[文章]：计算和entmake椭圆弧的方法和程序...

XDSoft

ical arcs, group code 41 is described as the "start parameter".
However, its association with known values for the arc is difficult to
determine. How is this value derived?

An elliptical arc is a special version of an arc that follows the
eccentricity of the ellipse. One way to generate this type of arc is to find
the parametric normal of the starting point. To do so, you must specify a
start angle that is different from the actual start angle of the drawn arc.
Group code 41 contains this parametric angle expressed in
radians.

What Is the Parametric Angle?

The parametric angle is generated from two concentric circles whose center is
the center of the ellipse and whose radii are the major and minor axes,
respectively. Every point of the ellipse lies either between or on these two
circles, and each elliptical point can be defined by a unique relation to
them. To discover this relationship, draw a line perpendicular to the major
axis from a point on the ellipse to the closest intersection with the circle
described by the major axis. Then do the same with the minor axis, starting
from the elliptical point and drawing perpendicular to the minor axis until
the line intersects the circle described by the minor axis. The two points of
intersection with the circles are colinear with the center of the ellipse.
The angle between the line containing these three points and the major axis
is the parametric angle specified by group code 41.

How Do I Calculate It from the True Start Angle?

To calculate the parametric angle from the true start angle, you must
first find the start point on the ellipse. This requires a simultaneous
solution to the equations for the line and the ellipse. In this example we
assume that the major axis of the ellipse lies on the x-axis with the origin
at the center of the ellipse. When this point is found, you
can use its y-value and the minor axis to solve the equation for the circle
whose radius is the minor axis value and whose center is the center of the
ellipse. This will provide the x,y point on the circle that dictates the
parametric angle from the center of the ellipse.

Below is an AutoLisp example that demonstrates how to use trigonometric
functions to determine the parametric angle:

2. 
   
3. (defun c:e_arc( / a b slope ang q1 q2 q3 q4 qmode x y a2)
   
4.   ;assuming 0,0 is at the center of the ellipse, major axis in x direction
   
5.   (setq ang (getangle '(0.0 0.0) "Choose start angle: "))
   
6.   (setq a 1
   
7.          b 0.5
   
8.          slope (/ (sin ang) (cos ang))
   
9.          q1 (/ pi 2.0)
   
10.          q2 pi
    
11.          q3 (/ (* 3 pi) 2.0)
    
12.          q4 (* 2.0 pi)
    
13.          qmode 'q1
    
14.   );setq
    
15. 
    
16.   (entmake (setq ent '((0 . "ELLIPSE")
    
17.                         (100 . "AcDbEntity")
    
18.                         (100 . "AcDbEllipse")
    
19.                         (10 0.0 0.0 0.0)
    
20.                         (11 1.0 0.0 0.0)
    
21.                         (40 . 0.5)
    
22.                         (62 . 1))
    
23.             );setq
    
24.   );entmake
    
25.    
    
26.   ;line equation is y = mx + 0, where m is the slope and 0 is the y-intercept
    
27.   ;ellipse equation is x^2/a^2 + y^2/b^2 = 1
    
28.   ;solve line and ellipse equations simultaneously to find x and y values
    
29.    
    
30.   (setq y (/ (* a b slope) (sqrt (+ (* (* slope slope) (* a a)) (* b b)))))
    
31.    
    
32.   ;minor axis circle equation is x^2 + y^2 = b^2
    
33.   ;solve circle equation where y = value calculated above
    
34.   (setq x (sqrt (- (* b b) (* y y))))
    
35.    
    
36.   ;calculate start angle trigonometrically
    
37.   (setq cos_a2 (/ x b)
    
38.          sin_a2 (/ y b)
    
39.   );setq
    
40. 
    
41. (if (/= cos_a2 0)
    
42.      (setq a2 (atan (/ sin_a2 cos_a2)))
    
43.      (setq a2 q1
    
44.       qmode 'q1)
    
45.   );if
    
46.    
    
47.   ;make a2 insensitive to quadrant
    
48.   (cond ((and (> ang q1) (< ang q2))
    
49.              (setq a2 (- pi (abs a2))
    
50.                   qmode 'q2
    
51.              );setq
    
52.          );statement 1
    
53.         ((and (> ang q2) (< ang q3))
    
54.             (setq a2 (+ (abs a2) pi)
    
55.                 qmode 'q3
    
56.            );setq
    
57.          );statement 2
    
58.         ((and (> ang q3) (< ang q4))
    
59.             (setq a2 (abs (- (* 2 pi) (abs a2)))
    
60.                  qmode 'q4
    
61.              );setq
    
62.         );statement 3
    
63.    
    
64.       ;special cases: angle = 0, 90, 180, 270 or 360 deg
    
65.       ((or (= ang 0) (= ang q1))
    
66.            (setq qmode 'q1)
    
67.         );statement 4
    
68.       ((= ang q2)
    
69.            (setq a2 pi qmode 'q1)
    
70.       );statement 5
    
71.       ((= ang q3)
    
72.            (setq a2 (- (/ pi 2.0) pi) qmode 'q1)
    
73.       );statement 6
    
74.            (t nil);default statement
    
75.   );cond
    
76.    
    
