1、)C 516 DEUTSCHE NORMEN March 1972 c E U .c C c .- E Geometrical Orientation pJ I 1312 I Geometrische Orientierung This Standard deale with the orientation of straight lines, planes and space, with the defini- tions traverse sense, rotation sense and screw aense to ether with the application of these
2、 definitions to Co-ordinate aystems and physical quantitfes. 1, Orientation 1:lt-orio4oa-o4Eai6h4_1-ioo If a traverse sense is marked out on a straight line, the line is said to be oriented. The trav- erse sense is made evident, e.g. by an ar ow (see In the same way as atraight lines, dietances and
3、serial extensions as well aa the majority of curves encountered may also be oriented. The traverse sense of a cloaed curve which does not intersect itself is also termed the revolution sense. If a rotation sense is marked out in a plane, the plane is said to be oriented. The rotation sense is made e
4、vident, e.g. by a circle with traverse sense (see Figure 2). In the same way a8 planea, the majority of surfaces encountered may also be oriented. If a screw sense is marked out in space, the space is said to be oriented. The screw sense is made evident, .g. by an oriented plane and an oriented stra
5、ight line perpendicular thereto (see Figure 3) or by a helical line (see Figure 4). S t i p u 1 a t i o n : trated in Fiere 3 and 4 is termed a right-hand screw sense; the screw sense illustrated in Figure 5 and 6 is termed a left-hand screw sense. Figure I) or by an ordered pair of points 7 1. I:2:
6、-!?reo4e!LElaot 1:z:eoEPsce The screw senae illus- 1 A finite set ie naid to be “ordered“ when there is no doubt about which of its elements is to be regarded as the first, which an the second etc. Instead of saying “ordered met of n elements“ it is auicker to Figure 1. Figure 2. Figure 3. Figure 4.
7、 Figure 5. Figure 6. say “n-tuple*gi instead of “%-tuple“, itpair“ is used, inatead of “3-tuple“. *tripletot nay also be used and eiiilarly “quadruple“ netead of “4-tuple“. In nathematic6 a opafrt alwagri Implies an ordered pair“. Continued on pages 2 to 6 Notes on pages 6 to 9 Alleinvorkauf der Nor
8、mbltter dur Phe same rotation sense also ariees when a circ rhich, from out of semi- p 1 a n e touches the straight line and Kt the point of contact hae impressed upon it- self the traverse sense of the straight line - see also under Section 2.5 a. Ience the figure in Figure 11 establishes the same
9、rotation sense a the figure in Figure 2. 1, if inurationa Screw sense A screw Bonae ia assigned to the three- dimeanional figures below: b) An ordered set ) of three oriented straight lines 81, g g which intersect at a point and do not fie fn the same plane. 1 93 Figure 8. S t i p u 1 a t i o n : Th
10、e intended screw sense results from the smallest rotation which converts gl into g2, subject to a superimposed motion in the traverse senee of g3. - Hence the three-dimensional figure in Figure 8 eatablishe the same screw senne as the figure in Figure 3. 1) An ordered set of three linearly independe
11、nt vector Pl. Phe same rotation sense also results from hag- hing that point PI denotes t h e f i r s t B e i - p 1 a n e Line P2P3i see also under Section 2.3 a. Lho same rotation sense also results from the rector pair (w vector pair ( d: 1 2 $1; see also under 3ection 2.2 a. ience the figure in F
12、igure I3 eotablishes the ame rotation sense a8 the fipe in Figure 2. of the oriented straight P) and similarly from the B) An oriented circle (or any other curve topo- logically related thereto). Phis figure has already been used in Section I., to describe the rotation sense. The association uith Se
13、ction 2.1 a reaults by denoting as P 0llows : g2 a tangent on which is impressed the traverse sense of the circle, and gl the normal which, at the point of contact, leads from inside to outside. Screw sense A screw sense is assigned to tho three- dimeniional figuren below: b) An ordered set of four
14、points P1, P2, P3, PI not lying in a plane. Figure 14. S t i p u 1 a t i o n : The intended screw sense results from imagining that point Pl de- notes the first semi-space of the oriented plane P93P4, Bee also under Section 2.3 b. The same screw sense also results from the vec- tor triplet (p1pI?, e
15、, m) and similarly from the vector triplet (w2, s, -11 see also under Section 2.2 b. Kence the three-dimensional figure reprreented in Figure 14 establiehes the same crew sense a the figure in Figure 3. b) An oriented spherical surface (or any other surface topologically related thereto). Figure 16.
