(O. H.)
III. Descriptive Geometry
This branch of geometry is concerned with the methods for representing solids and other figures in three dimensions by drawings in one plane. The most important method is that which was invented by Monge towards the end of the 18th century. It is based on parallel projections to a plane by rays perpendicular to the plane. Such a projection is called orthographic (seeProjection, § 18). If the plane is horizontal the projection is called the plan of the figure, and if the plane is vertical the elevation. In Monge’s method a figure is represented by its plan and elevation. It is therefore often called drawing in plan and elevation, and sometimes simply orthographic projection.
§ 1. We suppose then that we have two planes, one horizontal, the other vertical, and these we call the planes of plan and of elevation respectively, or the horizontal and the vertical plane, and denote them by the letters π1and π2. Their line of intersection is called the axis, and will be denoted by xy.If the surface of the drawing paper is taken as the plane of the plan, then the vertical plane will be the plane perpendicular to it through the axis xy. To bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane. This process of turning one plane down till it coincides with another is calledrabattingone to the other. Of course there is no necessity to have one of the two planes horizontal, but even when this is not the case it is convenient to retain the above names.Fig. 37.Fig. 38.The whole arrangement will be better understood by referring to fig. 37. A point A in space is there projected by the perpendicular AA1and AA2to the planes π1and π2so that A1and A2are the horizontal and vertical projections of A.If we remember that a line is perpendicular to a plane that is perpendicular to every line in the plane if only it is perpendicular to any two intersecting lines in the plane, we see that the axis which is perpendicular both to AA1and to AA2is also perpendicular to A1A0and to A2A0because these four lines are all in the same plane. Hence, if the plane π2be turned about the axis till it coincides with the plane π1, then A2A0will be the continuation of A1A0. This position of the planes is represented in fig. 38, in which the line A1A2is perpendicular to the axis x.Conversely any two points A1, A2in a line perpendicular to the axis will be the projections of some point in space when the plane π2is turned about the axis till it is perpendicular to the plane π1, because in this position the two perpendiculars to the planes π1and π2through the points A1and A2will be in a plane and therefore meet at some point A.Representation of Points.—We have thus the following method of representing in a single plane the position of points in space:—we take in the plane a line xy as the axis, and then any pair of points A1, A2in the plane on a line perpendicular to the axis represent a point A in space. If the line A1A2cuts the axis at A0, and if at A1a perpendicular be erected to the plane, then the point A will be in it at a height A1A = A0A2above the plane. This gives the position of the point A relative to the plane π1. In the same way, if in a perpendicular to π2through A2a point A be taken such that A2A = A0A1, then this will give the point A relative to the plane π2.Fig. 39.§ 2. The two planes π1, π2in their original position divide space into four parts. These are called the four quadrants. We suppose that the plane π2is turned as indicated in fig. 37, so that the point P comes to Q and R to S, then the quadrant in which the point A lies is called the first, and we say that in the first quadrant a point lies above the horizontal and in front of the vertical plane. Now we go round the axis in the sense in which the plane π2is turned and come in succession to the second, third and fourth quadrant. In the second a point lies above the plane of the plan and behind the plane of elevation, and so on. In fig. 39, which represents a side view of the planes in fig. 37 the quadrants are marked, and in each a point with its projection is taken. Fig. 38 shows how these are represented when the plane π2is turned down. We see thatA point lies in the first quadrant if the plan lies below, the elevation above the axis; in the second if plan and elevation both lie above; in the third if the plan lies above, the elevation below; in the fourth if plan and elevation both lie below the axis.If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself.If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself.If a point lies in the axis, both its plan and elevation lie in the axis and coincide with it.Of each of these propositions, which will easily be seen to be true, the converse holds also.§ 3.Representation of a Plane.—As we are thus enabled to represent points in a plane, we can represent any finite figure by representing its separate points. It is, however, not possible to represent a plane in this way, for the projections of its points completely cover the planes π1and π2, and no plane would appear different from any other. But any plane α cuts each of the planes π1, π2in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane α.A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely,any two lines which meet on the axis determine a plane.If the plane is parallel to the axis its traces are parallel to the axis.Of these one may be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. Thus a plane parallel to the horizontal plane of the plan has only one finite trace, viz. that with the plane of elevation.Fig. 40.If the plane passes through the axis both its traces coincide with the axis.This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is therefore introduced, which is best taken perpendicular to the other two. We call it simply the third plane and denote it by π3. As it is perpendicular to π1, it may be taken as the plane of elevation, its line of intersection γ with π1being the axis, and be turned down to coincide with π1. This is represented in fig. 40. OC is the axis xy whilst OA and OB are the traces of the third plane. They lie in one line γ. The plane is rabatted about γ to the horizontal plane. A plane α through the axis xy will then show in it a trace α3. In fig. 40 the lines OC and OP will thus be the traces of a plane through the axis xy, which makes an angle POQ with the horizontal plane.We can also find the trace which any other plane makes with π3. In rabatting the plane π3its trace OB with the plane π2will come to the position OD. Hence a plane β having the traces CA and CB will have with the third plane the trace β3, or AD if OD = OB.It also follows immediately that—If a plane α is perpendicular to the horizontal plane, then every point in it has its horizontal projection in the horizontal trace of the plane, as all the rays projecting these points lie in the plane itself.Any plane which is perpendicular to the horizontal plane has its vertical trace perpendicular to the axis.Any plane which is perpendicular to the vertical plane has its horizontal trace perpendicular to the axis and the vertical projections of all points in the plane lie in this trace.§ 4.Representation of a Line.—A line is determined either by two points in it or by two planes through it. We get accordingly two representations of it either by projections or by traces.First.—A line a is represented by its projectionsa1anda2on the two planesπ1andπ2. These may be any two lines, for, bringing the planes π1, π2into their original position, the planes through these lines perpendicular to π1and π2respectively will intersect in some line a which has a1, a2as its projections.Secondly.—A line a is represented by its traces—that is, by the points in which it cuts the two planesπ1, π2. Any two points may be taken as the traces of a line in space, for it is determined when the planes are in their original position as the line joining the two traces. This representation becomes undetermined if the two traces coincide in the axis. In this case we again use a third plane, or else the projections of the line.The fact that there are different methods of representing points and planes, and hence two methods of representing lines, suggests the principle of duality (section ii.,Projective Geometry, § 41). It is worth while to keep this in mind. It is also worth remembering that traces of planes or lines always lie in the planes or lines which they represent. Projections do not as a rule do this excepting when the point or line projected lies in one of the planes of projection.Having now shown how to represent points, planes and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties. It will be found that the former are very much simpler than the latter.Before we do this, however, we shall explain the notation used; for it is of great importance to have a systematic notation. We shall denote points in space by capitals A, B, C; planes in space by Greek letters α, β, γ; lines in space by small letters a, b, c; horizontal projections by suffixes 1, like A1, a1; vertical projections by suffixes 2, like A2, a2; traces by single and double dashes α′ α″, a′, a″. Hence P1will be the horizontal projection of a point P in space; a line a will have the projections a1, a2and the traces a′ and a″; a plane α has the traces α′ and α″.