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O. Box 402 St. Clair Shores, MI 48080-0402 (313) 343-0073 (Voice) CIS 76505,125 DRCS-VeCalc i TABLE OF CONTENTS COPYRIGHT NOTICE....................................... ii DISCLAIMER............................................. ii LICENSE AGREEMENT..................................... iii REGISTRATION AND ORDERING INFORMATION.................. vi ORDER FORM............................................. v PREFACE TO VERSION 1.11............................... vi 1. INTRODUCTION....................................... 1 1.1 VeCalc Description........................... 1 1.2 System Requirements.......................... 1 1.3 VeCalc Files................................. 1 2. STARTING VECALC.................................... 2 2.1 Starting a VeCalc Session.................... 2 3. VECALC USES........................................ 4 3.1 Three dimensional analysis................... 4 3.2 Graphic uses................................. 10 7. APPENDIX A -Product Support........................ 11 8. APPENDIX B -Acknowledgements....................... 12 DRCS-VeCalc ii Copyright DRCS-VeCalc is Copyrighted (c) 1989 by Dan Rubis. This document is Copyright (c) 1989 by Dan Rubis Creative Systems. No part of DRCS-VeCalc or this document may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise except as in accordance with the Licensing Agreement below, or without the prior written consent of Dan Rubis C.S. Disclaimer DRCS-VeCalc and all accompanying written materials (including instructions for use) are supplied "AS IS" without warranty of any kind. Dan Rubis C.S. does not warrant, guarantee or make any representations regarding the use, or the results of use, of DRCS-VeCalc or the written materials in terms of correctness, accuracy, reliability, currentness, or otherwise. Dan Rubis C.S. will not be responsible for any direct, indirect, consequential, or incidental damages (including damages for loss of business profits, business interruption, loss of business information, and the like) arising out of the use or inability to use DRCS-VeCalc even if Dan Rubis C.S. has been advised of the possibility of such damages. DRCS-VeCalc iii License Agreement DRCS-VeCalc is Copyrighted (c) 1989 by Dan Rubis C.S.. DRCS-VeCalc is not, and has never been public domain or free software. DRCS-VeCalc is distributed under the Shareware concept. Non-registered users of DRCS-VeCalc are granted a limited license to use DRCS-VeCalc for a trial period in order to determine whether DRCS-VeCalc suits their needs. Under no circumstances can an un-registered version of DRCS-VeCalc be used by a business, organization, or institution of any kind. All users are granted a limited license to distribute shareware versions of DRCS-VeCalc subject to the following restrictions: - DRCS-VeCalc is distributed in unmodified form, along with all program, documentation, and any other files present in the original package. - DRCS-VeCalc is not bundled with any other product. - No distribution fee is charged. Registered users may receive upgrades from Dan Rubis C.S. from time to time. Registered copies are NEITHER free software nor shareware. Users are granted unlimited use of registered copies of DRCS-VeCalc subject to the following restrictions: - DRCS-VeCalc may be installed on more than one computer at a time, so long as only ONE version is run at one time. - Users are not free to copy, modify, or redistribute registered copies of DRCS-VeCalc IN ANY WAY whatsoever. Bulletin Board Services are free to post DRCS-VeCalc for downloading to prospective users, providing that no fee is charged for such downloading, aside from possible connect charges. Any other groups, companies, or organizations who wish to distribute DRCS-VeCalc must obtain prior written approval from Dan Rubis C.S.. DRCS-VeCalc iv Registration and Ordering Information Registering DRCS-VeCalc will entitle you to continued use of DRCS-VeCalc beyond the trial period. Furthermore, it will entitle you to receive, free of charge, the latest upgrade to DRCS-VeCalc, as well as technical support. To register your copy of DRCS-VeCalc, fill out the order form on the following page, and send it, along with a check or money order for $10 Dan Rubis Creative Systems 21719 Harper Ave. P. O. Box 402 St. Clair Shores, MI 48080-0402 21719 Harper Ave. Distribution is not permitted outside of North America. Corporate users and other institutions should inquire at the above address regarding charges for site licensing of DRCS-VeCalc. Note that continued use of an unregistered copy is a violation of the licensing agreement. Support Shareware- register your copy today! DRCS-VeCalc v Order Form DRCS-VeCalc version 1.11 ------------------------------------------------------------ DRCS-VeCalc 1.11 registration ____ @ $10 U.S. ____ NAME_______________________________________________________ Company____________________________________________________ ADDRESS____________________________________________________ ____________________________________________________ ____________________________________________________ PHONE ____________________ Business ______________________ Where did you obtain DRCS-VeCalc? ______________________________________________________ Comments/Suggestion? ______________________________________________________ ______________________________________________________ ______________________________________________________ ______________________________________________________ DRCS-VeCalc vi PREFACE TO VERSION 1.11 DRCS-VeCalc was invented out of need for doing a quick vector calculation while working on engineering and computer graphics problems. The vectors calculated by DRCS-VeCalc are three dimensional. To perform a simple cross-product of two vectors, six multiplications and three subtractions are required. Using a hand calculator for a calculation like this is tedious and error prone. This program has been found to be a very handy tool for small calculations and "what if situations". Take for instance the angle between two vectors. A two dimension problem can be easily calculated using trigonometry, but a three dimensional space problem is a an exercise beyond comparison. A memory resident version (TSR) would be even more useful, because it could be popped-up over any program that needed a quick vector calculation. A TSR version is being developed and will be available to registered DRCS-VeCalc users for free when available. DRCS-VeCalc 1 1. INTRODUCTION 1.1 DRCS-VeCalc Description DRCS-VeCalc is a vector calculator. It is like a four function calculator for vectors. What is a vector? The use of the word vector in this context is to mean a single dimension array with three elements of floating point numbers. Vectors are used to represent a quantity that has both magnitude and direction. Two of the most common applications are in engineering to represent forces and accelerations, and in geometric modeling and computer graphics. 1.2 System Requirements DRCS-VeCalc 1.11 requires an 8080, 80286 or 80386 based computer running IBM or Microsoft DOS 3.x. A color monitor is desirable. 1.3 DRCS-VeCalc Files The DRCS-VeCalc distribution diskette contains the following files: - VECALC.DOC -program documentation - VECALC.EXE -executable code for DRCS-VeCalc - README.1ST -latest changes if any DRCS-VeCalc 2 2. STARTING VECALC 2.1 Starting the VeCalc Session Starting DRCS-VeCalc is as simple as typing "VECALC" at the DOS command prompt. When DRCS-VeCalc is run for the first time, a file is created to hold the values for vectors A, B, X, and S. There after, the program reads this file for default values for the vectors. In effect, old values are "remembered". - VEC.DAT -vector data You will be presented with screen with a window of vector calculation selections. Choose one by moving the highlight up or down with the arrow keys. Select function by pressing the enter key. At the top of the next the screen the title of the function you are using will be displayed: ■ Quit ■ Vector addition calculation ■ Vector subtraction calculation ■ Vector dot product calculation ■ Vector cross product calculation ■ Unit vector and magnitude calculation ■ Angle between two vectors in radians calculation ■ Three vector swapping functions ■ Stack vector roll down function ╔════════════════════════════════════╗ ║ The Ctrl-Break keyboard function ║ ║ is disabled. Select Quit from ║ ║ menu to end program. ║ ╚════════════════════════════════════╝ and an output window showing the previous values of vectors A, B, X, and S will be display to the right. To the left of the ? mark is the OLD value and to the right will be a blinking cursor waiting for a NEW value or a "return" to accept the OLD value as the new input. Some of the functions produce output that is a vector itself, and some produce a scalar. The vector outputs are stored in Vector X and scalar values in Stack S[0]. After each execution of a function that produces a scalar as output, it will be save in stack position S[0]. Therefore, unless you Roll Down the stack, the old value will be lost. With successive combinations of Roll Downs of the Stack and scalar producing vector functions, one can build up a required vector in the stack for a subsequent vector function. DRCS-VeCalc 3 For instance, let us say want to build a vector containing the magnitudes from three dot-products and then cross that with the unit vector of a known vector that you may want to enter in by hand. You would stack your three dot products in the stack, making sure that you rolled it down each time and the you would select the cross product of two vectors, enter your unit vector and you answer would be calculated and stored in vector X. A mixed-triple product, which gives you the moment of a force about an axis can also be calculated using DRCS-VeCalc. The procedure for that is left to the user. See the VeCalc uses Section 3. DRCS-VeCalc 4 3. VECALC USES ENGINEERING 3.1 3-D Analysis. General Equations and Principles. 