16、 6 t i p u 1 a t i o n : The intended screw monse results when a tangential plane is inado to contact the spherical muface, such plane having the rotation Bense of the spherical sur- face impressed upon it, and the normal is so oriented that at the point of contact it leads froi inside to outside. T
17、he figure of Figure 3 resul ts . e I) see on page I. 2, A set of rectors v., . , Etn of a vector apace is termed linearly independent if the equation kl - VI + .-. + see also Section 2.3 a. 3.2. In an oriented space the normal to a point on an oriented surface takes that traverse sense which, jointl
18、y with the rotation senae of the oriented surface according to Section 1.5 establishes the screw emse of the space. 4. StiDulations in vector analrsiq The vector- A x B is BO directed that the three vectors A, B, A x B (in this sequence) establish the right-hand screw sense in both the right-hand (s
19、ee Section 5.1). The parallelepiped product ABC of the three vectors A, B, C is positive according to Section 4.1 in both the right-hand a n d 1 e f t - h a n d (in this sequence) establish the right-hand screw sense. !:?;-!ector-E!EoaEc4 a n d 1 e f t - h a n d 8 y s t e m 8 !:2=-oarallegEIEeo_Eroo
20、Uct a y s t e m s if, and only if, A, B, O !?:2:,2oior The rotor E! of a vector field is so directed that the rotation sense of the vector field to- gether with the rotor establishes the right screw sense in both the right-hand hand systems. 5. Co-ordinate systems 5.1. According to Section 2.1 b a s
21、crew sense is assigned to every three-dimensional Cartesian Co-ordinate system. If the right-hand screw sense is assigned, the system is termed a right- hand system, in the other case it is a left-hand system, The screw sense of a curvilinear three-dimensional Co-ordinate system is the screw sense o
22、f the three tangential vectors which can be made to contact the lattice lines of the Co-ordinate sys- tem at some point. a n d 1 e f t ,2:-BecommeoOatiooe a) Where t h r e e - dimensional Co-ordinate systems are concerned - and the option exists - r i g h t - h a n d 8 y 8 t e m a should be adopted.
23、 b) Where t w o - dimensional Co-ordinate systems are concerned - and the option exists - those systems should be adopted which, jointly with a third Co-ordinate axis directed w a r d s t h e o b 8 e r v e r , conetitute a right-hand system. t o - 6. DesiKnation of the rotation sense of a rotary mot
24、ion The rotation sense of a rotary motion may be indicated a) by demonstrating, b) by describing; description is only intelligible to the person who knows the screw sense to which the descrip- tion refers, and the traverse senee of the rotation axis (see Figure 3 and 5). A rotary motion is termed if
25、 the rotation sense, jointly with the traverse sense of the rotation axis, establishes the s c r e w s e n 8 e m a r k e d o u t i n s p a c e ; the trace of the screw sense is usually indicated by a Co-ordinate system (see Section 5). A rotarg motion is termed r i g h t - h a n d if the rotation se
26、nse, jointly with the traverse sense of the rotation axis, establishes the r i g h t - h a n d “Rotation in the clockwise direction“ means the same as “right-hand rotation“. If no traverse sense is marked out on the rotation axis, the assumed traverse sense ia either the traverse sense of the Co-ord
27、inate axis coincident with it, or the traverse sense of the viewing direction coincident with it. hich of these two possibilities is adopted is indicated by use of the words “positive rotation/ negative rotation“ or “ right -hand rot at ion/l e f t-hand rot a tion“ : Use of the words ositive“, “nega
28、tive“ means: The rotation axis has the traverse sense of the Co-ordinate axfs coincident with it. (In other words the presence of an observer is not assumed in this case.) When the words “right“, “left“ are used this means: The rotation axis has the traverse sense of the viewing direction coincident
29、 with it. (In other words the presence of an observer is assumed in this case.) 6:1=_AeTereoce_to_a_8cfew_8eoee p o s i t i v e (a positive rotation), (a right-hand rotation), s c r e w a e n s e (see Section 1.3). 6:2:-S4iElatoo-of_4oe,4Eaefse_eeose_ol-lhe_ro4aion_axia 6:2=,S4Ela4ioo-of-4o-n6-ec4o
30、6.3.1. If the rotary motion can be obaerrrred f r o m o n 1 y o n e of the two semi-spaces there is no doubt about the viewing direction. This case always arises when the surface of opeque bodies (surface of a table or A sheet of paper) is observed. 6.3.2. If the rotary motion can be observed f r o
31、m b o t h semi-spaces it is necessary for the viewing direction to be specified, if necessary by special standards dealing with the particular fields concerned. (For combustion enginei, eee DIN 6265; for electrical machines in motor vehicles, see DI19 72256). 6.3.3. If the terms “Co-ordinate axis“ a
32、nd “viewing direction“ occur together it is laid down that tho direction of the ao-ordinate axis is opposed to the viewing direction. Therefore in a space with n e 6 a t i v e This determination is equivalent to the following determination: When the words are used, the semi-space in which the observ
