JP7272742B1 - Drawing method for CAD - Google Patents

Abstract

Kind Code: A1 A CAD drawing method for trisecting a practical arbitrary angle that can minimize a substantial error at an arbitrary angle from 0° to 180° is provided. SOLUTION: In a drawing method for CAD in which an intersection angle XOY between a straight line OX and a straight line OY is equally divided into three by a CAD system, the CAD system draws a point A on the straight line OX and a point B on the straight line OY. , OA=OB, connecting point A on straight line OX and point B on straight line OY to create triangle AOB, and creating equilateral triangle ABO1 with side AB as one side. and the step of drawing the circumscribed circle of the center O2 that circumscribes the equilateral triangle ABO1, and the angle (central angle θ) formed by ZA and ZB formed by the vertex Z on OO1, according to the value of the angle XOY, A step of calculating the value of the central angle θ by an approximation curve in which the central angle θ is the angle XOY as a variable; a step of obtaining Z that makes the angle AZB=θ on OO1; Finding H and I on AZ and BZ, respectively; finding the intersection point P between O1H and O2A and the intersection point Q between O1I and O2B; subtracting OP and OQ; constructing QOY, thereby dividing the angle XOY into three equal parts. [Selection diagram] Fig. 1

Description

The present invention relates to a drawing method for CAD, and more particularly, to a CAD related to trisection of a practical arbitrary angle capable of minimizing substantial errors as much as possible at an arbitrary angle from 0° to 180°. It relates to a drawing method.

It has been mathematically proven that, in particular, the method of constructing 3 halves of an angle is impossible, with respect to the rigorous method of constructing an equator of an angle using only a compass and a ruler.
In more detail, regarding the "angle trisection problem", one of the three major construction problems in Greece, the angle trisection problem means "It is possible to divide an arbitrary given angle into three equal parts. mosquito? It is known that it is generally impossible to construct the trisection of a given angle by the usual method of construction, i.e., using a compass and an unscaled ruler. . However, by changing the usage of tools and adding new tools, such as a drawing method using a ruler with a scale, a drawing method using origami, a drawing method using a prop that can draw a parabola, etc. It is also known that in some cases it is possible to construct halves.

For example, U.S. Pat. No. 6,200,301 discloses an angle trisection drawing aid. More specifically, for the purpose of learning how to draw trisections of any angle, and for the purpose of assisting in drawing trisections easily, one end edge is arcuate. A fan-shaped quadrant and a center line whose upper end is the center point of the fan-shaped quadrant are drawn on the upper surface of the circular arc side of the substantially rectangular transparent plate. Connected at the left corner of the quadrant. A corner trisection drawing aid is disclosed in which the above figure is displayed, and a pointing needle and a central axis for supporting the pointing needle are provided on a substrate at the center point of the quadrant. According to such an auxiliary device, on a sheet of paper such as a notebook, an arbitrary angle, a quadrant line, and one side of a straight line forming a right angle to the central angle are drawn. By placing a substrate 1 on which a trisecting line mark can be written and operating a pointer or the like, adding an angle of 1/12 to the quadrant line, which is 1/4 of an arbitrary angle, makes 3 You can write a mark that can draw an equisecting line. It is said that the construction described above facilitates drawing.

For example, Patent Literature 2 discloses an equisecting line drawing ruler. More specifically, for the purpose of providing a ruler with which an equisecting line can be obtained by a simple method, a triangular ruler consisting of two right-angled sides and an inclined side connecting the two sides is provided. This ruler for drawing equisecting lines is characterized in that the connecting point of one side and an inclined side is used as a base point, and a side is provided by a slit on a straight line connecting this base point and a point that equally divides the other side of two right-angled sides. . According to such an equisecting line drawing ruler, one side of two right-angled sides is aligned with one point between two points on the straight line to be measured, and the other side at that time intersects the inclined side. A straight measurement line perpendicular to the side is drawn, and the intersection point where this measurement line and the slit intersect becomes an equal dividing point.

However, any construction is nothing more than an equal division of an angle with a special instrument, and even if it is not possible to construct an exact division of an angle into three equal parts, a circle or It does not present a method of drawing by a simple procedure using only a compass that defines the length of a line segment and a ruler that draws a straight line. In particular, Patent Literature 2 is a drawing instrument that relates to equal division of line segments rather than equal division of angles.

Therefore, even in CAD, the drawing algorithm for equally dividing angles is not strictly equal. In the case of machining a regular pyramid, the 11 side surfaces are not strictly formed, so that, for example, the reflection of light on each of the 11 side surfaces is slightly different. For this reason, in applications where it is important to realize a precise optical path in an optical device, it is required to minimize the error.
In CAD, the thinner the thickness of the line for drawing a straight line or curve, the better. Therefore, in particular, as the drawing method for equally dividing angles becomes stricter, the thickness of the line to be drawn is required to be thinner.