1) Scalar product. The scalar product of two vectors P and Q is defined as the product of the magnitudes of P and Q and of the cosine of the angle φ formed by P an Q (Fig. 3.19). The scalar Q │ │ │ ── φ │ │ └────────────P Figure 3.19 product of P and Q is denoted by P∙Q. Therefore P∙Q = PQ cos φ Note that this expression is not a vector, but a scalar, which explains the name scalar product; because of the notation used, P∙Q is also referred to as the dot product of the vectors P and Q. Some important applications for the scalar product are finding the angle between two vectors and projecting a vector on an axis. In the particular case when the vector selected along OL is the unit vector i (Fig. 3.23) y │ L │ ∙ │ Q P │ ∙ ∙ │ ∙ φ ∙ │ ∙ ∙ │ i ∙ │∙∙ O└───────────────────x ∙ Figure 3.23 ∙ ∙ ∙ z we write POL = PQ cos φ or POL = P ∙ i DRCS-VeCalc 5 2) Vector product (Moment of a force about a point). The vector product of two vectors P and Q is defined as the vector V which satisfies the following conditions: 1. The line of action of V is perpendicular to the plane containing P and Q (Fig. 3.6). V = P X Q (where X means cross) │ │ Q │ ∙ │ ∙ │ ∙ │ ∙ │∙ ∙ Θ ∙ Figure 3.6 ∙ ∙ ∙ ∙ P 2. The magnitude of V is the product of the magnitudes of P and Q and of the sine of the angle theta formed by P and Q (the measure of which always be 180 degrees or less); we thus have V = PQ sin theta (3.1) 3. The sense of V is such that a man located at the tip of V will observe as counterclockwise the the rotation through theta which brings the vector P in line with the vector Q; note that if P and Q do not have a common point of application, they should first be redrawn from the same point. the three vectors P, Q, and V -- taken in that order--are said to form a right-handed triad. As stated above, the vector V satisfying these three conditions (which define it uniquely) is referred to as the vector product of P and Q; it is represented by the mathematical expression V = P X Q Because of the "X" notation used, the vector product of two vectors P and Q; it is also referred to as the cross product of P and Q. It follows from Eq.(3.1) that, when two vectors P and Q have either the same direction or opposite directions, their vector product is zero. The vector product V can be expressed in the following form: |i j k | V = |Px Py Pz| |Qx Qy Qz| where i, j, and k are unit vectors. DRCS-VeCalc 6 Moment of a Force about a Point. Consider a force F acting on a rigid body (Fig. 3.12). Mo │ │ │ F │ ∙ │ ∙ │ ∙ │ ∙ │ r ∙ φ o└─────────+────────── Figure 3.12 ∙ A d ∙ ∙ ∙ The force F is represented by a vector which defines its magnitude and direction. However, the effect of the force on the rigid body depends also upon its point of application A. The position of A may be conveniently defined by the vector r which joins the fixed reference point o with A; this vector is known as the position vector of A. The position vector r and the force F define the plane shown in Fig. 3.12. The moment of F about o is defined as the vector product of r and F: Mo = r X F According to the definition of the vector product given, the moment Mo must be perpendicular to the plane containing o and F; thus, the line of action of Mo represents the axis about which the body tends to rotate when attached at o and subjected to the force F. The sense of Mo is defined by the sense of the rotation which would bring the vector r in line with the vector F; but this rotation is the rotation that F tends to impart to the body. Thus the sense of the moment Mo characterizes the sense of the rotation that F tends to impart to the rigid body; this rotation will be observed as counterclockwise by an observer located at the tip of Mo. Another way of stating the relationship existing between the sense of Mo and the sense of rotation of the rigid body is furnished by the right-hand rule. Finally, denoting by φ the angle between the lines of action of the position vector r and the force F, we find that the magnitude of the moment of F about o is Mo = rF sin φ = Fd where d represents the perpendicular distance from o to the line of action of F. DRCS-VeCalc 7 3) Mixed triple product of three vectors. The mixed triple product of the three vectors S, P, and Q is S ∙ (P X Q) obtained by forming the scalar product of S with the vector product of P and Q. This expression may be written in a more compact form if we observe that it represents the expansion of a determinant: | Sx Sy Sz | S ∙ (P X Q) = | Px Py Pz | | Qx Qy Qz | P X Q │ S │ ∙ │ ∙ Q │ ∙ ∙ │ ∙ ∙ │ ∙ ∙ │ ∙ ∙ │∙∙ └───────────────────P Figure 3.24 The mixed triple product is thus equal, in absolute