33、er is situated (see Section 2.3 b) is regarded as semi-space 2. When the words are used, the semi-space in which the observer is situated (see Section 2.3 b) is regarded as semi-space I. 6.3.4. If reference to the surface of the earth is involved, rotary motions in a horizontal plane are described w
34、ith the assumption that the observer is looking at them from above. (Hence, the vertical Co-ordinate axis runs f r o m d o w n t o u p in agreement with Section 2.5 b). The traversing of a curve towards the right and the slewing of a crane to the right are right- hand rotations on basis of this stip
35、ulation. Exceptions: The Co-ordinate systems (geodetic or earth-perpendicular system; stable Co-ordinate system; aerodynamic or drift-proof Co-ordinate system; flight path Co-ordinate system; etc.) used in the mechanics of flight (Standard Sheet LN 9300) are directed d o w n 7. Devendence of the sin
36、n of vbaical auantities on orientation r i g h t - h a n d screw sense a right-hand rotation is always a rotation. “positive rotation, negative rotation“ “right-hand rotation, left-hand rotation“ f r o m u p t o along the vertical Co-ordinate axis. z:lz_sgos,oeEeooeot-e-eeoe 7.1.1. Line integrals In
37、 the case of scalar quantities which can be represented as a line integral it is part of the definition of the quantity to state in which sense the line is to be traversed. Examples: W o r k AI2 done in moving from point I to point 2 2 Al* = J“ F d 6; *I2 - -%I 1 UI2 V o 1 t a g e from point 1 to po
38、int 2 L u12 - -u21 Ula = E d 6; 1 The voltage from point I to point 2 of a conductor is therefore, inter alia, positive when the vector of the electric field strength in the conductor points from I to 2 (see DIN 5489). 7.1.2. Double integrals In the case of scalar quantities which can be represented
39、 as double integrals the sign depends on the traverse sense of the normal to the surface. In the denomination of these quantities the (arbitrarily) chosen traverse sense is generally not expressed ; the traverse sense must there- fore be stated separately. Example: Current intensity I-JS-dA (8 curre
40、nt density) The current intensity in a conductor is therefore move through the conductor in the arbitrarily apecrfied traverse sense, and the negative charge carriers in the opposite direction. 7.1.3. Co-ordinatee of a vector A vector equal to a vector I but with opposed direction is designated by -
41、I. Multiplication of the vector X by a negative real number -k is reduced to multiplying the vector -I( by the posi- tive real number k: The magnitude and sign of the Co-ordinates cx, depend on the arbitrary choice of the basis vec%cf, I, 2. 7.2.1. A r e a The area of a surface element lying in an o
42、riented plane and having an oriented boundary line is reckoned to be positive if the rotation sense of the oriented boundarg line is the same as the rotation sense marked out in the plane. Area A (d n ie the normal vector belonging to the curve element according to Section 3.1 and having the magnitu
43、de of the interval d 6). The orientation of a boundary line may come about, for example, by travel round the surface element. 7.2.2. Angle of rotation The angle of a positive rotary motion (see Section 6.1) is reckoned to be positive. See Note. 7.2.3. Geometric angle 7.2.3.1. Angle between two strai
44、ght liiles: lines g, and g2 in the oriented plane ia understood the smallest non-negative angle of rotation which converts 61 into 62. 1 see on page I. A ositive when the positive charge carriers -k I I k (-X). of the vector C = cx X + CS Y + cz 2 2:2:,SiBoe,aeooent_-oo_qhe_oqaloo_eeose 1/2 # r d u
45、the angle K of an ordered pair) of straight Page 6 DIB 1312 a) If the straight lines gl and 132 are o r i e n t e d Osa2 n (= 3600), see Figures 17 and 18. the angle is limited to It follows that +kql 62) - 2% - +(g2* 61) when 81 f g2. n o n o r i e n t e 3 the angle a is lim- ited to O 5 a c= x (-
46、IUOO), see Figures 19 and 20. when g? 82. gl b) If the straight lines g? and 132 are Piwe 17. Pigure 18. It fOl1GWS that 7k(gl, g2) - O 7.2.3.2. Angle between straight line and plane: By the angle a between an oriented straight line g and an oriented plane E is understood the angle Here, n is an ori
47、ented normal of plane E ac- cording to Section 3. The angle a is limited to 2:Z:-Si6oe_aeE8oaeoq-o- 7.3.1. The volume of a body with oriented surface situated in an oriented space is reckoned to be pos- itive if the screw sense assigned to the body according to Section 2.5 b is the same as the screw
48、 sense marked out in space. (d A is the normal vector belonging to the surface element according to Section 3.2 and having the magnitude of the surface element d A). 7.3.2. S o 1 i d a n g 1 e A solid angle is reckoned to be positive if the portion of the sphere describing the solid angle has a posi
49、tive volume according to Section 7.3.1. -+L O*: a = * - +(gr a). -=s a z+n. Piwe 19. Figure 20. V o 1 u m e Volume V = 1/3 I$ r d A Notes “he concept of orientation in mathematics Instead of the terms “traverse sense“ , “rotation sense1 and “screw sense1 the word “orientation“ is commonly used in mathematics. With this usage it is general practice to call the marked orientation the “positive orientation and the opposite sense “negative orientation“. For (mathematical) spaces of any imagined dimension