In this regard, the applicant of the present application has disclosed in Patent Document 3 that, in the drawing for equally dividing an arbitrary angle, the drawing for CAD, which has a simple procedure and has a smaller error than the conventional drawing method, is used for practical use. provide a way.
This drawing method for CAD is a drawing method for CAD _that divides _the intersection angle XOY between the straight line OX and the straight line OY into _three equal parts. a step of obtaining OB ₁ ; a step of connecting _point A ₁ on straight line OX and point B ₁ on straight line OY to form triangle A ₁ OB _{1 ;} The steps of creating a triangle A ₁ B ₁ O ' and the points where straight lines passing through the center O ₁₁ of the circumscribed circle of the equilateral triangle A ₁ B ₁ O ' and parallel to A ₁ B ₁ intersect with AO ' and BO ' respectively A ₂ and B ₂ , while taking the midpoint of A ₁ B ₁ as O ₁₂ , and the intersections C ₁ ′ of A ₂ B ₁ and B ₂ O ₁₂ , A ₁ B ₂ and A ₂ O ₁₂ respectively, and obtaining C ₁ ; and setting E 1 and E ₁ ' as points where extension lines of O'C ₁ and O'C ₁ ' intersect A ₁ B ₁ , _respectively , whereby E ₁ and E ₁ ' divide A ₁ B ₁ into three equal parts, and also the intersection F 1 of A ₂ B ₂ and O'C ₁ and the intersection F 1 ' of _{A 2} B ₂ and O'C _{1 '} obtaining each; and letting a and b be the points on the circumscribed circle that are the distances of F ₁ E ₁ ' or F ₁ 'E ₁ from A and B, respectively; and a O ' and b O ' respectively. and A ₁ B ₁ , and drawing straight lines OI and OJ, wherein the angles XOI, IOJ and JOY respectively correspond to thirds of the intersection angle XOY , is configured.

However, in this drawing method for CAD, when the crossing angle XOY between the straight line OX and the straight line OY is divided into three equal parts, the point _A1 on the straight line OX and the point _B1 on the straight line OY are set as follows: _OA1 = _OB1 Connect point A ₁ on straight line OX and point B ₁ on straight line OY to create triangle A ₁ OB ₁ , equilateral triangle A 1 B ₁ O with side A _{1 B 1} as one side ' and using points E ₁ and E ₁ ' that divide A ₁ B ₁ into three equal parts, the angles XOI, IOJ and At each angle JOY, the drawing error exceeded at least 1%, and it was not necessarily a practical drawing method for CAD.

Japanese Patent Application Laid-Open No. 2001-131593 Actual open No. 03-95901 Japanese Patent Application Laid-Open No. 2022-22928

In view of the above technical problems, it is an object of the present invention to provide a CAD system for practically trisecting arbitrary angles that can minimize substantial errors at arbitrary angles from 0° to 180°. To provide a drawing method for drawing.
In view of the above technical problems, it is an object of the present invention to reduce the error in plotting that equally divides an arbitrary angle from 0° to 180° in comparison with the conventional plotting method, so that it can be put to practical use. To provide a drawing method for CAD capable of making the thickness of drawing lines thinner than before.

In order to achieve the above objects, the nested drawing method for CAD of the present invention includes:
In the drawing method for CAD that divides the intersection angle XOY between the straight line OX and the straight line OY into three equal parts by the CAD system,
The CAD system is
obtaining a point A on the straight line OX and a point B on the straight line OY so that OA=OB;
connecting point A on line OX and point B on line OY to form triangle AOB;
creating an equilateral triangle ABO ₁ with side AB as one side;
constructing a circumscribed circle of the center _O2 that circumscribes the equilateral triangle _ABO1 ;
The angle (central angle θ) formed by ZA and ZB formed by the vertex Z on OO ₁ is calculated according to the value of the angle XOY using an approximate curve with the central angle θ as a variable and the angle XOY as the central angle θ. calculating a value;
finding Z such that the angle AZB=θ on OO ₁ ;
finding H and I on AZ and BZ such that O ₁ A//O ₂ H, O ₁ B//O ₂ I, respectively;
determining the intersection point P between O ₁ H and O ₂ A and the intersection point Q between O ₁ I and O ₂ B;
subtracting OP and OQ and constructing the angles XOP, POQ and QOY.

According to the CAD drawing method having the above configuration, when the intersection angle XOY between the straight line OX and the straight line OY is divided into three equal parts, OA=OB Regarding point A on straight line OX and point B on straight line OY, based on the circumscribed circle of equilateral triangle ABO ₁ with side AB as one side and center O ₂ circumscribing equilateral triangle ABO ₁ , By focusing on the angle between ZA and ZB formed by the vertex Z on OO ₁ (hereafter referred to as the central angle), any angle from 0° to 180° reduces the substantial error as much as possible. It becomes possible to divide a practical arbitrary angle into three equal parts.
More specifically, it relates to a CAD drawing method for trisecting arbitrary angles, based on an arbitrarily sized equilateral triangle whose two vertices are each on each line segment forming an arbitrary angle to be trisected, Focusing on the central angle corresponding to any angle to be divided into thirds, when drawing using the central angle, between 0° and 180°, at each finite number of angles, the thirds of the angle If the central angle is calculated using the angle itself corresponding to the angle, and an approximate curve is obtained based on these finite number of coordinate points (trisection symmetrical angle, central angle), this approximate curve is used. Therefore, even if it is proved mathematically impossible to draw three equal parts of an arbitrary angle, it is a purely theoretical scene with no thickness of line segments, and it is an actual situation. In the drawing method for CAD, it is essential to be visible, and for that reason, considering that the thickness of the line segments that make up the drawing is essential, at any angle from 0 ° to 180 ° It is possible to provide a drawing method for CAD related to practical trisection of an arbitrary angle that can minimize errors.