§ 5.If a point lies in a line, the projections of the point lie in the projections of the line.If a line lies in a plane, the traces of the line lie in the traces of the plane.These propositions follow at once from the definitions of the projections and of the traces.If a point lies in two lines its projections must lie in the projections of both. HenceIf two lines, given by their projections, intersect, the intersection of theirplanesand the intersection of their elevations must lie in a line perpendicular to the axis, because they must be the projections of the point common to the two lines.Similarly—If two lines given by their traces lie in the same plane or intersect, then the lines joining their horizontal and vertical traces respectively must meet on the axis, because they must be the traces of the plane through them.§ 6.To find the projections of a line which joins two points A, B given by their projectionsA1, A2andB1, B2, we join A1, B1and A2, B2; these will be the projections required. For example, the traces of a line are two points in the line whose projections are known or at all events easily found. They are the traces themselves and the feet of the perpendiculars from them to the axis.Fig. 41.Henceifa′ a″ (fig. 41)are the traces of a line a, and if the perpendiculars from them cut the axis inPandQrespectively, then the linea′Qwill be the horizontal anda″Pthe vertical projection of the line.Conversely, if the projections a1, a2of a line are given, and if these cut the axis in Q and P respectively, thenthe perpendicularsPa′andQa″to the axis drawn through these points cut the projectionsa1anda2in the tracesa′anda″.To find the line of intersection of two planes, we observe that this line lies in both planes; its traces must therefore lie in the traces of both. Hence the points where the horizontal traces of the given planes meet will be the horizontal, and the point where the vertical traces meet the vertical trace of the line required.§ 7.To decide whether a pointA,given by its projections, lies in a plane α, given by its traces, we draw a line p by joining A to some point in the plane α and determine its traces. If these lie in the traces of the plane, then the line, and therefore the point A, lies in the plane; otherwise not. This is conveniently done by joining A1to some point p′ in the trace α′; this gives p1; and the point where the perpendicular from p′ to the axis cuts the latter we join to A2; this gives p2. If the vertical trace of this line lies in the vertical trace of the plane, then, and then only, does the line p, and with it the point A, lie in the plane α.§ 8.Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by π1and by π2, in parallel lines.Parallel lines have parallel projections, because points at infinity are projected to infinity.If a line is parallel to a plane, then lines through the traces of the line and parallel to the traces of the plane must meet on the axis, because these lines are the traces of a plane parallel to the given plane.§ 9.To draw a plane through two intersecting lines or through two parallel lines, we determine the traces of the lines; the lines joining their horizontal and vertical traces respectively will be the horizontal and vertical traces of the plane. They will meet, at a finite point or at infinity, on the axis if the lines do intersect.To draw a plane through a line and a point without the line, we join the given point to any point in the line and determine the plane through this and the given line.To draw a plane through three points which are not in a line, we draw two of the lines which each join two of the given points and draw the plane through them. If the traces of all three lines AB, BC, CA be found, these must lie in two lines which meet on the axis.§ 10. We have in the last example got more points, or can easily get more points, than are necessary for the determination of the figure required—in this case the traces of the plane. This will happen in a great many constructions and is of considerable importance. It may happen that some of the points or lines obtained are not convenient in the actual construction. The horizontal traces of the lines AB and AC may, for instance, fall very near together, in which case the line joining them is not well defined. Or, one or both of them may fall beyond the drawing paper, so that they are practically non-existent for the construction. In this case the traces of the line BC may be used. Or, if the vertical traces of AB and AC are both in convenient position, so that the vertical trace of the required plane is found and one of the horizontal traces is got, then we may join the latter to the point where the vertical trace cuts the axis.The draughtsman must remember that the lines which he draws are not mathematical lines without thickness, and therefore every drawing is affected by some errors. It is therefore very desirable to be able constantly to check the latter. Such checks always present themselves when the same result can be obtained by different constructions, or when, as in the above case, some lines must meet on the axis, or if three points must lie in a line. A careful draughtsman will always avail himself of these checks.§ 11.To draw a plane through a given point parallel to a given plane α, we draw through the point two lines which are parallel to the plane α, and determine the plane through them; or, as we know that the traces of the required plane are parallel to those of the given one (§ 8), we need only draw one line l through the point parallel to the plane and find one of its traces, say the vertical trace l″; a line through this parallel to the vertical trace of α will be the vertical trace β″ of the required plane β, and a line parallel to the horizontal trace of α meeting β″ on the axis will be the horizontal trace β′.Fig. 42.Let A1A2(fig. 42) be the given point, α′ α″ the given plane, a line l1through A1, parallel to α′ and a horizontal line l2through A2will be the projections of a line l through A parallel to the plane, because the horizontal plane through this line will cut the plane α in a line c which has its horizontal projection c1parallel to α′.§ 12. We now come to the metrical properties of figures.A line is perpendicular to a plane if the projections of the line are perpendicular to the traces of the plane.We prove it for the horizontal projection. If a line p is perpendicular to a plane α, every plane through p is perpendicular to α; hence also the vertical plane which projects the line p to p1. As this plane is perpendicular both to the horizontal plane and to the plane α, it is also perpendicular to their intersection—that is, to the horizontal trace of α. It follows that every line in this projecting plane, therefore also p1, the plan of p, is perpendicular to the horizontal trace of α.To draw a plane through a given point A perpendicular to a given line p, we first draw through some point O in the axis lines γ′, γ″ perpendicular respectively to the projections p1and p2of the given line. These will be the traces of a plane γ which is perpendicular to the given line. We next draw through the given point A a plane parallel to the plane γ; this will be the plane required.Other metrical properties depend on the determination of the real size or shape of a figure.In general the projection of a figure differs both in size and shape from the figure itself. But figures in a plane parallel to a plane of projection will be identical with their projections, and will thus be given in their true dimensions. In other cases there is the problem, constantly recurring, either to find the true shape and size of a plane figure when plan and elevation are given, or, conversely, to find the latter from the known true shape of the figure itself. To do this, the plane is turned about one of its traces till it is laid down into that plane of projection to which the trace belongs. This is technically called rabatting the plane respectively into the plane of the plan or the elevation. As there is no difference in the treatment of the two cases, we shall consider only the case of rabatting a plane α into the plane of the plan. The plan of the figure is a parallel (orthographic) projection of the figure itself. The results of parallel projection (seeProjection, §§ 17 and 18) may therefore now be used. The trace α′ will hereby take the place of what formerly was called the axis of projection. Hence we see that corresponding points in the plan and in the rabatted plane are joined by lines which are perpendicular to the trace α′ and that corresponding lines meet on this trace. We also see that the correspondence is completely determined if we know for one point or one line in the plan the corresponding point or line in the rabatted plane.Before, however, we treat of this we consider some special cases.§ 13.To determine the distance between two points A, B given by their projectionsA1, B1andA2, B2,or, in other words, to determine the true length of a line the plan and elevation of which are given.Fig. 43.Fig. 44.Solution.—The two points A, B in space lie vertically above their plans A1, B1(fig. 43) and A1A = A0A2, B1B = B0B2. The four points A, B, A1, B1therefore form a plane quadrilateral on the base A1B1and having right angles at the base. This plane we rabatt about A1B1by drawing A1A and B1B perpendicular to A1B1and making A1A = A0A2, B1B = B0B2. Then AB will give the length required.The construction might have been performed in the elevation by making A2A = A0A1and B2B = B0B1on lines perpendicular to A2B2. Of course AB must have the same length in both cases.This figure may be turned into a model. Cut the paper along A1A, AB and BB1, and fold the piece A1ABB1over along A1B1till it stands upright at right angles to the horizontal plane. The points A, B will then be in their true position in space relative to π1. Similarly if B2BAA2be cut out and turned along A2B2through a right angle we shall get AB in its true position relative to the plane π2. Lastly we fold the whole plane of the paper along the axis x till the plane π2is at right angles to π1. In this position the two sets of points AB will coincide if the drawing has been accurate.Models of this kind can be made in many cases and their construction cannot be too highly recommended in order to realize orthographic projection.