The Museum of HP Calculators

Conics for the HP-41

This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.

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 )        with a >= b            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 ------------- a parabola
F03 ------------- a hyperbola

Subroutines: /

Program Listing

01 LBL "CONIC"
02 CF 01                          If you have an HP-41CX, lines 02-03-04 may be replaced by    CLX   X<> F
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            ( a three-byte GTO )
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?                      Lines 93 to 100 are only useful to produce a DATA ERROR line 96 if the conic is imaginary ( like x²+4y²+9=0 )
94 X<>Y                     Otherwise, these lines may be deleted.
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          ( a three-byte GTO )
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            ( a three-byte GTO )
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...

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