Further, in the CAD system, the step of creating the approximate curve includes:
drawing line segments OX, OL, OM, and OY starting from point O, based on a given angle φ, such that angles XOL, angles LOM, and MOY are angles φ, respectively;
obtaining a point A on the straight line OX and a point B on the straight line OY so that OA=OB;
connecting point A on line OX and point B on line OY to form triangle AOB;
creating an equilateral triangle ABO ₁ with side AB as one side;
constructing a circumscribed circle of the center _O2 that circumscribes the equilateral triangle _ABO1 ;
determining points of intersection P and Q of line segments OL and OM, respectively, and _O2A and _O2B ;
determining points H and I at which line segments passing through _O2 and parallel to _AO1 and _BO1 respectively intersect with an extension line of _O1P to the P side and an extension line of _O1Q to the Q side; ,
determining the central angle θ between AH and BI;
performing the step of determining the central angle θ for a plurality of angles φ;
plotting (3φ, θ) against the obtained central angle θ and angle φ to create an approximate curve.

Furthermore, the intersection angle XOY between the straight lines OX and OY is preferably greater than 0° and less than or equal to 180°.
Furthermore, when the angle 3φ is 90° and 180°, respectively, the central angle θ is 90° and 180°, and the approximate curve passes through at least these two points.
In addition, in 3φs corresponding to a plurality of given angles φ, the angular spacing of adjacent 3φs should be set every 10° to 20°.

In addition, in 3φ corresponding to a plurality of given angles φ, it is set for each angle interval of adjacent 3φ in the range of 10° to 20° according to the derivative of the approximated curve with respect to the angle 3φ. good.
Furthermore, in the drawing stage for dividing the angle XOY into three equal parts, it is preferable to use a zoom-up function to draw an enlarged drawing according to the value of the angle XOY.
Furthermore, the approximate curve may be approximated by a polygonal line connecting a plurality of coordinates (3φ, θ) defined by 3φ and the corresponding central angle θ.
In addition, when constructing an equilateral triangle ABO ₁ , it is preferable to position O ₁ so that it is on the opposite side of AB from O.

FIG. 1 is a functional block diagram showing part of the basic configuration of a CAD system that implements a drawing support method according to one embodiment of the present invention.
The input unit 10 is composed of input devices such as a mouse and a keyboard. For example, position information of a mouse cursor input from the input unit 10 is detected by the cursor position detection unit 20 . The cursor position information detected by the cursor position detection unit 20 is supplied to the display control unit 90, where the mouse cursor or the like is displayed at the detected position on the display screen of the display unit 80 composed of, for example, a CRT device or a liquid crystal display device. is displayed.
Further, the cursor position information from the cursor position detection unit 20 is given to the command execution processing unit 30, where the command selected from the cursor position detected by the cursor position detection unit 20 and the click information from the input unit 10 is processed. Then, the executed command processing is applied to the CAD processing unit 40 . When the command activation menu such as an icon is not displayed at the cursor position detected by the cursor position detection unit 20, the command execution processing unit 30 supplies the cursor position information to the CAD processing unit 40 in a through state. .
The CAD processing section 40 performs CAD processing based on information input via the command execution processing section 30 . The CAD processing unit 40 performs normal CAD processing, so a detailed description is omitted. It also plays the role of storing the graphic element information in the graphic information storage unit 50 after it has been processed.

Cursor position information from the cursor position detector 20 is also supplied to the target element position detector 60 . The target element position detection unit 60 detects that the cursor position information detected by the cursor position detection unit 20 captures the target element position. Here, the "target element position" means one or more drawn graphic elements stored in the graphic information storage unit 50, or one or more graphic elements currently being drawn by the CAD processing unit 40 (hereinafter referred to as These are called "response elements"), and the position of the target element (for example, the intersection point on the extension line, horizontal, vertical line, etc.) that has a specific positional relationship. Also, "target element position detection" means that the target element position is positioned within a predetermined neighborhood range centered on the cursor position. When the target element position detector 60 detects that the cursor has captured the target element position, the target element position detector 60 provides the display controller 90 with information on the target element, its position, and the reaction element.

Based on the information from the target element position detection unit 60 and preset display color information, the display control unit 90 changes the display form of the cursor and reaction elements displayed on the display unit 80, and changes the target from the reaction elements. Executes controls such as displaying auxiliary lines to elements.

FIG. 2 is a diagram showing an example of a screen displayed on the display section 80 of this CAD system. This screen is composed of a graphic display area, various menu display areas, and the like. First, when a menu for setting the "drawing support function" is selected from the menu display area 12 of the screen, a drawing support function setting dialog box is displayed on the screen as shown in FIG.

As shown in the figure, the drawing support function is enabled by clicking a drawing support function enable check box 140 with a mouse or the like to add a check mark. In this case, information indicating that the drawing support function has been selected is supplied from the cursor position detection unit 20 to the command execution processing unit 30, the target element position detection unit 60, and the display control unit 90, and the target element position detection unit 60 is activated. This will enable the function. The drawing support function commands that can be set in this drawing support function setting dialog box are as follows. Extend intersection commands 150 that change the display form and, secondly, the respective points of the reaction elements:
(1) For a straight line, end point/midpoint,
(2) For circle, ellipse, and hole symbols, the center point,
(3) For circular arcs and elliptical arcs, endpoints and center points,
(4) In the case of a free-form curve, the vertex,
Etc.X and Etc.Y commands 160 for changing the display form of the cursor when the target element is the intersection of straight lines extended along the X and Y axes from the pair of , and thirdly, the response element A horizontal/vertical command 170 for changing the display form of the cursor when the target element is a line extending horizontally or vertically from the drawing start point of the graphic element , and fourthly, a reaction element Angle increase command 180 for changing the display form of the cursor when the target element is a line that extends from the drawing start point of the graphic element in the direction of an angle that matches the value of the angle increase value that has been input in advance. By checking the check box of each command 150 to 180, the target element position detection unit 60 executes the checked command. The drawing support function setting dialog box also includes an angle increase value input box 190 for manually inputting an arbitrary angle in the angle increase command 180 using the keyboard of the input unit 10 or the like.
A division number input box 191 and a division target angle input box 192 are also provided.