§ 14.To find the angle between two given linesa, bof which the projectionsa1, b1anda2, b2are given.Solution.—Let a1, b1(fig. 44) meet in P1, a2, b2in T, then if the line P1T is not perpendicular to the axis the two lines will not meet. In this case we draw a line parallel to b to meet the line a. This is easiest done by drawing first the line P1P2perpendicular to the axis to meet a2in P2, and then drawing through P2a line c2parallel to b2; then b1, c2will be the projections of a line c which is parallel to b and meets a in P. The plane α which these two lines determine we rabatt to the plan. We determine the traces a′ and c′ of the lines a and c; then a′c′ is the trace α′ of their plane. On rabatting the point P comes to a point S on the line P1Q perpendicular to a′c′, so that QS = QP. But QP is the hypotenuse of a triangle PP1Q with a right angle P1. This we construct by making QR = P0P2; then P1R = PQ. The lines a′S and c′S will therefore include angles equal to those made by the given lines. It is to be remembered that two lines include two angles which are supplementary. Which of these is to be taken in any special case depends upon the circumstances.To determine the angle between a line and a plane, we draw through any point in the line a perpendicular to the plane (§ 12) and determine the angle between it and the given line. The complement of this angle is the required one.To determine the angle between two planes, we draw through any point two lines perpendicular to the two planes and determine the angle between the latter as above.In special cases it is simpler to determine at once the angle between the two planes by taking a plane section perpendicular to the intersection of the two planes and rabatt this. This is especially the case if one of the planes is the horizontal or vertical plane of projection.Thus in fig. 45 the angle P1QR is the angle which the plane α makes with the horizontal plane.§ 15. We return to the general case of rabatting a plane α of which the traces α′ α″ are given.Fig. 45.Here it will be convenient to determine first the position which the trace α″—which is a line in α—assumes when rabatted. Points in this line coincide with their elevations. Hence it is given in its true dimension, and we can measure off along it the true distance between two points in it. If therefore (fig. 45) P is any point in α″ originally coincident with its elevation P2, and if O is the point where α″ cuts the axis xy, so that O is also in α′, then the point P will after rabatting the plane assume such a position that OP = OP2. At the same time the plan is an orthographic projection of the plane α. Hence the line joining P to the plan P1will after rabatting be perpendicular to α′. But P1is known; it is the foot of the perpendicular from P2to the axis xy. We draw therefore, to find P, from P1a perpendicular P1Q to α′ and find on it a point P such that OP = OP2. Then the line OP will be the position of α″ when rabatted. This line corresponds therefore to the plan of α″—that is, to the axis xy, corresponding points on these lines being those which lie on a perpendicular to α′.We have thus one pair of corresponding lines and can now find for any point B1in the plan the corresponding point B in the rabatted plane. We draw a line through B1, say B1P1, cutting α′ in C. To it corresponds the line CP, and the point where this is cut by the projecting ray through B1, perpendicular to α′, is the required point B.Similarly any figure in the rabatted plane can be found when the plan is known; but this is usually found in a different manner without any reference to the general theory of parallel projection. As this method and the reasoning employed for it have their peculiar advantages, we give it also.Supposing the planes π1and π2to be in their positions in space perpendicular to each other, we take a section of the whole figure by a plane perpendicular to the trace α′ about which we are going to rabatt the plane α. Let this section pass through the point Q in α′. Its traces will then be the lines QP1and P1P2(fig. 9). These will be at right angles, and will therefore, together with the section QP2of the plane α, form a right-angled triangle QP1P2with the right angle at P1, and having the sides P1Q and P1P2which both are given in their true lengths. This triangle we rabatt about its base P1Q, making P1R = P1P2. The line QR will then give the true length of the line QP in space. If now the plane α be turned about α′ the point P will describe a circle about Q as centre with radius QP = QR, in a plane perpendicular to the trace α′. Hence when the plane α has been rabatted into the horizontal plane the point P will lie in the perpendicular P1Q to α′, so that QP = QR.If A1is the plan of a point A in the plane α, and if A1lies in QP1, then the point A will lie vertically above A1in the line QP. On turning down the triangle QP1P2, the point A will come to A0, the line A1A0being perpendicular to QP1. Hence A will be a point in QP such that QA = QA0.If B1is the plan of another point, but such that A1B1is parallel to α′, then the corresponding line AB will also be parallel to α′. Hence, if through A a line AB be drawn parallel to α′, and B1B perpendicular to α′, then their intersection gives the point B. Thus of any point given in plan the real position in the plane α, when rabatted, can be found by this second method. This is the one most generally given in books on geometrical drawing. The first method explained is, however, in most cases preferable as it gives the draughtsman a greater variety of constructions. It requires a somewhat greater amount of theoretical knowledge.If instead of our knowing the plan of a figure the latter is itself given, then the process of finding the plan is the reverse of the above and needs little explanation. We give an example.§ 16.It is required to draw the plan and elevation of a polygon of which the real shape and position in a given plane α are known.Fig. 46.We first rabatt the plane α (fig. 46) as before so that P1comes to P, hence OP1to OP. Let the given polygon in α be the figure ABCDE. We project, not the vertices, but the sides. To project the line AB, we produce it to cut α′ in F and OP in G, and draw GG1perpendicular to α′; then G1corresponds to G, therefore FG1to FG. In the same manner we might project all the other sides, at leastthose which cut OF and OP in convenient points. It will be best, however, first to produce all the sides to cut OP and α′ and then to draw all the projecting rays through A, B, C ... perpendicular to α′, and in the same direction the lines G, G1, &c. By drawing FG we get the points A1, B1on the projecting ray through A and B. We then join B to the point M where BC produced meets the trace α′. This gives C1. So we go on till we have found E1. The line A1E1must then meet AE in α′, and this gives a check. If one of the sides cuts α′ or OP beyond the drawing paper this method fails, but then we may easily find the projection of some other line, say of a diagonal, or directly the projection of a point, by the former methods. The diagonals may also serve to check the drawing, for two corresponding diagonals must meet in the trace α′.Having got the plan we easily find the elevation. The elevation of G is above G1in α″, and that of F is at F2in the axis. This gives the elevation F2G2of FG and in it we get A2B2in the verticals through A1and B1. As a check we have OG = OG2. Similarly the elevation of the other sides and vertices are found.§ 17. We proceed to give some applications of the above principles to the representation of solids and of the solution of problems connected with them.Of a pyramid are given its base, the length of the perpendicular from the vertex to the base, and the point where this perpendicular cuts the base; it is required first to develop the whole surface of the pyramid into one plane, and second to determine its section by a plane which cuts the plane of the base in a given line and makes a given angle with it.1. As the planes of projection are not given we can take them as we like, and we select them in such a manner that the solution becomes as simple as possible. We take the plane of the base as the horizontal plane and the vertical plane perpendicular to the plane of the section. Let then (fig. 47) ABCD be the base of the pyramid, V1the plan of the vertex, then the elevations of A, B, C, D will be in the axis at A2, B2, C2, D2, and the vertex at some point V2above V1at a known distance from the axis. The lines V1A, V1B, &c., will be the plans and the lines V2A2, V2B2, &c., the elevations of the edges of the pyramid, of which thus plan and elevation are known.We develop the surface into the plane of the base by turning each lateral face about its lower edge into the horizontal plane by the method used in § 14. If one face has been turned down, say ABV to ABP, then the point Q to which the vertex of the next face BCV comes can be got more simply by finding on the line V1Q perpendicular to BC the point Q such that BQ = BP, for these lines represent the same edge BV of the pyramid. Next R is found by making CR = CQ, and so on till we have got the last vertex—in this case S. The fact that AS must equal AP gives a convenient check.2. The plane α whose section we have to determine has its horizontal trace given perpendicular to the axis, and its vertical trace makes the given angle