value, to the volume of the parallelepiped having the vectors S, P, and Q for sides. S ∙ ∙ Q ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙∙ ────────────────────P Figure 3.25 Moment of a Force about a Given Axis. Consider again a force F acting on a rigid body and the moment Mo of that force about o (Fig. 3.27). DRCS-VeCalc 8 y L │ ∙ MO │ ∙ ∙ │ ∙ F ∙ │ C ∙ ∙ │ ∙ ∙ ∙ │ ∙ ∙ ∙ │ i ∙ ∙ │ ∙ ∙ φ O└─────r───+─────────────x Figure 3.27 ∙ A ∙ ∙ ∙ ∙ z Let OL be an axis through O; we define the moment Mol of F about OL as the projection OC of the moment Mo on the axis OL. Denoting by i the unit vector along OL, and recalling the expressions for obtaining the projection of a vector on a given axis and for the moment Mo of a force F, we write Mol = i ∙ (r X F) which shows that the moment Mol of F about the axis OL is the scalar obtained by forming the mixed triple product of i, r, and F. Expressing Mol in the form of a determinant: | ix iy iz | Mol = | x y z | | Fx Fy Fz | where ix, iy, iz = direction cosines of axis OL x, y, z = coordinates of point of application of F Fx, Fy, Fz = components of force F the moment Mol of F about OL measures the tendency of the force F to impart to the rigid body a motion of rotation about the fixed axis OL. More generally, the moment of a force F applied at A about axis which does not pass through the origin is obtained by choosing an arbitrary point B on the axis (Fig.3.29) and determining the projection on the axis BL of the moment Mb of F about B. DRCS-VeCalc 9 L y F ∙│ ∙ │∙ ∙ │ ∙ δ ∙ │ B∙∙∙∙∙∙∙∙∙∙∙∙∙A │ ∙ ∙ │ ∙ ∙ │ C∙ │ ∙ O└──────────────+─────────────x Figure 3.29 ∙ ∙ ∙ ∙ ∙ z Mbl = i ∙ Mb = i ∙ (δr X F) where δr = ra - rb represents the vector joining B and A. Expressing Mbl in the form of a determinant,: | ix iy iz | Mbl = | δx δy δz | | Fx Fy Fz | where ix, iy, iz = direction cosines of axis BL δx = xa -xb, δy = ya - yb, δz = za - zb Fx, Fy, Fz = components of force F DRCS-VeCalc 10 3.2 Graphics Uses. As mentioned before in the previous section, VeCalc can be used to calculate the angle between two vectors in three dimensional space. By calculation the sines and cosines of a desired angle of rotation, like for instance the rotation of a graphic image on the screen, one can transform a matrix of image points to be plotted on the screen. Admittedly, VeCalc cannot make this calculation by itself. It is a useful device for checking some of your work as you are developing you graphics plotting function. A three by three matrix of direction cosines of the direction vector will be all that is needed to calculate the need coordinates of the rotated image. If the is enough interest in DRCS-VeCalc, and its uses in graphic calculations, another function can be added called matrix times a vector, which would enable the direct transformation of a coordinate in rotation, translation, and shearing. By adding just one more function called scalar multiplication of a vector, one can zoom coordinates, and stretch them. Please let me know if you are interested in this kind of application. Unless you take time to contact me, even if it is by a post card, I will not know. DRCS-VeCalc 11 7. Appendix A - Product Support If any problems should come up that are not covered in the manuals, registered users may receive help from Dan Rubis C.S.'s Technical Support Division. When sending in a report of a problem, please make sure to include the following information: Computer make:_____________________________________________ Computer peripherals:______________________________________ _________________________________________________ Operating System:__________________________________________ Contents of CONFIG.SYS:____________________________________ Nature of problem:_________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ Is problem reproducible?:__________________________________ Other programs running:___________________________________ Registered DRCS-VeCalc users who have problems with DRCS-VeCalc can receive technical help by sending E-mail concerning the problem to Dan Rubis on CompuServe (user ID: 76505,125). Alternatively, you may write to Dan Rubis C.S. at the following address: Dan Rubis Creative Systems ATTN: Technical Support 21719 Harper Ave. P. O. Box 402 St. Clair Shores, MI 48080-0402 Finally, registered users may phone the following number: (313) 343-0073 (Voice) DRCS-VeCalc 12 8. Appendix B - Acknowledgements Dan Rubis Creative System would like to thank the unknown creator of the Detroit Skyline on the cover page of this manual. Please let me know who you are so I can give you due credit. Thanks to Randy for editing this manual and all of the suggestions and comments on version 1.0. Many of the enhancements seen in version 1.11 were the direct result of user comments. Please continue to voice and opinions or suggestions that would increase the usefulness of this program to you.