When the drawing support function is enabled, the display control unit 90 changes the mouse cursor from the normal mode mouse cursor 250a to the drawing support function mode mouse cursor 250b, and displays it on the display unit 80. Control. Also, when the mouse cursor 250b captures the target element position, an information display cursor 250c specifying the target element is displayed diagonally below the mouse cursor 250b. The shape of the information display cursor 250c is set to a shape that facilitates visual recognition of the content of the target element. For example, when the mouse cursor 250b captures an existing point such as an end point or point element on the reaction element, a cross is displayed as the information display cursor to indicate the existing point. When an extended intersection command or an equal X or equal Y command is executed, an X shape is displayed as an information display cursor. When the mouse cursor 250b captures a point on the reaction element, an information display cursor is displayed indicating that the point is on the element. Similarly, when a horizontal/vertical command and an angle increase command are executed, horizontal lines, vertical lines and angle marks are displayed as information display cursors 250c, respectively.

Further, the drawing support function setting dialog box has a display color setting function for setting the display color of the graphic element displayed in the graphic display area when the drawing support function is enabled. By checking the drawing support graphic display check box, the color display setting function in the display control unit 90 is enabled. There are two types of display colors for graphic elements that can be set by this display color setting function: the display color for graphic elements when the operator is performing an input operation using a mouse or the like, and the display color for reaction elements. In this case, an arbitrary display color can be set from a pull-down menu for the display color of the graphic element being input, and from a pull-down menu for the display color of the reaction element.

Next, the operation when each command 150 to 180 is executed with the drawing support function enabled will be described. FIG. 4 is a diagram showing an image when the extended intersection command 150 is executed. First, the mouse cursor 250b is superimposed on an arbitrary position on the linear elements 240a and 240b of the two graphic elements displayed in the display window 230 opened in the graphic display area, and the graphic elements 240a and 240b are displayed by mouse clicks or the like. Select/determine. This selection/determining operation is performed to specify the object and eliminate the amount of calculation, and if there is no problem with the amount of calculation, this operation may not be performed. At this time, the information display cursor 250c is not displayed unless the extended intersection component of the straight line elements 240a and 240b is within the fixed area range 250e from the center position 250d of the mouse cursor 250b.

Next, when the mouse cursor 250b is moved in the direction of the arrow in the drawing, the extended intersection point P1 of the linear elements 240a and 240b comes within the fixed area range 250e of the mouse cursor 250b. When the extended intersection point P1 enters the fixed area range 250e of the mouse cursor 250b, the intersection display shape is displayed near the mouse cursor 250b as an information display cursor 250c, and the position of the mouse cursor 250b is at the extended intersection point of the linear elements 240a and 240b. For example, if the operator inputs a decision to confirm in this state, rubber bands 260a and 260b are displayed as auxiliary lines on extension lines of the linear elements 240a and 240b, and the linear elements 240a and 240b are displayed accurately. It is possible to easily create a figure drawn by extending it to the extension intersection point P1. At this time, the linear elements 240a and 240b change to the preset color of the reaction elements, and the rubber bands 260a and 260b also display differently from other graphic elements.

FIG. 5 is a diagram showing an image when the equal X and equal Y command 160 is executed. First, as described above, the mouse cursor 250b is superimposed on an arbitrary position on the linear elements 240a and 240b of the two displayed graphic elements, and the graphic elements 240a and 240b are selected/determined by a mouse click or the like. This operation is also arbitrary as described above. At this time, the intersection of lines extending in the X-axis direction and the Y-axis direction from the end points a and b of the linear elements 240a and 240b must be included within the range 250e of the fixed area from the center position 250d of the mouse cursor 250b. For example, the information display cursor 250c is not displayed.

Next, as the mouse cursor 250b is moved in the direction of the arrow in the figure, the intersection point P2 of the lines extending in the X-axis direction and the Y-axis direction from the end points a and b of the straight line elements 240a and 240b is the fixed area range of the mouse cursor 250b. Comes in 250e. At this time, an intersection display shape is displayed near the mouse cursor 250b as the shape of the information display cursor 250c, and the position of the mouse cursor 250b is equal to the X coordinate value or the Y coordinate value of the end points a and b of the linear elements 240a and 240b. to show that it is in In this way, it is possible to use the positions of the end points a and b of the straight line elements 240a and 240b as references, and to draw a figure or the like for equal X and Y coordinate values therefrom.

FIG. 6 is a diagram showing an image when the horizontal/vertical command 170 is executed. First, a starting point s for drawing a graphic element is set at an arbitrary position in the display window 230 by clicking the mouse cursor 250b or the like. At this time, the information display cursor 250c is not displayed unless the center position 250d of the mouse cursor 250b or the fixed area range 250e therefrom does not have a horizontal/vertical positional relationship with the starting point s.

