The Museum of HP Calculators

Quadrics 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 Quadric ( or quadratic surface ) is defined by its cartesian equation: a.x² + b.y² + c.z² + d.x.y + e.x.z + f.y.z + g.x + h.y + i.z = j
where at least one of the 6 coefficients a , b , c , d , e , f is different from zero.
-After a suitable transformation of coordinates, this equation is reduced to one of the 17 standard forms:

1°)    x²/a² + y²/b² + z²/c² = 1    Ellipsoid
2°)    x²/a² + y²/b² - z²/c² = 1    Hyperboloid of one sheet
3°)    x²/a² - y²/b² - z²/c² = 1     Hyperboloid of two sheets
4°)    x²/a² + y²/b² = z              Elliptic Paraboloid
5°)    x²/a² - y²/b² = z               Hyperbolic Paraboloid

6°)    x²/a² + y²/b² - z²/c² = 0    Cone
7°)    x²/a² + y²/b² = 1              Elliptic Cylinder
8°)    x²/a² - y²/b² = 1               Hyperbolic Cylinder
9°)    x²/a² = y                          Parabolic Cylinder
10°) -x²/a² - y²/b² - z²/c² = 1     Imaginary Ellipsoid
11°) x²/a² + y²/b² + z²/c² = 0    One Point ( more precisely an Imaginary Cone )
12°) -x²/a² - y²/b² = 1               Imaginary Elliptic Cylinder
13°) x²/a² = 0                            One Plane
14°) x²/a² = 1                            Two Parallel Planes
15°) -x²/a² = 1                          Two Imaginary Parallel Planes
16°) x²/a² + y²/b² = 0              One Straight-Line ( more precisely Two Imaginary Intersecting Planes )
17°) x²/a² - y²/b² = 0               Two Intersecting Planes

-The following program uses the Jacobi's iterating method to determine these reduced equations and the type of the quadric.

Data Registers: R00: unused ( Registers R01 thru R10 are to be initialized before executing "QUADRIC" )

• R01 = a • R02 = b • R03 = c • R04 = d • R05 = e
• R06 = f • R07 = g • R08 = h • R09 = i • R10 = j

When the program stops, the coefficients of the reduced form are in R01 thru R10
Flags: /
Subroutines: /

01 LBL "QUADRIC"
02 LBL 01
03 VIEW 04
04 RCL 01
05 RCL 04
06 RCL 01
07 RCL 02
08 ST+ T
09 -
10 R-P
11 RDN
12 X<> 04
13 LASTX
14 R-P
15 X<>Y
16 RDN
17 ST- Z
18 +
19 2
20 ST/ 04
21 ST/ Z
22 /
23 STO 01
24 X<>Y
25 STO 02
26 RCL 04
27 RCL 07
28 P-R
29 RCL 04
30 RCL 08
31 P-R
32 R^
33 -
34 STO 08
35 RDN
36 +
37 STO 07
38 RCL 04
39 RCL 05
40 P-R
41 RCL 04
42 RCL 06
43 P-R
44 R^
45 -
46 STO 06
47 RDN
48 +
49 STO 05
50 R^
51 R-P
52   E-8
53 X>Y?
54 GTO 03
55 LASTX
56 ABS
57 RCL 05
58 ABS
59 X>Y?
60 GTO 02
61 RCL 01
62 X<> 03
63 STO 01
64 CLX
65 X<> 06
66 STO 04
67 RCL 07
68 X<> 09
69 STO 07
70 GTO 01
71 LBL 02
72 RCL 02
73 X<> 03
74 STO 02
75 CLX
76 X<> 05
77 STO 04
78 RCL 08
79 X<> 09
80 STO 08
81 GTO 01
82 LBL 03
83 CLX
84 STO 04
85 XEQ 05
86 RCL 01
87 X#0?
88 GTO 02
89 X<> 02
90 STO 01
91 RCL 07
92 X<> 08
93 STO 07
94 LBL 02
95 RCL 01
96 X#0?
97 GTO 02
98 X<> 03
99 STO 01
100 RCL 07
101 X<> 09
102 STO 07
103 LBL 02
104 RCL 02
105 X#0?
106 GTO 02
107 X<> 03
108 STO 02
109 RCL 08
110 X<> 09
111 STO 08
112 LBL 02
113 RCL 07
114 X^2
115 RCL 01
116 /
117 RCL 08
118 X^2
119 RCL 02
120 X=0?
121 GTO 04                    ( it should be a three-byte GTO )
122 /
123 +
124 RCL 09
125 X^2
126 RCL 03
127 X=0?
128 GTO 03
129 /
130 +
131 4
132 /
133 ST+ 10
134 CLX
135 STO 07
136 STO 08
137 STO 09
138 XEQ 05
139 RCL 10
140 X=0?
141 GTO 02
142 ST/ 01
143 ST/ 02
144 ST/ 03
145 ST/ 10
146 LBL 02
147 RCL 01
148 SIGN
149 RCL 02
150 SIGN
151 +
152 RCL 03
153 SIGN
154 +
155 RCL 10
156 X=0?
157 GTO 02
158 " IM ELLIPS"          Imaginary Ellipsoid
159 SIGN
160 X=Y?
161 " 1-SH HYP"           One-sheeted Hyperboloid
162 X<Y?
163 " ELLIPSOID"
164 CHS
165 X=Y?
166 " 2-SH HYP"           Two-sheeted Hyperboloid
167 PROMPT
168 LBL 02
169 SIGN
170 X<>Y
171 ABS
172 " POINT"                ( More precisely: Imaginary Cone with only one real point )
173 X=Y?
174 " CONE"
175 PROMPT
176 LBL 03
177 X<> Z
178 4
179 /
180 ST+ 10
181 CLX
182 STO 07
183 STO 08
184 XEQ 05
185 RCL 01
186 RCL 02
187 *
188 RCL 09
189 X=0?
190 GTO 02
191 ST/ 01
192 ST/ 02
193 ST/ 09
194 CLX
195 STO 10
196 X<>Y
197 " ELL PARAB"       Elliptic Paraboloid
198 X<0?
199 " HYP PARAB"      Hyperbolic Paraboloid
200 PROMPT
201 LBL 02
202 CLX
203 RCL 10
204 X=0?
205 GTO 02
206 ST/ 01
207 ST/ 02
208 ST/ 10
209 X<>Y
210 " HYP CYL"           Hyperbolic Cylinder
211 X<0?
212 PROMPT
213 RCL 01
214 " ELL CYL"            Elliptic Cylinder
215 X<0?
216 " IM ELL CYL"      Imaginary Elliptic Cylinder
217 PROMPT
218 LBL 02
219 X<>Y
220 " STR LINE"           One Straight-Line ( In fact, 2 intersecting imaginary planes )
221 X<0?
222 " 2 PLANES"          Two intersecting planes
223 PROMPT
224 LBL 04
225 X<> Z
226 4
227 /
228 ST+ 10
229 CLX
230 STO 07
231 XEQ 05
232 RCL 08
233 RCL 09
234 R-P
235 X=0?
236 GTO 02
237 RCL 01
238 ST/ 01
239 /
240 STO 08
241 CLX
242 STO 09
243 STO 10
244 " PARAB CYL"        Parabolic Cylinder
245 PROMPT
246 LBL 02
247 " PLANE"                 One Plane
248 RCL 10
249 X=0?
250 PROMPT
251 ST/ 01
252 ST/ 10
253 " 2 IM PLANES"      Two imaginary parallel Planes
254 RCL 01
255 X>0?
256 " 2//PLANES"           Two parallel Planes
257 PROMPT
258 LBL 05
259 10                              or 1 10^X ( thus saving one byte )
260   E-7                           or another "small" number
261 LBL 06
262 RCL IND Y
263 ABS
264 X<Y?
265 CLX
266 X=0?
267 STO IND Z
268 RDN
269 DSE Y
270 GTO 06
271 END

