Interval Angles and the Fortran ATAN2 Intrinsic Function G. William Walster Sun Microsystems, Inc. 901SanAntonioRoad Palo Alto, CA 94303 1 (800) 786.7638 1.512.434.1511 Copyright 2002 Sun Microsystems, Inc., 901 San Antonio Road, Palo Alto, California 94303 U.S.A. All rights reserved. This product or document is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation. Nopartofthis product or document may be reproduced in any form by any means without prior written authorization of Sun and its licensors, if any. Third-party software, including font technology, is copyrighted and licensed from Sun suppliers. Parts of the product may be derived from Berkeley BSD systems, licensed from the University of California. UNIX is a registered trademark in the U.S. and other countries, exclusively licensed through X/Open Company, Ltd. Sun, Sun Microsystems, the Sun logo, EJB, EmbeddedJava, Enterprise JavaBeans, Forte, iPlanet, Java, JavaBeans, Java Blend, JavaServer Pages, JDBC, JDK, J2EE, J2SE, and Solaris trademarks, registered trademarks, or service marks of Sun Microsystems, Inc. in the U.S. and other countries. All SPARC trademarks are used under license and are trademarks or registered trademarks of SPARC International, Inc. in the U.S. and other countries. Products bearing SPARC trademarks are based upon an architecture developed by Sun Microsystems, Inc. TheOPENLOOKandSunTM Graphical User Interface was developed by Sun Microsystems, Inc. for its users and licensees. Sun acknowledges the pioneering efforts of Xerox in researching and developing the concept of visual or graphical user interfaces for the computer industry. Sun holds a non-exclusive license from Xerox to the Xerox Graphical User Interface, which license also covers Sun’s licensees who implement OPEN LOOK GUIs and otherwise comply with Sun’s written license agreements. RESTRICTED RIGHTS: Use, duplication, or disclosure by the U.S. Government is subject to restrictions of FAR 52.227-14(g)(2)(6/87) and FAR 52.227- 19(6/87), or DFAR 252.227-7015(b)(6/95) and DFAR 227.7202-3(a). DOCUMENTATION IS PROVIDED “AS IS” AND ALL EXPRESS OR IMPLIED CONDITIONS, REPRESENTATIONS AND WARRANTIES, IN- CLUDING ANY IMPLIED WARRANTY OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE OR NON-INFRINGEMENT, ARE DISCLAIMED, EXCEPT TO THE EXTENT THAT SUCH DISCLAIMERS ARE HELD TO BE LEGALLY INVALID. Copyright 2002 Sun Microsystems, Inc., 901 San Antonio Road, Palo Alto, Californie 94303 Etats-Unis. Tous droits réservés. Ce produit ou document est protégé par un copyright et distribué avec des licences qui en restreignent l’utilisation, la copie, la distribution, et la décompilation. Aucune partie de ce produit ou document ne peut être reproduite sous aucune forme, par quelque moyen que ce soit, sans l’autorisation préalable et écrite de Sun et de ses bailleurs de licence, s’il y en a. Le logiciel détenu par des tiers, et qui comprend la technologie relative aux polices de caractères, est protégé par un copyright et licencié par des fournisseurs de Sun. Des parties de ce produit pourront être dérivées des systèmes Berkeley BSD licenciés par l’Université de Californie. UNIX est une marque déposée aux Etats-Unis et dans d’autres pays et licenciée exclusivement par X/Open Company, Ltd. Sun, Sun Microsystems, le logo Sun, EJB, EmbeddedJava, Enterprise JavaBeans, Forte, iPlanet, Java, JavaBeans, Java Blend, JavaServer Pages, JDBC, JDK, J2EE, J2SE, et Solaris sont des marques de fabrique ou des marques déposées, ou marques de service, de Sun Microsystems, Inc. aux Etats-Unis et dans d’autres pays. Toutes les marques SPARC sont utilisées sous licence et sont des marques de fabrique ou des marques déposées de SPARC International, Inc. aux Etats-Unis et dans d’autres pays. Les produits portant les marques SPARC sont basés sur une architecture développée par Sun Microsystems, Inc. L’interface d’utilisation graphique OPEN LOOK et SunTM a été développée par Sun Microsystems, Inc. pour ses utilisateurs et licenciés. Sun reconnaît les efforts de pionniers de Xerox pour la recherche et le développement du concept des interfaces d’utilisation visuelle ou graphique pour l’industrie de l’informatique. Sun détient une licence non exclusive de Xerox sur l’interface d’utilisation graphique Xerox, cette licence couvrant également les licenciés de Sun qui mettent en place l’interface d’utilisation graphique OPEN LOOK et qui en outre se conforment aux licences écrites de Sun. CETTE PUBLICATION EST FOURNIE “EN L’ETAT” ET AUCUNE GARANTIE, EXPRESSE OU IMPLICITE, N’EST ACCORDEE, Y COMPRIS DES GARANTIES CONCERNANT LA VALEUR MARCHANDE, L’APTITUDE DE LA PUBLICATION A REPONDRE A UNE UTILISATION PARTIC- ULIERE, OU LE FAIT QU’ELLE NE SOIT PAS CONTREFAISANTE DE PRODUIT DE TIERS. CE DENI DE GARANTIE NE S’APPLIQUERAIT PAS, DANS LA MESURE OU IL SERAIT TENU JURIDIQUEMENT NUL ET NON AVENU. Contents Introduction.................................................................................... 