4 Planes in 3-D Descriptive Geometry

Transcription

1 4 Planes in 3- escripive Geomery 4. SPEIFYING PLNES Formally, for any wo lines ha inersec, he se of all poins ha lie on any line specified by wo poins one from each line specifies a plane defined by hese wo lines. m P 4- Illusraing he definiion of a plane l Q s a corollary: efiniion 4-: Plane spaial figure is a plane whenever, for any wo poins on he figure, he line specified by he poins also lies on he figure. plane surface has he characerisic being fla a surface on which a line or sraigh edge may lie in any disposiion. Tha is, every poin on he line is in conac wih he plane surface. Informally, he simples way of specifying a plane can specified by moving a line parallel o iself or by roaing a line abou a poin as shown in Figure 4-.

2 Line moving parallel o iself will generae a plane 4- Generaing a plane surface by moving a line or roaing a line Line roaed abou a poin form a secor of a plane circle Formally, a plane surfaces can be represened in four basic ways.. y wo inersecing lines This follows direcly from efiniion 4-. Y Plane creaed by wo inersecing lines and. s line XY moves along and i remains in conac wih boh lines X. y hree disinc non-collinear poins (poins no in a sraigh line) Two lines formed by he poins define wo inersecing lines and hus define he plane. Y X Plane creaed by hree non-collinear poins, and. Line XY passes hrough poin and is always in conac wih he line connecing poins and 6

3 3. y a line and poin no on he line The poin and any wo poins on he line define wo inersecing lines and hus define a plane. X Y Plane creaed by poin and line. Line XY passes hrough poin and is always in conac wih as i roaes abou. 4. y wo parallel lines This reduces o he above case by a selecing a poin on one line and wo on he oher, again forming a plane by definiion. X Y Plane creaed by wo parallel lines and. s line XY moves along and, i remains in conac wih boh lines Noe ha planes are always drawn o have limied size. In principle, a plane has indefinie exen. s a plane is compleely and uniquely defined by hree non-collinear poins on he plane, we can use hese poins which form he corners of a riangle ha belongs o he plane o delineaes a bounded porion of he plane. Views of such riangles are generally used in descripive geomery o define a plane. In pracical applicaions, however, we ofen know more han hree poins on a plane. n example is a planar roof surface, which is defined by is four corners; any hree of hese poins suffice o define he plane of he surface. The consrucions ha follow can be applied using any hree convenien non-collinear poins in hese cases. 4. POINT ON PLNE Since a poin lies in a plane surface whenever any line hrough he poin lies on he surface, we can use his propery consruc he view any poin on a plane surface. 7

4 X P Y Y LIne XY lies on plane and passes hrough poin P, which is also in he plane X P Y X P 4-3 Posiion of a poin on a plane The following consrucion deermines he posiion of a poin given a plane in adjacen views and a view of he poin in a picure plane. onsrucion 4- Locaing he posiion of a poin in a plane We are given a plane in wo adjacen views and a view of a poin. We are required o find he posiion of he poin in he oher view Suppose P is he given poin in fron view (view ). (Righ) Problem configuraion P 8

5 There are hree seps: X. raw any line X Y in view passing hrough P and inersecing wo sides of he plane represened by a R and S.. Projec he poins ino view by consrucing perpendicular projecors o locae poins R and S, hereby, locaing he projecion of line X Y in view. 3. Pick P up ino view. The poin of inersecion of he projecor and he projecion of X Y is he required poin P. R R P S S P Y Y 4-4 onsrucing a view of a poin on a plane X 4.. Edge view of a plane We know from Propery -5 (on page 58; see Figure -0) ha if a plane a, ha is projeced ino a picure plane p by a line family ƒ, conains a leas one projecion line in ƒ, is image is he line where planes a and p mee; his line is called he edge view (EV) of plane a. The consrucion given below generaes he edge view of a plane given by hree non-collinear poins or by he corners of a riangle. In order o see a plane in edge view he viewer mus assume a posiion in space where a line in he plane appears as a poin. Line Z appears as a poin plane as an edge Z 4-5 ifferen ways of seeing a plane in edge view Y Line Y appears as a poin plane as an edge X Line X appears as a poin plane as an edge 9

6 The edge view of a plane is needed for various invesigaions in descripive geomery. I can, for example, be used o deermine he disance beween a poin and a plane. ll one has o do is generae an edge view of he plane and projec he poin ino he same view; he disance beween he poin and he edge view, which is a line, is he required disance. We will explore such kinds of spaial relaionships in he nex chaper. onsrucion 4- Edge view of a plane Suppose we are given a plane surface represened by in wo adjacen views, and. We are required o find an edge view of (and hence, he plane). There are four seps:. Selec a view, say, and draw an inerior segmen of, parallel o ; in many cases, i is convenien o selec one corner of as an endpoin of his segmen; in Figure 4-6, corner was seleced and a segmen X drawn. This and he following sep can be omied if in one view, a side of is parallel o. This side will play he same role as he segmen in sep 3. Projec he endpoins of X ino view ; X now appears in TL. (Why?) 3. Selec a folding line 3 perpendicular o X o define an auxiliary view Projec from ino 3. Poins, and will be collinear, and (and he plane defined by i) appear in edge view in view 3. The consrucion is illusraed in Figure 4-6. Plane seen as an edge 3 TLHL X Horizonal line seen in TL 3 Line X seen as a poin,x 3 3 HL X Horizonal line in plane 4-6 onsrucing he edge view (EV) of a plane surface 30