77.   (command "zoom" "c" "0,0" 3)
    
78.   (setq ent (append ent (list (cons 41 a2) (cons 42 (+ a2 (/ pi 2.0))))))
    
79.   (setq ent (subst '(62 . 5) (assoc 62 ent) ent))
    
80.   (entmake ent)
    
81. 
    
82.   (setq a2
    
83.       (cond ((= qmode 'q1) a2)
    
84.            ((= qmode 'q2) a2)
    
85.            ((= qmode 'q3) (- a2 q4))
    
86.            ((= qmode 'q4) (- a2 q4))
    
87.            (t nil)
    
88.       );cond
    
89.   );setq
    
90. 
    
91.   (princ "\nParametric angle in radians: ")
    
92.   (princ a2)
    
93.   (princ "\nParametric angle in degrees: ")
    
94.   (princ (/ (* 180 a2) pi))
    
95.   (princ)
    
96.   );e_arc
    

<br /> (defun c:e_arc( / a b slope ang q1 q2 q3 q4 qmode x y a2)<br />   ;assuming 0,0 is at the center of the ellipse, major axis in x direction<br />   (setq ang (getangle '(0.0 0.0) "Choose start angle: "))<br />   (setq a 1 <br />          b 0.5 <br />          slope (/ (sin ang) (cos ang))<br />          q1 (/ pi 2.0)<br />          q2 pi<br />          q3 (/ (* 3 pi) 2.0)<br />          q4 (* 2.0 pi)<br />          qmode 'q1<br />   );setq<br /> <br />   (entmake (setq ent '((0 . "ELLIPSE") <br />                         (100 . "AcDbEntity") <br />                         (100 . "AcDbEllipse") <br />                         (10 0.0 0.0 0.0) <br />                         (11 1.0 0.0 0.0)<br />                         (40 . 0.5) <br />                         (62 . 1))<br />             );setq<br />   );entmake<br />    <br />   ;line equation is y = mx + 0, where m is the slope and 0 is the y-intercept<br />   ;ellipse equation is x^2/a^2 + y^2/b^2 = 1<br />   ;solve line and ellipse equations simultaneously to find x and y values<br />    <br />   (setq y (/ (* a b slope) (sqrt (+ (* (* slope slope) (* a a)) (* b b)))))<br />    <br />   ;minor axis circle equation is x^2 + y^2 = b^2<br />   ;solve circle equation where y = value calculated above<br />   (setq x (sqrt (- (* b b) (* y y))))<br />    <br />   ;calculate start angle trigonometrically<br />   (setq cos_a2 (/ x b)<br />          sin_a2 (/ y b)<br />   );setq<br /> <br /> (if (/= cos_a2 0)<br />      (setq a2 (atan (/ sin_a2 cos_a2)))<br />      (setq a2 q1<br />       qmode 'q1) <br />   );if<br />    <br />   ;make a2 insensitive to quadrant<br />   (cond ((and (> ang q1) (< ang q2))<br />              (setq a2 (- pi (abs a2))<br />                   qmode 'q2<br />              );setq<br />          );statement 1<br />         ((and (> ang q2) (< ang q3))<br />             (setq a2 (+ (abs a2) pi)<br />                 qmode 'q3<br />            );setq<br />          );statement 2<br />         ((and (> ang q3) (< ang q4))<br />             (setq a2 (abs (- (* 2 pi) (abs a2)))<br />                  qmode 'q4<br />              );setq <br />         );statement 3<br />    <br />       ;special cases: angle = 0, 90, 180, 270 or 360 deg<br />       ((or (= ang 0) (= ang q1))<br />            (setq qmode 'q1)<br />         );statement 4<br />       ((= ang q2) <br />            (setq a2 pi qmode 'q1)<br />       );statement 5<br />       ((= ang q3)<br />            (setq a2 (- (/ pi 2.0) pi) qmode 'q1)<br />       );statement 6<br />            (t nil);default statement<br />   );cond<br />    <br />   (command "zoom" "c" "0,0" 3) <br />   (setq ent (append ent (list (cons 41 a2) (cons 42 (+ a2 (/ pi 2.0))))))<br />   (setq ent (subst '(62 . 5) (assoc 62 ent) ent))<br />   (entmake ent)<br /> <br />   (setq a2 <br />       (cond ((= qmode 'q1) a2)<br />            ((= qmode 'q2) a2)<br />            ((= qmode 'q3) (- a2 q4))<br />            ((= qmode 'q4) (- a2 q4))<br />            (t nil)<br />       );cond<br />   );setq<br /> <br />   (princ "\nParametric angle in radians: ")<br />   (princ a2)<br />   (princ "\nParametric angle in degrees: ")<br />   (princ (/ (* 180 a2) pi))<br />   (princ)<br />   );e_arc<br />

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