Next, the mouse cursor 250b is moved in the direction of the arrow (1) or (2) in the figure, and the mouse cursor is placed at a position where the starting point s and the center position 250d of the mouse cursor 250b are in a horizontal/vertical positional relationship. When the fixed area range 250e of 250b is entered, a horizontal display shape or a vertical display shape is displayed near the mouse cursor 250b as the shape of the information display cursor 250c. If a line segment or the like is created when the information display cursor 250c is displayed horizontally or vertically, the horizontal line segment or the vertical line segment can be easily drawn without using an auxiliary line or the like.

FIG. 7 is a diagram showing an image when the angle increase command 180 is executed. First, a starting point s for drawing a graphic element is set at an arbitrary position in the display window 230 by clicking the mouse cursor 250b or the like. Next, an arbitrary angle is entered in the angle increase value input box 190 of the drawing support function setting dialog box. For example, when the angle is set to 45°, the information display cursor 250c is not displayed unless the center position 250d of the mouse cursor 250b or the fixed area range 250e from there is in the positional relationship of the set angle 45° with the starting point s. .

Next, the mouse cursor 250b is moved in the arrow direction in the figure. Each time the fixed area range 250e of the mouse cursor 250b enters a position where the start point s and the center position 250d of the mouse cursor 250b have a positional relationship of 45°, the angle increase display shape of the information display cursor 250c is changed to that of the mouse cursor 250b. displayed in the vicinity. If a line segment or the like is created when the information display cursor 250c is in the angle increasing display, it is possible to create the line segment or the like accurately at an arbitrary angle.

A drawing procedure flow will be described with reference to FIG.
As shown in FIG. 1, first, in step 1, φmin and φmax of a given trisecting angle φ and the notch angle width Δφ are set.
Next, in step 2, a given trisection angle φ is set to φmin.
Next, in step 3, a central angle θ for a given triangular angle φ is obtained by a drawing method to be described later.
Next, in step 4, it is judged whether or not the given triangular angle φ is equal to or greater than φmax. If it exceeds φmax, proceed to step 6;
Next, in step 5, the given triangular angle φ is set to φ+Δφ, and the process advances to step 3 to obtain the central angle θ with respect to the given triangular angle φ by the later-described drawing method.
Next, in step 6, an approximate curve is created from a plurality of points (3φ, θ) based on the calculated combination of φ and θ.
Next, in step 7, a trisection target angle XOY is set.
Next, in step 8, based on the approximated curve created in step 6, a central angle θ corresponding to the trisection target angle XOY is obtained.
Next, in step 9, based on the obtained central angle θ, plotting of the triangular angles of the triangular object angle XOY is performed by the plotting method described later.

In the drawing method for CAD that divides the intersection angle XOY between the straight line OX and the straight line OY into three equal parts by the CAD system,
The CAD system is
obtaining a point A on the straight line OX and a point B on the straight line OY so that OA=OB;
connecting point A on line OX and point B on line OY to form triangle AOB;
creating an equilateral triangle ABO ₁ with side AB as one side;
constructing a circumscribed circle of the center _O2 that circumscribes the equilateral triangle _ABO1 ;
The angle (central angle θ) formed by ZA and ZB formed by the vertex Z on OO ₁ is calculated according to the value of the angle XOY using an approximate curve with the central angle θ as a variable and the angle XOY as the central angle θ. calculating a value;
finding Z such that the angle AZB=θ on OO ₁ ;
finding H and I on AZ and BZ such that O ₁ A//O ₂ H, O ₁ B//O ₂ I, respectively;
determining the intersection point P between O ₁ H and O ₂ A and the intersection point Q between O ₁ I and O ₂ B;
subtracting OP and OQ and constructing the angles XOP, POQ and QOY, thereby dividing the angle XOY into three equal parts.

According to the CAD drawing method having the above configuration, when the intersection angle XOY between the straight line OX and the straight line OY is divided into three equal parts, OA=OB Regarding point A on straight line OX and point B on straight line OY, based on the circumscribed circle of equilateral triangle ABO ₁ with side AB as one side and center O ₂ circumscribing equilateral triangle ABO ₁ , By focusing on the angle between ZA and ZB formed by the vertex Z on OO ₁ (hereafter referred to as the central angle), any angle from 0° to 180° reduces the substantial error as much as possible. It becomes possible to divide a practical arbitrary angle into three equal parts.
More specifically, it relates to a CAD drawing method for trisecting arbitrary angles, based on an arbitrarily sized equilateral triangle whose two vertices are each on each line segment forming an arbitrary angle to be trisected, Focusing on the central angle corresponding to any angle to be divided into thirds, when drawing using the central angle, between 0° and 180°, at each finite number of angles, the thirds of the angle If the central angle is calculated using the angle itself corresponding to the angle, and an approximate curve is obtained based on these finite number of coordinate points (trisection symmetrical angle, central angle), this approximate curve is used. Therefore, even if it is proved mathematically impossible to draw three equal parts of an arbitrary angle, it is a purely theoretical scene with no thickness of line segments, and it is an actual situation. In the drawing method for CAD, it is essential to be visible, and for that reason, considering that the thickness of the line segments that make up the drawing is essential, at any angle from 0 ° to 180 ° It is possible to provide a drawing method for CAD related to practical trisection of an arbitrary angle that can minimize errors.