7 Insead of consrucing a horizonal line as in sep, we could have chosen an arbirary line. In his case, we would need o creae an addiional auxiliary view ha showed his line in TL before consrucing an edge view of he plane. See Figure 4-7. Plane seen as an edge in view #4 4 Line RS seen as a poin in view #4 R 4,S Line RS seen in TL in view #3 S 3 3 TL 3 3 R 3 S R R S 4-7 Edge view of a plane 3

8 4.. Worked example House on sloping ground seen in edge view We know ha a plane can be described by a pair of parallel lines, which lie on he surface of he plane. This is a convenien way of describing sloping grounds such as hillsides. The example, shown in Figure 4-8, shows a house on a uniformly sloping ground. The problem is o find a view of he house showing he ground in edge view. l m f l a b m 4-8 The problem: Finding a view of he house showing he sloping ground in edge view In Figure 4-8 we are given he op () and fron (f) views of he house and he sloping ground indicaed by parallel lines l and m. Noice ha he fron view of he house shows where wo exerior walls of he base of house mee he sloping ground. Lines, a and b, can, in fac, be consruced from he op view and an incomplee fron view using echniques ha will be discussed in subsequen chapers. However, for now we will assume ha hese lines are known. The soluion is shown in Figure 4-9. The firs sep is o find riangles ha are coplanar wih he sloping ground. The simples way is o choose poins,, ha lies on l and m. This guaranees ha he riangle is coplanar wih he sloping ground. We 3

9 noice ha is parallel o he folding line f. n edge view can be obained by choosing a folding perpendicular o l or m. l m f a b l a b, m 4-9 pplicaion of onsrucion 4- o find he edge view of a house on a sloping plane specified by parallel lines 4.3 TRUE SLOPE NGLE (IP) OF PLNE The rue slope angle of a plane, also called dip, is he angle he plane makes wih he horizonal plane. The rue slope angle of a plane can be seen in a view, which shows simulaneously he edge view of he plane as well an edge of he horizonal projecion plane. Tha is, he rue slope angle can only be seen in an elevaion view. 33

10 3 X #3 is an elevaion view TLHL 3 True slope angle,x 3 3 Mus be seen ogeher in he same elevaion in order for rue slope o appear { Plane seen in edge view Horizonal projecion plane seen in edge view HL X # is an elevaion view 4-0 True slope angle of a plane 4.4 NORML (TRUE SHPE) VIEW OF PLNE noher imporan view one migh wan o consruc is a normal view of a plane, which is a view in a picure plane parallel o he plane. We know from Propery -5 (on page 58: see Figure -0) ha in such a projecion, disances are preserved, and a normal view consequenly shows any figure in he plane in is rue shape and size. This view is herefore of general pracical ineres. efore he rue shape of he given plane can be deermined, we mus consruc a view in which he plane appears as an edge. This EV of he plane can hen projeced ono a parallel plane on which he rue shape projecion of he plane will appear. See he consrucion below illusraed in Figure 4-. onsrucion 4-3 Normal view of a plane Given in wo adjacen views, and, find a normal view of (ha is, a view whose picure plane is parallel o he plane defined by ). There are hree seps.. onsruc an edge view of in an auxiliary view 3.. Place folding line 3 4 in view 3 parallel o he edge view o define an auxiliary view Projec from view 3 ino view 4. This is he desired normal view. 34

11 Using a horizonal line in plan X Fronal Line True shape of plane 4 TL 3 X 4 X 4,X Using a horizonal line in elevaion ,X 3 X X 4 TLHL True shape of plane HL X 4- Normal view or rue shape of a plane 35

12 4.4. Worked example True size of a roof shape Normal views of a plane are very imporan because hey show every planar figure in he plane in is rue size and shape. This is demonsraed by he following example. Figure 4- shows a building wih a hipped roof in op and fron view, T and F, respecively. Neiher view shows he major roof surface in rue shape and size. The figure demonsraes how wo successive auxiliary views resul in a projecion depicing he surface in normal view ha shows is rue shape and size. This view can be used, for example, o compue is surface area or o design a iling paern, which can hen be projeced back ino he op and fron views. T F F EV True shape of he roof 4- True shape of a roof The figure shows he complee auxiliary views; readers are encouraged o sudy hese views in deail in order o deepen heir undersanding of he consrucions under review and o develop heir abiliies o visualize shapes in hree dimensions. 36