with the axis. This determines it. To find the section of the pyramid by this plane there are two methods applicable: we find the sections of the plane either with the faces or with the edges of the pyramid. We use the latter.As the plane α is perpendicular to the vertical plane, the trace α″ contains the projection of every figure in it; the points E2, F2, G2, H2where this trace cuts the elevations of the edges will therefore be the elevations of the points where the edges cut α. From these we find the plans E1, F1, G1, H1, and by joining them the plan of the section. If from E1, F1lines be drawn perpendicular to AB, these will determine the points E, F on the developed face in which the plane α cuts it; hence also the line EF. Similarly on the other faces. Of course BF must be the same length on BP and on BQ. If the plane α be rabatted to the plan, we get the real shape of the section as shown in the figure in EFGH. This is done easily by making F0F = OF2, &c. If the figure representing the development of the pyramid, or better a copy of it, is cut out, and if the lateral faces be bent along the lines AB, BC, &c., we get a model of the pyramid with the section marked on its faces. This may be placed on its plan ABCD and the plane of elevation bent about the axis x. The pyramid stands then in front of its elevations. If next the plane α with a hole cut out representing the true section be bent along the trace α′ till its edge coincides with α″, the edges of the hole ought to coincide with the lines EF, FG, &c., on the faces.§ 18. Polyhedra like the pyramid in § 17 are represented by the projections of their edges and vertices. But solids bounded by curved surfaces, or surfaces themselves, cannot be thus represented.For a surface we may use, as in case of the plane, its traces—that is, the curves in which it cuts the planes of projection. We may also project points and curves on the surface. A ray cuts the surface generally in more than one point; hence it will happen that some of the rays touch the surface, if two of these points coincide. The points of contact of these rays will form some curve on the surface, and this will appear from the centre of projection as the boundary of the surface or of part of the surface. The outlines of all surfaces of solids which we see about us are formed by the points at which rays through our eye touch the surface. The projections of these contours are therefore best adapted to give an idea of the shape of a surface.Fig. 47.Thus the tangents drawn from any finite centre to a sphere form a right circular cone, and this will be cut by any plane in a conic. It is often called the projection of a sphere, but it is better called the contour-line of the sphere, as it is the boundary of the projections of all points on the sphere.If the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle, and if the projection is, as in our case, orthographic, then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere. We get such a circle in the plan and another in the elevation, their centres being plan and elevation of the centre of the sphere.Similarly the rays touching a cone of the second order will lie in two planes which pass through the vertex of the cone, the contour-line of the projection of the cone consists therefore of two lines meeting in the projection of the vertex. These may, however, be invisible if no real tangent rays can be drawn from the centre of projection; and this happens when the ray projecting the centre of the vertex lies within the cone. In this case the traces of the cone are of importance. Thus in representing a cone of revolution with a vertical axis we get in the plan a circular trace of the surface whose centre is the plan of the vertex of the cone, and in the elevation the contour, consisting of a pair of lines intersecting in the elevation of the vertex of the cone. The circle in the plan and the pair of lines in the elevation do not determine the surface, for an infinite number of surfaces might be conceived which pass through the circular trace and touch two planes through the contour lines in the vertical plane. The surface becomes only completely defined if we write down to the figure that it shall represent a cone. The same holds for allsurfaces. Even a plane is fully represented by its traces only under the silent understanding that the traces are those of a plane.§ 19. Some of the simpler problems connected with the representation of surfaces are the determination of plane sections and of the curves of intersection of two such surfaces. The former is constantly used in nearly all problems concerning surfaces. Its solution depends of course on the nature of the surface.To determine the curve of intersection of two surfaces, we take a plane and determine its section with each of the two surfaces, rabatting this plane if necessary. This gives two curves which lie in the same plane and whose intersections will give us points on both surfaces. It must here be remembered that two curves in space do not necessarily intersect, hence that the points in which their projections intersect are not necessarily the projections of points common to the two curves. This will, however, be the case if the two curves lie in a common plane. By taking then a number of plane sections of the surfaces we can get as many points on their curve of intersection as we like. These planes have, of course, to be selected in such a way that the sections are curves as simple as the case permits of, and such that they can be easily and accurately drawn. Thus when possible the sections should be straight lines or circles. This not only saves time in drawing but determines all points on the sections, and therefore also the points where the two curves meet, with equal accuracy.§ 20. We give a few examples how these sections have to be selected. A cone is cut by every plane through the vertex in lines, and if it is a cone of revolution by planes perpendicular to the axis in circles.A cylinder is cut by every plane parallel to the axis in lines, and if it is a cylinder of revolution by planes perpendicular to the axis in circles.A sphere is cut by every plane in a circle.Hence in case of two cones situated anywhere in space we take sections through both vertices. These will cut both cones in lines. Similarly in case of two cylinders we may take sections parallel to the axis of both. In case of a sphere and a cone of revolution with vertical axis, horizontal sections will cut both surfaces in circles whose plans are circles and whose elevations are lines, whilst vertical sections through the vertex of the cone cut the latter in lines and the sphere in circles. To avoid drawing the projections of these circles, which would in general be ellipses, we rabatt the plane and then draw the circles in their real shape. And so on in other cases.Special attention should in all cases be paid to those points in which the tangents to the projection of the curve of intersection are parallel or perpendicular to the axis x, or where these projections touch the contour of one of the surfaces.
§ 1. We suppose then that we have two planes, one horizontal, the other vertical, and these we call the planes of plan and of elevation respectively, or the horizontal and the vertical plane, and denote them by the letters π1and π2. Their line of intersection is called the axis, and will be denoted by xy.
If the surface of the drawing paper is taken as the plane of the plan, then the vertical plane will be the plane perpendicular to it through the axis xy. To bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane. This process of turning one plane down till it coincides with another is calledrabattingone to the other. Of course there is no necessity to have one of the two planes horizontal, but even when this is not the case it is convenient to retain the above names.
The whole arrangement will be better understood by referring to fig. 37. A point A in space is there projected by the perpendicular AA1and AA2to the planes π1and π2so that A1and A2are the horizontal and vertical projections of A.
If we remember that a line is perpendicular to a plane that is perpendicular to every line in the plane if only it is perpendicular to any two intersecting lines in the plane, we see that the axis which is perpendicular both to AA1and to AA2is also perpendicular to A1A0and to A2A0because these four lines are all in the same plane. Hence, if the plane π2be turned about the axis till it coincides with the plane π1, then A2A0will be the continuation of A1A0. This position of the planes is represented in fig. 38, in which the line A1A2is perpendicular to the axis x.
Conversely any two points A1, A2in a line perpendicular to the axis will be the projections of some point in space when the plane π2is turned about the axis till it is perpendicular to the plane π1, because in this position the two perpendiculars to the planes π1and π2through the points A1and A2will be in a plane and therefore meet at some point A.
Representation of Points.—We have thus the following method of representing in a single plane the position of points in space:—we take in the plane a line xy as the axis, and then any pair of points A1, A2in the plane on a line perpendicular to the axis represent a point A in space. If the line A1A2cuts the axis at A0, and if at A1a perpendicular be erected to the plane, then the point A will be in it at a height A1A = A0A2above the plane. This gives the position of the point A relative to the plane π1. In the same way, if in a perpendicular to π2through A2a point A be taken such that A2A = A0A1, then this will give the point A relative to the plane π2.
