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・ sin (t) + y (n) ・ cos (t) (where t is the angle to rotate and move the nth point to the n + 1th point, and the center of the circle is the coordinate origin. ) Is indicated by the recurrence formula, the information indicating the center of the arc, the start point and the end point of the arc is designated, and based on the designated information, the coordinates of the point sequence of the arc are displayed by the recurrence formula. Is obtained using only addition and subtraction of integers, and the arc is displayed by displaying the point sequence at the position of each obtained coordinate, and the circle representing the spherical surface is horizontally aligned along the scanning line. Three-dimensional graphics image display method, characterized in that the sphere hidden surface processing and the shadow processing are realized by successively performing the filling while obtaining the normal vector and the depth coordinate of the point sequence of various pixels. 5. The three-dimensional graphics image display method according to claim 4, wherein the arc is a circle, and sin (t) and cos (t) in the recurrence formula are expressed by t) = t cos (t) = 3 -dimensional graphic image display method characterized by approximated by 1-t ^2/2. 6. A three-dimensional graphics image display method for calculating and displaying coordinates of a sequence of points forming an elliptical arc for display, wherein the sequence of points on the display screen is related to the sequence of points forming the elliptical arc. The coordinate x (n +
1) and u (n + 1) are expressed by the following equation x (n + 1) = x (n) ・ cos (t) -y (n) ・ sin (t) y (n + 1) = x (n) ・sin (t) + y (n) ・ cos (t) u (n + 1) = b / a ・ y (n + 1) (where a is the length of the x-axis of the ellipse and b is the y-axis) The length and the center of the ellipse are used as the origin), and the information indicating the center of the ellipse, the starting point and the end point of the arc of the ellipse is designated, and the recurrence formula is based on the designated information. With each coordinate on the display screen of the point sequence of the ellipse is obtained using only addition and subtraction of integers, the ellipse is displayed by displaying the point sequence at the position of each obtained coordinate, It is characterized by realizing the hidden surface processing and the shadow processing of the sphere by successively filling the circle representing the sphere while obtaining the normal vector and the depth coordinate of the point sequence of horizontal pixels along the scanning line. 3D graphics image display method.