In the CAD system, creating the approximate curve includes:
drawing line segments OX, OL, OM, and OY starting from point O, based on a given angle φ, such that angles XOL, angles LOM, and MOY are angles φ, respectively;
obtaining a point A on the straight line OX and a point B on the straight line OY so that OA=OB;
connecting point A on line OX and point B on line OY to form triangle AOB;
creating an equilateral triangle ABO ₁ with side AB as one side;
constructing a circumscribed circle of the center _O2 that circumscribes the equilateral triangle _ABO1 ;
determining points of intersection P and Q of line segments OL and OM, respectively, and _O2A and _O2B ;
determining points H and I at which line segments passing through _O2 and parallel to _AO1 and _BO1 respectively intersect with an extension line of _O1P to the P side and an extension line of _O1Q to the Q side; ,
determining the central angle θ between AH and BI;
performing the step of determining the central angle θ for a plurality of angles φ;
and plotting (3φ, θ) with respect to the obtained central angle θ and angle φ to create an approximate curve.

The intersection angle XOY between the straight lines OX and OY is greater than 0° and less than or equal to 180°.
When the angle 3φ is 90° and 180°, respectively, the central angle θ is 90° and 180°, and the approximate curve passes through at least these two points.
In 3φ corresponding to a plurality of given angles φ, the angular spacing of adjacent 3φ is set every 10° to 20°.
At 3φ corresponding to a plurality of given angles φ, it is set for every adjacent 3φ angle interval in the range of 10° to 20° according to the derivative of the approximated curve with respect to the angle 3φ.
In the drawing step of dividing the angle XOY into three equal parts, the drawing is enlarged by using the zoom-up function according to the value of the angle XOY.
When drawing an equilateral triangle ABO ₁ approximated by a polygonal line connecting a plurality of coordinates (3φ, θ) with 3φ and the corresponding central angle θ, O ₁ is on the opposite side of AB with respect to O. Position it so that it is located at

According to the drawing procedure as shown in FIG. 9, the result of obtaining the central angle θ according to the angle for dividing into three equal parts is as follows.
10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90°, 100°, 110°, 120°, 130°, For 140°, 150°, 160°, and 170° respectively, the central angle θ is 110.72°, 102.93°, 96.60°, 91.83°, 88.55° to two decimal places. , 86.72°, 86.36°, 87.43°, 90.00°, 94.12°, 99.87°, 107.31°, 116.47°, 127.26°, 139.48° , 152.71°, and 166.44°.
This result is divided into three equal parts on the horizontal axis (equivalent to 3φ), and the central angle θ on the vertical axis. be.
As shown in FIG. 10, the approximation curve is a downward convex curve with a minimum value in the range of 60° to 70°.
Hereinafter, using this approximation curve, the center angle θ is read to the first decimal place using the approximation curve for the angle divided into three equal parts other than the above 17 points, and using this center angle θ, It shows that it is possible to draw within a practically error-free range by drawing three equal parts.

FIG. 11 and FIG. 12 show the drawing procedure for the case where the angle XOY is 67°. A drawing corresponds to a drawing screen for CAD.
As shown in Fig. 11, in (1) and (2), point A on straight line OX and point B on straight line OY are obtained so that OA = OB, and point A on straight line OX and straight line OY Connect with point B above to form triangle AOB.
Next, in (3), an equilateral triangle _ABO1 having side AB as one side is created.
Next, in (4), draw a circumscribed circle of the center _O2 that circumscribes the equilateral triangle _ABO1 .
Next, the angle (central angle θ) formed by ZA and ZB formed by the vertex Z on OO ₁ is calculated according to the value of the angle XOY using an approximate curve with the central angle θ and the angle XOY as variables. After calculating the value of θ, in (5), as shown in _FIG .
Then, in (6) and (7), H and I that result in O ₁ A//O ₂ H and O ₁ B//O ₂ I are obtained on AZ and BZ, respectively.
Next, in (8), find the intersection point P between O ₁ H and O ₂ A and the intersection point Q between O ₁ I and O ₂ B, subtract OP and OQ, and construct angles XOP, POQ and QOY. .

FIG. 13 presents a plotting result when the angle XOY is 67°, as a plotting method for 3-division CAD using the approximate curve described above.
According to the approximation curve, the central angle θ can be read as 86.4° to one decimal place. 22.33°), which are completely matched up to two decimal places, and correspond to three equal parts of the angle XOY 67°.
FIG. 14 presents a plotting result when the angle XOY is 4°, as a plotting method for 3-division CAD using the approximate curve described above.
According to the approximation curve, the central angle θ can be read as 116.1° to one decimal place. 1.33°), they are completely in agreement up to two decimal places, and correspond to three equal parts of the angle XOY4°.

FIG. 15 presents a plotting result when the angle XOY is 7°, as a plotting method for 3-division CAD using the approximate curve described above.
According to the approximation curve, the central angle θ can be read as 113.4° to the first decimal place. 2.33°), they are completely in agreement up to two decimal places, and correspond to three equal parts of the angle XOY 7°.
FIG. 16 presents a plotting result when the angle XOY is 37°, as a plotting method for 3-division CAD using the approximate curve described above.
According to the approximation curve, the central angle θ can be read as 93.1° to one decimal place. 12.33°), which are completely matched up to two decimal places, and correspond to three equal parts of the angle XOY of 37°.
FIG. 17 presents a plotting result in the case where the angle XOY is 97° as a plotting method for 3-part CAD using the approximate curve described above.
According to the approximation curve, the central angle θ can be read as 92.7° to the first decimal place. 32.33°), which are completely matched up to two decimal places, and correspond to three equal parts of the angle XOY97°.

