hp41programs

Conics Conics for the HP-41

Overview

-A conic (C) may be defined by its cartesian equation: A.x² + B.xy + C.y² + D.x + E.y + F = 0

-Knowing these 6 coefficients, the following program determines the specifications of the curve, namely:

    e = eccentricity,
    p = parameter,
    a & b = semi-major & semi-minor axis ( ellipse and hyperbola )
    C = center ( except for a parabola )
    F & F' = focuses ( only F for a parabola )
    (Ax) & (Ax') = Axis of symmetry ( only (Ax) for a parabola )
    (D) & (D') =directrices ( only (D) for a parabola )
    V, V' , V" , V''' = vertices ( only V , V' for a hyperbola, only V for a parabola )
    (As) & (As') = asymptotes ( for a hyperbola only )

-A rotation by an angle µ is performed to eliminate the term in "xy", and a translation leads to the standard expressions:

    x²/a² + y²/b² = 1   ( ellipse )                  0 <= e < 1         ( e = 0 and a = b if (C) is a circle )
    y² = 2px               ( parabola )                        e = 1
    x²/a² - y²/b² = 1   ( hyperbola )                      e > 1

                                         (Ax')
       (D)                               |                                 (D')
         |                                *|V'''                              |
         |                 *               |             *                    |
         |          *                      |                     *            |
       -|---V-*--F----------C |-----------F'--*-V'--- |--(Ax)                    ELLIPSE
         |          *                      |                     *            |
         |                  *              |             *                    |
         |                                *| V''                              |

        (D)                                             *
          |                        *
          |            *
          |        *
        -|---V*---F------------------------------------ (Ax)                       PARABOLA
          |        *
          |             *
          |                        *
                                                          *

                                    (D)    (D')
                               (As) |(Ax')   (As')
                             *   \    |   | |      / *
                               *   \ |   | |    / *
                                 *   \|   | | / *
                                  *   |\ | |/   *
          -------------F- *V|-\|/-|V'*---F'-------------------(Ax)                  HYPERBOLA
                                  *   | /|\ |    *
                                 *    |/ | |\    *
                                *   / |   | | \   *
                              *   /   |   | |    \   *
                                       |      |

-We assume the basis is orthonormal.
-The "degenerate" cases ( one point, one or two straight lines ) are not taken into account ( for example if A = B = C = 0 ).

Data Registers: R00: µ ( the rotation angle )
• R01 = A • R02 = B • R03 = C • R04 = D • R05 = E • R06 = F ( these 6 registers are to be initialized before executing "CONIC" )

-when the program stops, the specifications of the conic are stored in registers R07 to R34 as shown in the examples below.

Flags: F01 is set if (C) is an ellipse
F02 is set if (C) is a parabola
F03 is set if (C) is a hyperbola

Subroutines: /

Program Listing

-If you have an HP-41CX, lines 02-03-04 may be replaced by CLX X<> F
-Line 43 is a three-byte GTO 03
-Lines 93 to 100 are only useful to produce a DATA ERROR line 96 if the conic is imaginary ( like x²+4y²+9=0 ). Otherwise, these lines may be deleted.
-Lines 193-225 are three-byte GTO 04

01 LBL "CONIC"
02 CF 01
03 CF 02
04 CF 03
05 RCL 02
06 X^2
07 RCL 01
08 RCL 03
09 *
10 4
11 *
12 -
13 STO 33
14 X#0?
15 SIGN
16 2
17 +
18 SF IND X
19 RCL 01
20 RCL 03
21 +
22 ENTER ^
23 ENTER^
24 X^2
25 RCL 33
26 +
27 SQRT
28 ST+ Z
29 -
30 STO 23
31 X<>Y
32 STO 24
33 RCL 02
34 RCL 01
35 RCL 03
36 -
37 R-P
38 CLX
39 2
40 /
41 STO 00
42 FS? 02
43 GTO 03
44 RCL 03
45 RCL 04
46 *
47 ST+ X
48 RCL 02
49 RCL 05
50 *
51 -
52 STO 34
53 RCL 33
54 /
55 STO 11
56 STO 13
57 STO 15

58 STO 25
59 STO 27
60 STO 29
61 STO 31
62 RCL 04
63 *
64 RCL 01
65 RCL 05
66 *
67 ST+ X
68 RCL 02
69 RCL 04
70 *
71 -
72 RCL 33
73 /
74 STO 12
75 STO 14
76 STO 16
77 STO 26
78 STO 28
79 STO 30
80 STO 32
81 RCL 05
82 *
83 +
84 RCL 06
85 ST+ X
86 +
87 STO 09
88 RCL 23
89 /
90 RCL 09
91 RCL 24
92 /
93 X>Y?
94 X<>Y
95 CHS
96 LN
97 X<> L
98 CHS
99 X<Y?
100 X<>Y
101 FS? 01
102 X<=Y?
103 CHS
104 X>0?
105 GTO 01
106 1
107 ASIN
108 ST- 00
109 X<> Z
110 LBL 01
111 ABS
112 SQRT
113 STO 09
114 X<>Y