§ 2. The two planes π1, π2in their original position divide space into four parts. These are called the four quadrants. We suppose that the plane π2is turned as indicated in fig. 37, so that the point P comes to Q and R to S, then the quadrant in which the point A lies is called the first, and we say that in the first quadrant a point lies above the horizontal and in front of the vertical plane. Now we go round the axis in the sense in which the plane π2is turned and come in succession to the second, third and fourth quadrant. In the second a point lies above the plane of the plan and behind the plane of elevation, and so on. In fig. 39, which represents a side view of the planes in fig. 37 the quadrants are marked, and in each a point with its projection is taken. Fig. 38 shows how these are represented when the plane π2is turned down. We see that
A point lies in the first quadrant if the plan lies below, the elevation above the axis; in the second if plan and elevation both lie above; in the third if the plan lies above, the elevation below; in the fourth if plan and elevation both lie below the axis.
If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself.If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself.If a point lies in the axis, both its plan and elevation lie in the axis and coincide with it.
Of each of these propositions, which will easily be seen to be true, the converse holds also.
§ 3.Representation of a Plane.—As we are thus enabled to represent points in a plane, we can represent any finite figure by representing its separate points. It is, however, not possible to represent a plane in this way, for the projections of its points completely cover the planes π1and π2, and no plane would appear different from any other. But any plane α cuts each of the planes π1, π2in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane α.
A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely,any two lines which meet on the axis determine a plane.
If the plane is parallel to the axis its traces are parallel to the axis.Of these one may be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. Thus a plane parallel to the horizontal plane of the plan has only one finite trace, viz. that with the plane of elevation.
If the plane passes through the axis both its traces coincide with the axis.This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is therefore introduced, which is best taken perpendicular to the other two. We call it simply the third plane and denote it by π3. As it is perpendicular to π1, it may be taken as the plane of elevation, its line of intersection γ with π1being the axis, and be turned down to coincide with π1. This is represented in fig. 40. OC is the axis xy whilst OA and OB are the traces of the third plane. They lie in one line γ. The plane is rabatted about γ to the horizontal plane. A plane α through the axis xy will then show in it a trace α3. In fig. 40 the lines OC and OP will thus be the traces of a plane through the axis xy, which makes an angle POQ with the horizontal plane.
We can also find the trace which any other plane makes with π3. In rabatting the plane π3its trace OB with the plane π2will come to the position OD. Hence a plane β having the traces CA and CB will have with the third plane the trace β3, or AD if OD = OB.
It also follows immediately that—
If a plane α is perpendicular to the horizontal plane, then every point in it has its horizontal projection in the horizontal trace of the plane, as all the rays projecting these points lie in the plane itself.
Any plane which is perpendicular to the horizontal plane has its vertical trace perpendicular to the axis.
Any plane which is perpendicular to the vertical plane has its horizontal trace perpendicular to the axis and the vertical projections of all points in the plane lie in this trace.
§ 4.Representation of a Line.—A line is determined either by two points in it or by two planes through it. We get accordingly two representations of it either by projections or by traces.
First.—A line a is represented by its projectionsa1anda2on the two planesπ1andπ2. These may be any two lines, for, bringing the planes π1, π2into their original position, the planes through these lines perpendicular to π1and π2respectively will intersect in some line a which has a1, a2as its projections.
Secondly.—A line a is represented by its traces—that is, by the points in which it cuts the two planesπ1, π2. Any two points may be taken as the traces of a line in space, for it is determined when the planes are in their original position as the line joining the two traces. This representation becomes undetermined if the two traces coincide in the axis. In this case we again use a third plane, or else the projections of the line.
The fact that there are different methods of representing points and planes, and hence two methods of representing lines, suggests the principle of duality (section ii.,Projective Geometry, § 41). It is worth while to keep this in mind. It is also worth remembering that traces of planes or lines always lie in the planes or lines which they represent. Projections do not as a rule do this excepting when the point or line projected lies in one of the planes of projection.
Having now shown how to represent points, planes and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties. It will be found that the former are very much simpler than the latter.
Before we do this, however, we shall explain the notation used; for it is of great importance to have a systematic notation. We shall denote points in space by capitals A, B, C; planes in space by Greek letters α, β, γ; lines in space by small letters a, b, c; horizontal projections by suffixes 1, like A1, a1; vertical projections by suffixes 2, like A2, a2; traces by single and double dashes α′ α″, a′, a″. Hence P1will be the horizontal projection of a point P in space; a line a will have the projections a1, a2and the traces a′ and a″; a plane α has the traces α′ and α″.
§ 5.If a point lies in a line, the projections of the point lie in the projections of the line.
If a line lies in a plane, the traces of the line lie in the traces of the plane.
These propositions follow at once from the definitions of the projections and of the traces.
If a point lies in two lines its projections must lie in the projections of both. Hence
If two lines, given by their projections, intersect, the intersection of theirplanesand the intersection of their elevations must lie in a line perpendicular to the axis, because they must be the projections of the point common to the two lines.
Similarly—If two lines given by their traces lie in the same plane or intersect, then the lines joining their horizontal and vertical traces respectively must meet on the axis, because they must be the traces of the plane through them.
§ 6.To find the projections of a line which joins two points A, B given by their projectionsA1, A2andB1, B2, we join A1, B1and A2, B2; these will be the projections required. For example, the traces of a line are two points in the line whose projections are known or at all events easily found. They are the traces themselves and the feet of the perpendiculars from them to the axis.
Henceifa′ a″ (fig. 41)are the traces of a line a, and if the perpendiculars from them cut the axis inPandQrespectively, then the linea′Qwill be the horizontal anda″Pthe vertical projection of the line.
Conversely, if the projections a1, a2of a line are given, and if these cut the axis in Q and P respectively, thenthe perpendicularsPa′andQa″to the axis drawn through these points cut the projectionsa1anda2in the tracesa′anda″.
To find the line of intersection of two planes, we observe that this line lies in both planes; its traces must therefore lie in the traces of both. Hence the points where the horizontal traces of the given planes meet will be the horizontal, and the point where the vertical traces meet the vertical trace of the line required.
§ 7.To decide whether a pointA,given by its projections, lies in a plane α, given by its traces, we draw a line p by joining A to some point in the plane α and determine its traces. If these lie in the traces of the plane, then the line, and therefore the point A, lies in the plane; otherwise not. This is conveniently done by joining A1to some point p′ in the trace α′; this gives p1; and the point where the perpendicular from p′ to the axis cuts the latter we join to A2; this gives p2. If the vertical trace of this line lies in the vertical trace of the plane, then, and then only, does the line p, and with it the point A, lie in the plane α.
§ 8.Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by π1and by π2, in parallel lines.
Parallel lines have parallel projections, because points at infinity are projected to infinity.
If a line is parallel to a plane, then lines through the traces of the line and parallel to the traces of the plane must meet on the axis, because these lines are the traces of a plane parallel to the given plane.
§ 9.To draw a plane through two intersecting lines or through two parallel lines, we determine the traces of the lines; the lines joining their horizontal and vertical traces respectively will be the horizontal and vertical traces of the plane. They will meet, at a finite point or at infinity, on the axis if the lines do intersect.
To draw a plane through a line and a point without the line, we join the given point to any point in the line and determine the plane through this and the given line.
To draw a plane through three points which are not in a line, we draw two of the lines which each join two of the given points and draw the plane through them. If the traces of all three lines AB, BC, CA be found, these must lie in two lines which meet on the axis.
