Appendix II: Calculator Routines

1Before the introduction of the ‘‘scientific’’ pocket calculator in 1972, spherical coordinates were quite impractical. The basic distance formula (for chord factors) required looking up six different trig functions, manipulating these unwieldy numbers, and taking a square root. Even if a slide rule gave sufficient accuracy (which it doesn’t, except for crude geodesic structures) the need to add or subtract would drive its user to pencil and paper at least three times in the course of computing one chord factor. The calculator gives multiplace accuracy in less than a minute.

2 Routines for handling our principal equations on the Hewlett-Packard HP-35 are given below. This is a comparatively simple machine with a 4-level stack but only one storage register, and one must sometimes manipulate the stack a good deal to avoid reentering data. The reader may find better procedures of his own. The routines can also be used on other Hewlett-Packard machines such as the HP-21 and HP-45, with the following modifications. (1) Remember to insert shift prefixes where appropriate. (2) Use the alternate routines provided for Equations 13.3,13.4 and 13.7, to accommodate a power function of the form $y^{x}$ instead of the HP-35’s $x^{y}$ . (3) On the HP-45 with its addressable storage registers, remember to supply an address for each STO. You may want to rewrite the routines using more than one register and avoiding all data reentry.

3 The HP-35 routines are read left to right. Each block is either the name of a piece of data or the name of a machine key (t means ‘‘enter’’). Thus the first seven items in the Equation 9.2 (chord factors of spheres) routine tell you to key the value of 9_r, press ‘‘enter,’’ press ‘‘cos,’’ key the value of 0₂, press ‘‘STO,’’ press ‘‘cos,’’ press ‘‘X.’’

4Chord Factors of Spheres (Equation 9.2)

5 011 cos 0₂ STO cos X $𝑥𝑦$ sin RCL sin X 0i t 0₂ - ⁰⁰⁸ X + 2 X

6 CHS t 2 + V

7Chord Factors of Nonspherical Structures (Equation 9.1)

8 $𝑂𝑖$ t cos 0₂ STO cos X xt*y sin RCL sin X 0j t 0₂ - cos X + 2 X

9 CHS STO $𝑟𝑖$ t t RCLX STO R| t X r₂ t t RCL X STO R| t X + RCL

10⁺ V

11Radius of an Ellipse (Equation 13.1)

12 E t X 11 0 STO sin t X X RCL cos t X +

13Radius of a Superellipse (Equation 13.3)

5.: t E x^y 111 2.5 t 0 cos STO arc cos sin x^y X 2.5 RCLx + *

5.: ^l!xx*+y x^y

14 Alternative form, for y^x machines:

15 E t 2.5 y* t t 0 STO sin 2.5 y* X RCL cos 2.5 y^x+2.5 ¹ /x y^x

16Radius of a Supercircle (Equation 13.4)

5.: 111 0 STO sin x^y RCL cos R| $𝑥𝑦$ Rl Rl R| x^y+ ’/x x±;yx^y

17 Alternative form, for y* machines:

18 0 STO sin 2.5 y* RCL cos 2.5 y* + ^l!x 2.5 ^l/x y^x

19Radius When Two System Radii Are Unequal (Equation 13.7)

20 $n$ t 11 0 STO cos RCL sin E_x X Rl R| R| xy x^y + 0 STO sin

21 Rl Rl R| $𝑥𝑦$ E₂ X x^y X $n$ RCL cos E_x X x^y+ ’(x STO $n$ 11 E_x

22 t E₂ X x^y RCL X $𝑥𝑦$

23¹/x x*>y x^y

24 Alternative form, for y^x machines:

25 0 STO cos n y^X RCL sin E_x X n y* + 0 STO sin E₂ X n / X RCL

26 cos E_x STO X n y^x + E₂ RCL X n y^xx±>y n ⁱ/x

27Rotation Formulae (Equation 14.1, Equation 14.2)

28 0 t cos 0x STO cos X x*±y sin RCL sin X 0 t 0x - cos X + arc cos

29 STO

30
31(Write down value of 0₂ and continue:)

32 sin 0 t - sin X RCL sin arc sin

33
34(Value of 0₂ is now on display.)

35Face Angles (Equation 17.4, Equation 17.5, Equation 17.6)

36 at t CHS X ft STO t X + $c$ 11 X R| R4- Rl +x+>y + 2 -RCLarc

37 cos STO

38
39(Write down value of Face Angle $A$ and continue:)

40 sin $b$ X $a$ + arc sin

41
42(Write down value of Face Angle $B$ and continue:)

43 RCL+ ISOxy-

44
45(Value of Face Angle Cis now on display.)

46Dihedral Angles of Spheres (Equation 17.7)

47 B cos c X STO t XRCLa XSTO - 1 +/ 2 X cXB sin X ’/xc t X

48 RCL - X arc cos