FIG. 18 presents a plotting result when the angle XOY is 103°, as a plotting method for 3-part CAD using the approximate curve described above.
According to the approximation curve, the central angle θ can be read as 95.7° to the first decimal place. 34.33°), which are completely matched up to two decimal places, and correspond to three equal parts of the angle XOY of 103°.
FIG. 19 presents a plotting result when the angle XOY is 149° as a plotting method for 3 equal divisions using the approximate curve described above.
According to the approximation curve, the central angle θ can be read as 138.2° to one decimal place. 49.67°), which are completely matched up to two decimal places, and correspond to three equal parts of the angle XOY of 149°.
FIG. 20 presents a plotting result when the angle XOY is 179°, as a plotting method for 3-division CAD using the approximate curve described above.
According to the approximation curve, the central angle θ can be read as 178.7° to one decimal place. 59.67°), which are completely matched up to two decimal places, and correspond to three equal parts of the angle XOY of 179°.

13 to 20, prime angles are selected as the angles of the angle XOY to be divided equally into three, but the angles XOP (φ1), POQ (φ2) and QOY (φ3) are , as an angle corresponding to 3 equal parts of the angle XOY, which is a perfect match up to two decimal places, and considering the thickness of the line in drawing, an arbitrary angle of 3, etc. with practically no drawing error This indicates that minute drawing is possible. Conversely, in the drawing method for CAD using the CAD system, it contributes to further thinning of the thickness of the line for drawing.
In the above-mentioned CAD system, for each 10° (3φ, θ), the adjacent coordinates are approximated by a straight line, the equation of this straight line is calculated and stored, and based on the angle XOY to be divided into three , first, select which straight line equation to apply, second, based on the selected straight line equation, calculate the central angle θ, and use the calculated central angle θ to construct the above drawing method It has been confirmed that it is possible to draw a trisection with substantially no error by drawing by . In this case, it is preferable to store the combination of the angle range of 3φ and the linear equation as a database in the CAD system.

According to the CAD drawing method having the above configuration, when the intersection angle XOY between the straight line OX and the straight line OY is divided into three equal parts, OA=OB Regarding point A on straight line OX and point B on straight line OY, based on the circumscribed circle of equilateral triangle ABO ₁ with side AB as one side and center O _{2 circumscribing equilateral triangle ABO 1 ,} By focusing on the angle between ZA and ZB formed by the vertex Z on OO ₁ (hereafter referred to as the central angle), any angle from 0° to 180° reduces the substantial error as much as possible. It becomes possible to divide a practical arbitrary angle into three equal parts.
More specifically, it relates to a CAD drawing method for trisecting arbitrary angles, based on an arbitrarily sized equilateral triangle whose two vertices are each on each line segment forming an arbitrary angle to be trisected, Focusing on the central angle corresponding to any angle to be divided into thirds, when drawing using the central angle, between 0° and 180°, at each finite number of angles, the thirds of the angle If the central angle is calculated using the angle itself corresponding to the angle, and an approximate curve is obtained based on these finite number of coordinate points (trisection symmetrical angle, central angle), this approximate curve is used. Therefore, even if it is proved mathematically impossible to draw three equal parts of an arbitrary angle, it is a purely theoretical scene with no thickness of line segments, and it is an actual situation. In the drawing method for CAD, it is essential to be visible, and for that reason, considering that the thickness of the line segments that make up the drawing is essential, at any angle from 0 ° to 180 ° It is possible to provide a drawing method for CAD related to practical trisection of an arbitrary angle that can minimize errors.

1 is a diagram showing an outline of a CAD system used in a drawing method according to an embodiment of the present invention; FIG. FIG. 2 is a diagram showing an example of a screen displayed on a display unit in a CAD system used in the drawing method according to the embodiment of the present invention; FIG. 5 is a diagram showing a drawing support function setting dialog box on a screen displayed on a display unit in the CAD system used in the drawing method according to the embodiment of the present invention; FIG. 3 is a diagram showing an image when an extended intersection command displayed on the screen of FIG. 2 is executed in the CAD system used in the drawing method according to the embodiment of the present invention; FIG. 3 is a diagram showing an image when the equal X and equal Y commands displayed on the screen of FIG. 2 are executed in the CAD system used in the drawing method according to the embodiment of the present invention; FIG. 3 is a diagram showing an image when horizontal and vertical commands displayed on the screen of FIG. 2 are executed in the CAD system used in the drawing method according to the embodiment of the present invention; FIG. 3 is a diagram showing an image when an angle increase command displayed on the screen of FIG. 2 is executed in the CAD system used in the drawing method according to the embodiment of the present invention; 1 is a flow chart showing the procedure of a CAD drawing method according to the present invention; FIG. FIG. 5 is a diagram showing drawing for obtaining a central angle θ when a given trisection angle Δ is 20° in the CAD drawing method according to the present invention; In the CAD drawing method according to the present invention, for each finite number of angles, the central angle is calculated using the angle itself corresponding to the angle that divides the angle into three equal parts, and these finite number of coordinate points (trisection symmetric angle , central angle). FIG. 10 is a diagram showing a drawing procedure ((1) to (4)) for drawing three equal parts with an angle XOY of 67° in the CAD drawing method according to the present invention; FIG. 10 is a diagram showing a drawing procedure ((5) to (8)) for drawing three equal parts with an angle XOY of 67° in the CAD drawing method according to the present invention; FIG. 10 is a diagram showing the result of plotting three equal parts with an angle XOY of 67° in the CAD plotting method according to the present invention; FIG. 14 is a diagram similar to FIG. 13 showing the result of drawing three equal parts with an angle XOY of 4° in the CAD drawing method according to the present invention; FIG. 14 is a diagram similar to FIG. 13 showing the result of drawing three equal parts with an angle XOY of 7° in the CAD drawing method according to the present invention; FIG. 14 is a diagram similar to FIG. 13 showing the result of drawing three equal parts with an angle XOY of 37° in the CAD drawing method according to the present invention; FIG. 14 is a diagram similar to FIG. 13, showing the result of drawing three equal parts with an angle XOY of 97° in the CAD drawing method according to the present invention; FIG. 14 is a diagram similar to FIG. 13 showing the result of drawing three equal parts with an angle XOY of 103° in the CAD drawing method according to the present invention; FIG. 14 is a diagram similar to FIG. 13, showing the result of drawing three equal parts with an angle XOY of 149° in the CAD drawing method according to the present invention; FIG. 14 is a diagram similar to FIG. 13 showing the result of drawing three equal parts with an angle XOY of 179° in the CAD drawing method according to the present invention;