§ 10. We have in the last example got more points, or can easily get more points, than are necessary for the determination of the figure required—in this case the traces of the plane. This will happen in a great many constructions and is of considerable importance. It may happen that some of the points or lines obtained are not convenient in the actual construction. The horizontal traces of the lines AB and AC may, for instance, fall very near together, in which case the line joining them is not well defined. Or, one or both of them may fall beyond the drawing paper, so that they are practically non-existent for the construction. In this case the traces of the line BC may be used. Or, if the vertical traces of AB and AC are both in convenient position, so that the vertical trace of the required plane is found and one of the horizontal traces is got, then we may join the latter to the point where the vertical trace cuts the axis.
The draughtsman must remember that the lines which he draws are not mathematical lines without thickness, and therefore every drawing is affected by some errors. It is therefore very desirable to be able constantly to check the latter. Such checks always present themselves when the same result can be obtained by different constructions, or when, as in the above case, some lines must meet on the axis, or if three points must lie in a line. A careful draughtsman will always avail himself of these checks.
§ 11.To draw a plane through a given point parallel to a given plane α, we draw through the point two lines which are parallel to the plane α, and determine the plane through them; or, as we know that the traces of the required plane are parallel to those of the given one (§ 8), we need only draw one line l through the point parallel to the plane and find one of its traces, say the vertical trace l″; a line through this parallel to the vertical trace of α will be the vertical trace β″ of the required plane β, and a line parallel to the horizontal trace of α meeting β″ on the axis will be the horizontal trace β′.
Let A1A2(fig. 42) be the given point, α′ α″ the given plane, a line l1through A1, parallel to α′ and a horizontal line l2through A2will be the projections of a line l through A parallel to the plane, because the horizontal plane through this line will cut the plane α in a line c which has its horizontal projection c1parallel to α′.
§ 12. We now come to the metrical properties of figures.
A line is perpendicular to a plane if the projections of the line are perpendicular to the traces of the plane.We prove it for the horizontal projection. If a line p is perpendicular to a plane α, every plane through p is perpendicular to α; hence also the vertical plane which projects the line p to p1. As this plane is perpendicular both to the horizontal plane and to the plane α, it is also perpendicular to their intersection—that is, to the horizontal trace of α. It follows that every line in this projecting plane, therefore also p1, the plan of p, is perpendicular to the horizontal trace of α.
To draw a plane through a given point A perpendicular to a given line p, we first draw through some point O in the axis lines γ′, γ″ perpendicular respectively to the projections p1and p2of the given line. These will be the traces of a plane γ which is perpendicular to the given line. We next draw through the given point A a plane parallel to the plane γ; this will be the plane required.
Other metrical properties depend on the determination of the real size or shape of a figure.
In general the projection of a figure differs both in size and shape from the figure itself. But figures in a plane parallel to a plane of projection will be identical with their projections, and will thus be given in their true dimensions. In other cases there is the problem, constantly recurring, either to find the true shape and size of a plane figure when plan and elevation are given, or, conversely, to find the latter from the known true shape of the figure itself. To do this, the plane is turned about one of its traces till it is laid down into that plane of projection to which the trace belongs. This is technically called rabatting the plane respectively into the plane of the plan or the elevation. As there is no difference in the treatment of the two cases, we shall consider only the case of rabatting a plane α into the plane of the plan. The plan of the figure is a parallel (orthographic) projection of the figure itself. The results of parallel projection (seeProjection, §§ 17 and 18) may therefore now be used. The trace α′ will hereby take the place of what formerly was called the axis of projection. Hence we see that corresponding points in the plan and in the rabatted plane are joined by lines which are perpendicular to the trace α′ and that corresponding lines meet on this trace. We also see that the correspondence is completely determined if we know for one point or one line in the plan the corresponding point or line in the rabatted plane.
Before, however, we treat of this we consider some special cases.
§ 13.To determine the distance between two points A, B given by their projectionsA1, B1andA2, B2,or, in other words, to determine the true length of a line the plan and elevation of which are given.
Solution.—The two points A, B in space lie vertically above their plans A1, B1(fig. 43) and A1A = A0A2, B1B = B0B2. The four points A, B, A1, B1therefore form a plane quadrilateral on the base A1B1and having right angles at the base. This plane we rabatt about A1B1by drawing A1A and B1B perpendicular to A1B1and making A1A = A0A2, B1B = B0B2. Then AB will give the length required.
The construction might have been performed in the elevation by making A2A = A0A1and B2B = B0B1on lines perpendicular to A2B2. Of course AB must have the same length in both cases.
This figure may be turned into a model. Cut the paper along A1A, AB and BB1, and fold the piece A1ABB1over along A1B1till it stands upright at right angles to the horizontal plane. The points A, B will then be in their true position in space relative to π1. Similarly if B2BAA2be cut out and turned along A2B2through a right angle we shall get AB in its true position relative to the plane π2. Lastly we fold the whole plane of the paper along the axis x till the plane π2is at right angles to π1. In this position the two sets of points AB will coincide if the drawing has been accurate.
Models of this kind can be made in many cases and their construction cannot be too highly recommended in order to realize orthographic projection.
§ 14.To find the angle between two given linesa, bof which the projectionsa1, b1anda2, b2are given.
Solution.—Let a1, b1(fig. 44) meet in P1, a2, b2in T, then if the line P1T is not perpendicular to the axis the two lines will not meet. In this case we draw a line parallel to b to meet the line a. This is easiest done by drawing first the line P1P2perpendicular to the axis to meet a2in P2, and then drawing through P2a line c2parallel to b2; then b1, c2will be the projections of a line c which is parallel to b and meets a in P. The plane α which these two lines determine we rabatt to the plan. We determine the traces a′ and c′ of the lines a and c; then a′c′ is the trace α′ of their plane. On rabatting the point P comes to a point S on the line P1Q perpendicular to a′c′, so that QS = QP. But QP is the hypotenuse of a triangle PP1Q with a right angle P1. This we construct by making QR = P0P2; then P1R = PQ. The lines a′S and c′S will therefore include angles equal to those made by the given lines. It is to be remembered that two lines include two angles which are supplementary. Which of these is to be taken in any special case depends upon the circumstances.
To determine the angle between a line and a plane, we draw through any point in the line a perpendicular to the plane (§ 12) and determine the angle between it and the given line. The complement of this angle is the required one.
To determine the angle between two planes, we draw through any point two lines perpendicular to the two planes and determine the angle between the latter as above.
In special cases it is simpler to determine at once the angle between the two planes by taking a plane section perpendicular to the intersection of the two planes and rabatt this. This is especially the case if one of the planes is the horizontal or vertical plane of projection.
Thus in fig. 45 the angle P1QR is the angle which the plane α makes with the horizontal plane.
§ 15. We return to the general case of rabatting a plane α of which the traces α′ α″ are given.
Here it will be convenient to determine first the position which the trace α″—which is a line in α—assumes when rabatted. Points in this line coincide with their elevations. Hence it is given in its true dimension, and we can measure off along it the true distance between two points in it. If therefore (fig. 45) P is any point in α″ originally coincident with its elevation P2, and if O is the point where α″ cuts the axis xy, so that O is also in α′, then the point P will after rabatting the plane assume such a position that OP = OP2. At the same time the plan is an orthographic projection of the plane α. Hence the line joining P to the plan P1will after rabatting be perpendicular to α′. But P1is known; it is the foot of the perpendicular from P2to the axis xy. We draw therefore, to find P, from P1a perpendicular P1Q to α′ and find on it a point P such that OP = OP2. Then the line OP will be the position of α″ when rabatted. This line corresponds therefore to the plan of α″—that is, to the axis xy, corresponding points on these lines being those which lie on a perpendicular to α′.
