hp41programs

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: /

-Line 121 is a three-byte  GTO 04
 
 

 
   ( 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 the following instructions after line 270
  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
 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 lines 259-260  by  9   CHS

-This program uses 10^-7 ( line 261 ) 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 )