( 503 bytes / SIZE 011 )

Example1: (Q): 2xy + 3xz + 4yz = 7

0 STO 01 STO 02 STO 03 2 STO 04 3 STO 05 4 STO 06 0 STO 07 STO 08 STO 09 7 STO 10
XEQ "QUADRIC" the HP-41 displays the successive contents of register R04 ( which tend to zero )

and eventually displays " 2-SH HYP" (Q) is an hyperboloid of 2 sheets

-Recalling the new coefficients in registers R01 thru R10 we find: 0.433905256 x² - 0.299134471 y² - 0.134770785 z² = 1
which may be re-written: x²/a² - y²/b² - z²/c² = 1 with a = 1.518107565 , b = 1.828381291 , c = 2.723968753

-The signs may be + - - ; - + - or - - + for a 2-sheeted hyperboloid.

Example2: (Q): 2x² + y² + 2z² - 3xy + 4xz -3yz + 7x + y + 5z = 10

2 STO 01 1 STO 02 ................ 10 STO 10 XEQ "QUADRIC" >>>> " HYP PARAB" (Q) is an hyperbolic Paraboloid

and the new coefficients in R01 to R10 yield: -3.604884263 x² + 0.069350354 y² + z = 0
which may be re-written: x²/a² - y²/b² = z with a = 0.526689108 , b = 3.797306556

Remarks:

-The numbers displayed line 04 may seem to increase in the beginning, but they soon tend to zero.
-Of course, execution time depends on the coefficients, but it's usually less than 1 minute.
-When the program stops, R04 = R05 = R06 = 0
-There will be a DATA ERROR line 116 if a = b = c = d = e = f = 0 ( the equation reduces to g.x + h.y + i.z = j )
-If you want to take this case into account,
add   RTN LBL 07 RCL 07 RCL 08 R-P RCL 09 R-P " PLANE" X#0? PROMPT " EMPTY SET" RCL 10 X=0? " SPACE" PROMPT after line 270
and add   X=0? GTO 07 after line 115
-Practically, lines 158-216-253 may replaced with " EMPTY SET"
-Many bytes can be saved if you use more abbreviated alpha strings.
-Register R00 may be used instead of R10 but in this case, replace line 269 by ISG Y and line 259 by 9   CHS

-This program uses 10^-7 ( line 260 ) to identify and delete tiny elements.
-Change this value if, for instance, the coefficients are of the order of 10⁶ ( or divide them by 1 million )

Go back to the HP-41 software library
Go back to the general software library
Go back to the main exhibit hall