We have thus one pair of corresponding lines and can now find for any point B1in the plan the corresponding point B in the rabatted plane. We draw a line through B1, say B1P1, cutting α′ in C. To it corresponds the line CP, and the point where this is cut by the projecting ray through B1, perpendicular to α′, is the required point B.
Similarly any figure in the rabatted plane can be found when the plan is known; but this is usually found in a different manner without any reference to the general theory of parallel projection. As this method and the reasoning employed for it have their peculiar advantages, we give it also.
Supposing the planes π1and π2to be in their positions in space perpendicular to each other, we take a section of the whole figure by a plane perpendicular to the trace α′ about which we are going to rabatt the plane α. Let this section pass through the point Q in α′. Its traces will then be the lines QP1and P1P2(fig. 9). These will be at right angles, and will therefore, together with the section QP2of the plane α, form a right-angled triangle QP1P2with the right angle at P1, and having the sides P1Q and P1P2which both are given in their true lengths. This triangle we rabatt about its base P1Q, making P1R = P1P2. The line QR will then give the true length of the line QP in space. If now the plane α be turned about α′ the point P will describe a circle about Q as centre with radius QP = QR, in a plane perpendicular to the trace α′. Hence when the plane α has been rabatted into the horizontal plane the point P will lie in the perpendicular P1Q to α′, so that QP = QR.
If A1is the plan of a point A in the plane α, and if A1lies in QP1, then the point A will lie vertically above A1in the line QP. On turning down the triangle QP1P2, the point A will come to A0, the line A1A0being perpendicular to QP1. Hence A will be a point in QP such that QA = QA0.
If B1is the plan of another point, but such that A1B1is parallel to α′, then the corresponding line AB will also be parallel to α′. Hence, if through A a line AB be drawn parallel to α′, and B1B perpendicular to α′, then their intersection gives the point B. Thus of any point given in plan the real position in the plane α, when rabatted, can be found by this second method. This is the one most generally given in books on geometrical drawing. The first method explained is, however, in most cases preferable as it gives the draughtsman a greater variety of constructions. It requires a somewhat greater amount of theoretical knowledge.
If instead of our knowing the plan of a figure the latter is itself given, then the process of finding the plan is the reverse of the above and needs little explanation. We give an example.
§ 16.It is required to draw the plan and elevation of a polygon of which the real shape and position in a given plane α are known.
We first rabatt the plane α (fig. 46) as before so that P1comes to P, hence OP1to OP. Let the given polygon in α be the figure ABCDE. We project, not the vertices, but the sides. To project the line AB, we produce it to cut α′ in F and OP in G, and draw GG1perpendicular to α′; then G1corresponds to G, therefore FG1to FG. In the same manner we might project all the other sides, at leastthose which cut OF and OP in convenient points. It will be best, however, first to produce all the sides to cut OP and α′ and then to draw all the projecting rays through A, B, C ... perpendicular to α′, and in the same direction the lines G, G1, &c. By drawing FG we get the points A1, B1on the projecting ray through A and B. We then join B to the point M where BC produced meets the trace α′. This gives C1. So we go on till we have found E1. The line A1E1must then meet AE in α′, and this gives a check. If one of the sides cuts α′ or OP beyond the drawing paper this method fails, but then we may easily find the projection of some other line, say of a diagonal, or directly the projection of a point, by the former methods. The diagonals may also serve to check the drawing, for two corresponding diagonals must meet in the trace α′.
Having got the plan we easily find the elevation. The elevation of G is above G1in α″, and that of F is at F2in the axis. This gives the elevation F2G2of FG and in it we get A2B2in the verticals through A1and B1. As a check we have OG = OG2. Similarly the elevation of the other sides and vertices are found.
§ 17. We proceed to give some applications of the above principles to the representation of solids and of the solution of problems connected with them.
Of a pyramid are given its base, the length of the perpendicular from the vertex to the base, and the point where this perpendicular cuts the base; it is required first to develop the whole surface of the pyramid into one plane, and second to determine its section by a plane which cuts the plane of the base in a given line and makes a given angle with it.
1. As the planes of projection are not given we can take them as we like, and we select them in such a manner that the solution becomes as simple as possible. We take the plane of the base as the horizontal plane and the vertical plane perpendicular to the plane of the section. Let then (fig. 47) ABCD be the base of the pyramid, V1the plan of the vertex, then the elevations of A, B, C, D will be in the axis at A2, B2, C2, D2, and the vertex at some point V2above V1at a known distance from the axis. The lines V1A, V1B, &c., will be the plans and the lines V2A2, V2B2, &c., the elevations of the edges of the pyramid, of which thus plan and elevation are known.
We develop the surface into the plane of the base by turning each lateral face about its lower edge into the horizontal plane by the method used in § 14. If one face has been turned down, say ABV to ABP, then the point Q to which the vertex of the next face BCV comes can be got more simply by finding on the line V1Q perpendicular to BC the point Q such that BQ = BP, for these lines represent the same edge BV of the pyramid. Next R is found by making CR = CQ, and so on till we have got the last vertex—in this case S. The fact that AS must equal AP gives a convenient check.
2. The plane α whose section we have to determine has its horizontal trace given perpendicular to the axis, and its vertical trace makes the given angle with the axis. This determines it. To find the section of the pyramid by this plane there are two methods applicable: we find the sections of the plane either with the faces or with the edges of the pyramid. We use the latter.
As the plane α is perpendicular to the vertical plane, the trace α″ contains the projection of every figure in it; the points E2, F2, G2, H2where this trace cuts the elevations of the edges will therefore be the elevations of the points where the edges cut α. From these we find the plans E1, F1, G1, H1, and by joining them the plan of the section. If from E1, F1lines be drawn perpendicular to AB, these will determine the points E, F on the developed face in which the plane α cuts it; hence also the line EF. Similarly on the other faces. Of course BF must be the same length on BP and on BQ. If the plane α be rabatted to the plan, we get the real shape of the section as shown in the figure in EFGH. This is done easily by making F0F = OF2, &c. If the figure representing the development of the pyramid, or better a copy of it, is cut out, and if the lateral faces be bent along the lines AB, BC, &c., we get a model of the pyramid with the section marked on its faces. This may be placed on its plan ABCD and the plane of elevation bent about the axis x. The pyramid stands then in front of its elevations. If next the plane α with a hole cut out representing the true section be bent along the trace α′ till its edge coincides with α″, the edges of the hole ought to coincide with the lines EF, FG, &c., on the faces.
§ 18. Polyhedra like the pyramid in § 17 are represented by the projections of their edges and vertices. But solids bounded by curved surfaces, or surfaces themselves, cannot be thus represented.
For a surface we may use, as in case of the plane, its traces—that is, the curves in which it cuts the planes of projection. We may also project points and curves on the surface. A ray cuts the surface generally in more than one point; hence it will happen that some of the rays touch the surface, if two of these points coincide. The points of contact of these rays will form some curve on the surface, and this will appear from the centre of projection as the boundary of the surface or of part of the surface. The outlines of all surfaces of solids which we see about us are formed by the points at which rays through our eye touch the surface. The projections of these contours are therefore best adapted to give an idea of the shape of a surface.
Thus the tangents drawn from any finite centre to a sphere form a right circular cone, and this will be cut by any plane in a conic. It is often called the projection of a sphere, but it is better called the contour-line of the sphere, as it is the boundary of the projections of all points on the sphere.