θ: Central angle φ: Trisection angle XOY (3φ): Arbitrary acute angle
ABO ₁ : equilateral triangle
O _2: the center of the circumscribed circle of the equilateral triangle ABO ₁
Z: Angle between ZA and ZB formed by vertex Z on OO ₁ H: A line segment passing through O ₂ and parallel to AO ₁ intersects the extension line of O ₁ P to the P side I: O ₂ and the line segment parallel to BO ₁ intersects the extension line of O ₁ Q to the Q side

Claims (9)

In a drawing method for CAD in which a CAD system divides the intersection angle XOY between a straight line OX and a straight line OY into three equal parts, the CAD system places a point A on the straight line OX and a point B on the straight line OY, respectively, OA=OB connecting point A on straight line OX and point B on straight line OY to create triangle AOB; creating equilateral triangle ABO1 with side AB as one side; equilateral triangle The step of drawing the circumscribed circle of the center O2 that circumscribes ABO1, and the angle (central angle θ) formed by ZA and ZB formed by the vertex Z on OO1, and determining the central angle θ according to the value of the angle XOY A step of calculating the value of the central angle θ from a pre-created approximation curve with the angles XOY as variables ; O2I, H and I on AZ and BZ, respectively; finding the intersection point P between O1H and O2A and the intersection point Q between O1I and O2B; subtracting OP and OQ, the angle XOP, the angle constructing POQ and angle QOY, thereby dividing angle XOY into three equal parts. In the CAD system, the step of creating the approximation curve includes, based on a given angle φ, starting from point O, line segments OX, a step of drawing OL, OM and OY; a step of obtaining a point A on the straight line OX and a point B on the straight line OY so that OA=OB; a point A on the straight line OX and a point on the straight line OY; A stage of creating an equilateral triangle ABO1 with the side AB as one side A stage of drawing a circumscribed circle of the center O2 that circumscribes the equilateral triangle ABO1 A line segment OL and A step of obtaining intersections P and Q of O2A and O2B with each of OM, and a line segment passing through O2 and parallel to each of AO1 and BO1 is an extension line of O1P to the P side and an extension line of O1Q to the Q side. determining the points H and I intersecting with AH and BI; determining the central angle θ between AH and BI; performing the determining the central angle θ for a plurality of angles φ; 3. The drawing method for CAD for dividing the angle XOY into three equal parts according to claim 1, wherein (3φ, θ) is plotted against the angle φ to create an approximate curve. 2. The CAD drawing method for dividing the angle XOY into three equal parts according to claim 1, wherein the intersection angle XOY between the straight line OX and the straight line OY is greater than 0° and less than or equal to 180°. 3. When the angle 3φ is 90° and 180°, respectively, the central angle θ is 90° and 180°, and the approximate curve passes through at least these two points. Drawing method for CAD that divides equally. 5. The CAD drawing method for dividing the angle XOY into three equal parts according to claim 4, wherein in 3φ corresponding to a plurality of given angles φ, the angular interval of adjacent 3φ is set every 10° to 20°. . At 3φ corresponding to a plurality of given angles φ, it is set for each adjacent 3φ angle interval in the range of 10° to 20° according to the derivative of the approximated curve with respect to the angle 3φ. 5. A drawing method for CAD that divides the angle XOY according to 5 into 3 equal parts. 7. The CAD device for dividing the angle XOY of 25 into three equal parts according to claim 6, wherein the approximated curve is approximated by a polygonal line connecting between a plurality of coordinates (3φ, θ) by 3φ and the corresponding central angle θ. Drawing method. 3. The drawing method for CAD according to claim 2, wherein, in the drawing step of dividing the angle XOY into three equal parts, the drawing is enlarged by using a zoom-up function according to the value of the angle XOY. 3. The drawing method for CAD according to claim 2, wherein when drawing an equilateral triangle ABO1, O1 is positioned on the opposite side of AB with respect to O.

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