1 Point Angles................................................................................... 1 Interval angles ................................................................................ 6 Conclusion ..................................................................................... 11 Appendix A. References....................................................................... 12 Interval Angles and the Fortran ATAN2 Intrinsic Function Introduction From James and James [?]: “There are two commonly used signed measures of directed angles. If a circle is drawn with unit radius and center at the vertex of a directed angle, then a radian measure of the angle is the length of an arc that extends counterclockwise along the circle from the initial side to the terminal side of the angle, or the negative of the length of an arc that extends clockwise along the circle from the initial side to the terminal side. The arc may wrap around the circle any number 1 π, 1 π + π, of times. For example, if an angle has radian measure 2 it also has radian measure 2 2 1 π + π, 1 π − π, 1π − π, 2 4 etc., or 2 2 2 4 etc. A rotation angle consists of a directed angle and a signed measure of the angle. The angle is a positive angle or a negative angle according as the measure is positive or negative. Equal rotation angles are rotation angles that have the same measure. Usually, angle means rotation angle. A rotation angle can be thought of as being a directed angle together with a description of how the angle is formed by rotating a ray from an initial position (on the initial side) to a terminal position (on the terminal side).” 1 Point Angles Before considering conventions needed to deal with interval angles, it is helpful to review the conventions commonly used for points. Radian measures of rotation and directed angles Radian measures of rotation angles are real numbers and can be mathematically manipulated in any meaningful way. The directed angle corresponding to a given rotation angle does not contain the number of full rotations a ray takes between its initial and the terminal position. A directed angle may or may not retain infor- mation about the direction of rotation of the ray used to create its corresponding rotation angle. Respectively, let ω and θ denote the radian measures of a rotation angle and a possible corresponding directed angle. Then ω and θ are related as follows: ω = θ + 2kπ, where k is an integer function of ω. If the direction of rotation of the ray used to define ω is preserved in θ, ω and θ can be related as follows: ω θ = ω − trunc 2π, (1) 2π in which case ω k = trunc , (2) 2π where x , if x ≥ 0, and trunc (x) = (3) x , if x < 0. 2 θ 10π ω -10π 10π -10π θ = ω − ω π Figure 1: trunc 2π 2 If the direction of rotation is not preserved in θ, ω and θ can be related in a variety of ways, including the following three: 1. ω θ = ω mod 2π = ω − 2π, (4a) 2π in which case ω k = , or (4b) 2π θ 10π ω -10π 10π π -10 θ = ω − ω π Figure 2: 2π 2 Interval Angles and the FortranATAN2 Intrinsic Function 3 2. ω θ =−(−ω mod 2π) = ω − 2π, (5a) 2π in which case ω k = , or (5b) 2π θ 10π ω -10π 10π π -10 θ = ω − ω π Figure 3: 2π 2 3. ω + π θ = (ω ± π)mod 2π − π = ω − 2π (6a) 2π ω − π =−(− (ω ± π)mod 2π) + π = ω − 2π (6b) 2π in which case ω + π k = (6c) 2π ω − π = . (6d) 2π 4 θ 10π ω -10π 10π -10π θ = ω − ω+π π Figure 4: 2π 2 The choice of how to relate rotation and direction angles is a matter of convention and convenience. Directed angles are inherently ambiguous. Depending on the value of the rotation angle and the chosen relation between rotation and direction angles, different ro- tation angles can have the same or different directed angles. For example, the two 1 π 5π rotation angles with radian measures 2 and 2 can have the same or different 1 π 1 π directed angles. Even the rotation angle 2 can have the directed angle 2 or −3π 1 π −1 π, 2 . The rotation angle 2 is greater than the rotation angle 2 but some def- initions of the corresponding direction angles can erroneously lead to the opposite −1π 3 π. conclusion. For example, the rotation angle 2 can have the direction angle 2 Therefore, when comparing radian measures of directed angles, care must be taken to add or subtract the appropriate multiple of 2π to guarantee that the difference between two directed angles, θ1 − θ2, is the same as the corresponding difference between their respective rotation angles. Trigonometric function conventions Trigonometric functions of rotation angles contain even worse ambiguity than di- rected angles because they are not one-to-one mappings within a single 2π rotation of a ray.