115 ABS
116 SQRT
117 STO 10
118 RCL 00
119 LASTX
120 RCL 09
121 /
122 STO 08
123 SIGN
124 RCL 10
125 RCL 09
126 /
127 X^2
128 FS? 01
129 CHS
130 +
131 SQRT
132 STO 07
133 RCL 09
134 *
135 STO 24
136 P-R
137 ST+ 13
138 ST- 15
139 RDN
140 ST+ 14
141 ST- 16
142 CLX
143 RCL 09
144 P-R
145 ST+ 25
146 ST- 27
147 RDN
148 ST+ 26
149 ST- 28
150 CLX
151 RCL 10
152 P-R
153 ST+ 30
154 ST- 32
155 X<>Y
156 ST- 29
157 ST+ 31
158 RCL 00
159 1
160 P-R
161 STO 18
162 STO 20
163 RCL 11
164 *
165 X<>Y
166 STO 21
167 CHS
168 STO 17
169 RCL 12
170 *
171 -

172 STO 22
173 STO 23
174 X<> 24
175 RCL 09
176 X^2
177 X<>Y
178 X#0?
179 /
180 X=0?
181 E99
182 ST+ 23
183 ST- 24
184 RCL 12
185 RCL 20
186 *
187 RCL 11
188 RCL 21
189 *
190 -
191 STO 19
192 FS? 01
193 GTO 04
194 1
195 STO 30
196 RCL 02
197 RCL 02
198 RCL 33
199 SQRT
200 ST+ Z
201 -
202 RCL 03
203 X=0?
204 GTO 02
205 ST+ X
206 ST/ Z
207 /
208 STO 29
200 X<>Y
210 STO 32
211 RCL 05
212 CHS
213 LASTX
214 /
215 STO 31
216 X<> 34
217 LASTX
218 1
219 X<> 33
220 SQRT
221 *
222 /
223 ST- 31
224 ST+ 34
225 GTO 04
226 LBL 02
227 STO 33
228 SIGN

229 STO 32
230 RCL 01
231 RCL 02
232 /
233 STO 29
234 RCL 01
235 RCL 05
236 *
237 RCL 02
238 RCL 04
239 *
240 -
241 RCL 02
242 X^2
243 /
244 STO 31
245 RCL 05
246 CHS
247 RCL 02
248 /
249 STO 34
250 GTO 04
251 LBL 03
252 RCL 24
253 X=0?
254 GTO 03
255 STO 23
256 1
257 ASIN
258 ST- 00
259 LBL 03
260 RCL 00
261 RCL 05
262 P-R
263 RCL 00
264 RCL 04
265 P-R
266 ST+ T
267 RDN
268 -
269 STO 11
270 X<>Y
271 STO 12
272 CHS
273 STO 08
274 RCL 00
275 RCL 11
276 X^2
277 RCL 23
278 ST/ 11
279 ST+ X
280 ST/ 08
281 /
282 RCL 06
283 -
284 RCL 12
285 /

286 P-R
287 RCL 00
288 RCL 11
289 P-R
290 ST- T
291 RDN
292 +
293 STO 09
294 STO 11
295 X<>Y
296 STO 10
297 RCL 00
298 RCL 08
299 P-R
300 ST+ 11
301 RDN
302 +
303 STO 12
304 RCL 00
305 1
306 STO 07
307 P-R
308 STO 14
309 STO 16
310 RCL 09
311 *
312 X<>Y
313 STO 17
314 CHS
315 STO 13
316 RCL 10
317 *
318 -
319 RCL 08
320 -
321 STO 18
322 RCL 09
323 RCL 13
324 *
325 RCL 10
326 RCL 14
327 *
328 +
329 STO 15
330 RCL 08
331 ST+ X
332 ABS
333 STO 08
334 LBL 04
335 RCL 08
336 RCL 07
337 END

( 448 bytes / SIZE 035 )

STACK	INPUTS	OUTPUTS
Y	/	p
X	/	e

-Execution time = 9 seconds

Example1: (C): 29.x² + 24.xy + 36.y² - 98.x - 264.y + 305 = 0

-Store 29 24 36 -98 -264 305 into registers R01 R02 R03 R04 R05 R06 respectively

and XEQ "CONIC" >>>> e = 0.7454 = R07 the conic is an ellipse ( F01 is set )
X<>Y p = 1.3333 = R08 and we have in registers R09 thru R32