13 4.5 WORKE EXMPLES 4.5. Larges inscribed circle The figure on he righ gives op and fron views of a riangle. The problem is o deermine is slope and he diameer of he larges circle ha can be inscribed in his riangle showing he circle in boh op and plan views. f (Righ) Problem configuraion Le us firs solve for he slope angle for which we will need o find he edge view of he riangle in elevaion. EV slope = 4 f 37

14 Nex we find he rue shape of he riangle in order o inscribe a circle. slope = 4 f rue shape of Lasly, we consruc he inscribed circle, and back projec o he fron and back views. The (in)-cener of he circle is he inersecion of he angle bisecors, is radius equaling he perpendicular disance from he in-cener o any side. See Figures 4-3 and 4-4, he laer using he echnique for consrucing an ellipse wihin an oblique recangle. slope = 4 f incircle seen in rue shape consrucing ellipse by using ransfer disances from views & 4-3 Slope and in-circle of a riangle 38

15 slope = 4 f in-circle seen in rue shape using consrucion for an ellipse wihin an oblique recangle 4-4 In-circle of a riangle using consrucion for an ellipse wihin an oblique recangle 4.5. ompleing he views of a plane Suppose a plane is given by diagonal lines, say and. Suppose hree of he poins, say, and are given by heir quad paper coordinaes, for example, (, ½, 5½), (3,, 5), and (, ¼, 3¾). See Figure The problem saemen 4 39

16 In order o deermine we will need furher consrains. Suppose he diagonals are of equal lengh, ha is, = ; suppose furher ha hey inersec a righ angles. Given ha we are using quad paper, le us assume ha he quad coordinaes represen a scale, say " = 4'. The problem is o deermine he slope and rue shape of he plane ; o find he rue lengh and bearing of ; and o complee he op and fron views of he plane. We consruc an auxiliary view from he op view in which plane is seen in edge view. We can obain he slope of he plane in his view. We consruc a second view showing in rue shape. In his view is in rue lengh. onsruc a line from perpendicular o and mark off such ha =. is in rue lengh in his view. (Noe ha we will have o muliply he lengh measured off he quad paper by he scale for he drawing.) Projec back o he firs auxiliary view and hen o op and hen fron views using ransfer disances and in he second and firs auxiliary views respecively. We obain he bearing of from he op view. The complee consrucion is given in Figure Norh 7 6 bearing of = N E TL rue shape of plane consrains = 5 slope = 33 54' rue lengh of = 8'-6" 4 edge view of plane ompleing he views 40

17 4.5.3 True shape of a runcaed face We revisi problem in Figure 3-35 (on page 8) of consrucing he op and fron views of a runcaed pyramid shown on he lef in Figure 4-7. We are now ineresed in a view ha shows he rue shape of he runcaed face. Since he runcaed face is he resul of a secional plane hrough he pyramid, i follows ha he runcaed face lies on he plane. onsequenly, seeing he plane in rue shape will give a view of he runcaed face in rue shape. Since he plane is seen in edge view in fron elevaion, all we need o do is ake an auxiliary view normal o his edge view. The consrucion is given in he righ hand drawing in Figure 4-7. Noice ha we have chosen convenienly locaed folding lines. We have also numbered he poins o illusrae he consrucion. The rue shape of he runcaed face is shown shaded , 8 7, ,5,4,3 Problem onsrucion 4-7 onsrucing he rue shape of a runcaed face 4

18 4.5.4 isance beween parallel lines We revisi consrucion in Figure 3-7 (on page 0) of deermining he disance beween wo parallel lines, however, insead of a rue lengh consrucion we employ he fac ha parallel lines define a plane and use an edge view consrucion. The problem saemen is given on he righ in Figure 4-8, and is consrucion is given below in Figure isance beween parallel lines problem configuraion 3 parallel lines seen in edge view, ie., as collinear lines d d d X d 3 4 d d HL X disance beween lines 4 5,, poin view of he parallel lines in view #5 4-9 isance beween parallel lines he consrucion using planes The firs sep in he consrucion is o deermine a riangle ha defines he plane specified by he wo parallel lines. In fron view, draw a horizonal line from o mee 4

19 (in his case, exended a X). Then!X is coplanar wih he plane formed by lines and. y following onsrucion 4-3 (on page 34), we can consruc he plane in rue shape, which is shown in view #4. Since folding line 3 so chosen lies o he lef of, i is imporan o use he ransfer disance d o he appropriae side of folding line 3 4. Since he plane is shown in rue shape, lines and are in rue lengh; he disances can be easily read from his view. Or from he poin view shown in view # ngle beween wo inersecing lines We can employ he consrucions described in his chaper o solve for he rue size for he angle of inersecion of wo inersecing lines (see figure on he righ). The fac ha he lines do, in fac, inersec can be esablished by having a single projecion line connec he poin of inersecion in he wo views. The consrucion is shown in Figure 4-0 wihou furher explanaion. True angle of inersecion TL TL Edge view of he plane f f HL Two inersecing lines define a plane Problem configuraion onsrucion 4-0 ngle of inersecion of wo inersecing lines 43

20 How would you deermine he angle of inersecion in his case? f 44