   R09 = a = 3       R11 = 0.2            R12 = 3.6         the center is C(0.2;3.6)
   R10 = b = 2       R13 = 1.9889      R14 = 2.2584   focus F(1.9889;2.2584)
                             R15 = -1.5889    R16 = 4.9416   focus F'(-1.5889;4.9416)

R17 = 0.6 R18 = 0.8 R19 = 3 Axis (Ax): 0.6 x + 0.8 y = 3
R20 = 0.8 R21 = -0.6 R22 = -2 Axis (Ax'): 0.8 x - 0.6 y = -2 (Ax) and (Ax') are always perpendicular.

R23 = 2.0249 directrix (D): 0.8 x - 0.6 y = 2.0249
R24 = -6.0249 directrix (D'): 0.8 x - 0.6y = -6.0249 we always have: (Ax') // (D) // (D')

   R25 =   2.6     R26 = 1.8    vertex V(2.6;1.8)
   R27 = -2.2     R28 = 5.4    vertex V'(-2.2;5.4)
   R29 = 1.4     R30 = 5.2    vertex V"(1.4;5.2)
   R31 = -1      R32 = 2      vertex V'''(-1;2)

Example2: (C): 17.x² + 312.xy + 108.y² - 1130.x - 840.y + 2525 = 0

-Store these 6 coefficients into registers R01 to R06

and R/S >>>> e = 1.2019 = R07 the conic is a hyperbola ( F03 is set )
X<>Y p = 1.3333 = R08 and we have in registers R09 thru R34

   R09 = a = 3       R11 = 0.2            R12 = 3.6         the center is C(0.2;3.6)
   R10 = b = 2       R13 = 3.0844      R14 = 1.4367   focus F(3.0844;1.4367)
                             R15 = -2.6844    R16 = 5.7633   focus F'(-2.6844;5.7633)

R17 = 0.6 R18 = 0.8 R19 = 3 Axis (Ax): 0.6 x + 0.8 y = 3
R20 = 0.8 R21 = -0.6 R22 = -2 Axis (Ax'): 0.8 x - 0.6 y = -2 (Ax) and (Ax') are always perpendicular.

R23 = 0.4962 directrix (D): 0.8 x - 0.6 y = 0.4962
R24 = -4.4962 directrix (D'): 0.8 x - 0.6y = -4.4962 we always have: (Ax') // (D) // (D')

R25 = 2.6 R26 = 1.8 vertex V(2.6;1.8)
R27 = -2.2 R28 = 5.4 vertex V'(-2.2;5.4)

R29 = 0.0556 R30 = 1 R31 = 3.6111 Asymptote (As): 0.0556.x + y = 3.6111
R32 = 2.8333 R33 = 1 R34 = 4.1667 Asymptote (As'): 2.8333.x + y = 4.1667

Example3: (C): 9.x² + 24.xy + 16.y² - 150.x - 75.y + 75 = 0

-Store these 6 coefficients into registers R01 thru R06

and R/S >>>> e = 1 = R07 the conic is a parabola ( F02 is set )
X<>Y p = 1.5 = R08 and we have in registers R09 thru R18

R09 = 0.2 R10 = 3.6 vertex V(0.2;3.6) ( a parabola has no center )
R11 = 0.8 R12 = 3.15 focus F(0.8;3.15)

. R13 = 0.6 R14 = 0.8 R15 = 3 Axis (Ax): 0.6 x + 0.8 y = 3
R16 = 0.8 R17 = -0.6 R18 = -2 Directrix (D): 0.8 x - 0.6 y = -2.75

Notes:

-In all these examples, R00 = µ = Arc tan(-3/4) = -36.8699°
-If (C) is a sphere ( e = 0 ) this program yields 4 vertices but in fact, any point of the circle is a vertex.
Furthermore, in this case, the HP-41 produces y = + E99 and y = - E99 for the directrices:
(D) & (D') are regarded as "infinitely" far from the circle even if, strictly speaking, no directrix exists.
-If the conic is a hyperbola and C = 0 ( no term in y² ), the asymptotes are given by different formulas - lines 226 to 249 -
because one of them is parallel to the y-axis.

for instance, with 12.x² - 4.xy + 5.x + y + 3 = 0 we find in registers R29 thru R34

-3 1 2 (As): -3.x + y = 2
and 1 0 0.25 (As'): x = 1/4

-I let you add text lines at the end of the routine if you want more explicit outputs...