If the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle, and if the projection is, as in our case, orthographic, then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere. We get such a circle in the plan and another in the elevation, their centres being plan and elevation of the centre of the sphere.
Similarly the rays touching a cone of the second order will lie in two planes which pass through the vertex of the cone, the contour-line of the projection of the cone consists therefore of two lines meeting in the projection of the vertex. These may, however, be invisible if no real tangent rays can be drawn from the centre of projection; and this happens when the ray projecting the centre of the vertex lies within the cone. In this case the traces of the cone are of importance. Thus in representing a cone of revolution with a vertical axis we get in the plan a circular trace of the surface whose centre is the plan of the vertex of the cone, and in the elevation the contour, consisting of a pair of lines intersecting in the elevation of the vertex of the cone. The circle in the plan and the pair of lines in the elevation do not determine the surface, for an infinite number of surfaces might be conceived which pass through the circular trace and touch two planes through the contour lines in the vertical plane. The surface becomes only completely defined if we write down to the figure that it shall represent a cone. The same holds for allsurfaces. Even a plane is fully represented by its traces only under the silent understanding that the traces are those of a plane.
§ 19. Some of the simpler problems connected with the representation of surfaces are the determination of plane sections and of the curves of intersection of two such surfaces. The former is constantly used in nearly all problems concerning surfaces. Its solution depends of course on the nature of the surface.
To determine the curve of intersection of two surfaces, we take a plane and determine its section with each of the two surfaces, rabatting this plane if necessary. This gives two curves which lie in the same plane and whose intersections will give us points on both surfaces. It must here be remembered that two curves in space do not necessarily intersect, hence that the points in which their projections intersect are not necessarily the projections of points common to the two curves. This will, however, be the case if the two curves lie in a common plane. By taking then a number of plane sections of the surfaces we can get as many points on their curve of intersection as we like. These planes have, of course, to be selected in such a way that the sections are curves as simple as the case permits of, and such that they can be easily and accurately drawn. Thus when possible the sections should be straight lines or circles. This not only saves time in drawing but determines all points on the sections, and therefore also the points where the two curves meet, with equal accuracy.
§ 20. We give a few examples how these sections have to be selected. A cone is cut by every plane through the vertex in lines, and if it is a cone of revolution by planes perpendicular to the axis in circles.
A cylinder is cut by every plane parallel to the axis in lines, and if it is a cylinder of revolution by planes perpendicular to the axis in circles.
A sphere is cut by every plane in a circle.
Hence in case of two cones situated anywhere in space we take sections through both vertices. These will cut both cones in lines. Similarly in case of two cylinders we may take sections parallel to the axis of both. In case of a sphere and a cone of revolution with vertical axis, horizontal sections will cut both surfaces in circles whose plans are circles and whose elevations are lines, whilst vertical sections through the vertex of the cone cut the latter in lines and the sphere in circles. To avoid drawing the projections of these circles, which would in general be ellipses, we rabatt the plane and then draw the circles in their real shape. And so on in other cases.
Special attention should in all cases be paid to those points in which the tangents to the projection of the curve of intersection are parallel or perpendicular to the axis x, or where these projections touch the contour of one of the surfaces.
(O. H.)
IV. Analytical Geometry
1. In the namegeometrythere is a lasting record that the science had its origin in the knowledge that two distances may be compared by measurement, and in the idea that measurement must be effectual in the dissociation of different directions as well as in the comparison of distances in the same direction. The distance from an observer’s eye of an object seen would be specified as soon as it was ascertained that a rod, straight to the eye and of length taken as known, could be given the direction of the line of vision, and had to be moved along it a certain number of times through lengths equal to its own in order to reach the object from the eye. Moreover, if a field had for two of its boundaries lines straight to the eye, one running from south to north and the other from west to east, the position of a point in the field would be specified if the rod, when directed west, had to be shifted from the point one observed number of times westward to meet the former boundary, and also, when directed south, had to be shifted another observed number of times southward to meet the latter. Comparison by measurement, the beginning of geometry, involved counting, the basis of arithmetic; and the science of number was marked out from the first as of geometrical importance.
But the arithmetic of the ancients was inadequate as a science of number. Though a length might be recognized as known when measurement certified that it was so many times a standard length, it was not every length which could be thus specified in terms of the same standard length, even by an arithmetic enriched with the notion of fractional number. The idea of possible incommensurability of lengths was introduced into Europe by Pythagoras; and the corresponding idea of irrationality of number was absent from a crude arithmetic, while there were great practical difficulties in the way of its introduction. Hence perhaps it arose that, till comparatively modern times, appeal to arithmetical aid in geometrical reasoning was in all possible ways restrained. Geometry figured rather as the helper of the more difficult science of arithmetic.
2. It was reserved for algebra to remove the disabilities of arithmetic, and to restore the earliest ideas of the land-measurer to the position of controlling ideas in geometrical investigation. This unified science of pure number made comparatively little headway in the hands of the ancients, but began to receive due attention shortly after the revival of learning. It expresses whole classes of arithmetical facts in single statements, gives to arithmetical laws the form of equations involving symbols which may mean any known or sought numbers, and provides processes which enable us to analyse the information given by an equation and derive from that equation other equations, which express laws that are in effect consequences or causes of a law started from, but differ greatly from it in form. Above all, for present purposes, it deals not only with integral and fractional number, but with number regarded as capable of continuous growth, just as distance is capable of continuous growth. The difficulty of the arithmetical expression of irrational number, a difficulty considered by the modern school of analysts to have been at length surmounted (seeFunction), is not vital to it. It can call the ratio of the diagonal of a square to a side, for instance, or that of the circumference of a circle to a diameter, a number, and let a or x denote that number, just as properly as it may allow either letter to denote any rational number which may be greater or less than the ratio in question by a difference less than any minute one we choose to assign.
Counting only, and not the counting of objects, is of the essence of arithmetic, and of algebra. But it is lawful to count objects, and in particular to count equal lengths by measure. The widened idea is that even when a or x is an irrational number we may speak of a or x unit lengths by measure. We may give concrete interpretation to an algebraical equation by allowing its terms all to mean numbers of times the same unit length, or the same unit area, or &c. and in any equation lawfully derived from the first by algebraical processes we may do the same. Descartes in hisGéométrie(1637) was the first to systematize the application of this principle to the inherent first notions of geometry; and the methods which he instituted have become the most potent methods of all in geometrical research. It is hardly too much to say that, when known facts as to a geometrical figure have once been expressed in algebraical terms, all strictly consequential facts as to the figure can be deduced by almost mechanical processes. Some may well be unexpected consequences; and in obtaining those of which there has been suggestion beforehand the often bewildering labour of constant attention to the figure is obviated. These are the methods of what is now calledanalytical, or sometimesalgebraical,geometry.
3. The modern use of the term “analytical” in geometry has obscured, but not made obsolete, an earlier use, one as old as Plato. There is nothing algebraical in this analysis, as distinguished from synthesis, of the Greeks, and of the expositors of pure geometry. It has reference to an order of ideas in demonstration, or, more frequently, in discovering means to effect the geometrical construction of a figure with an assigned special property. We have to suppose hypothetically that the construction has been performed, drawing a rough figure which exhibits it as nearly as is practicable. We then analyse or critically examine the figure, treated as correct, and ascertain other properties which it can only possess in association with the one in question. Presently one of these properties will often be found which is of such a character that the construction of a figure possessing it is simple. The means of effecting synthetically a construction such as was desired is thus brought to light by what Plato calledanalysis. Or again, being asked to prove a theorem A, we ascertain that it must be true if another theorem B is, that B must be if C is, and so on, thus eventually finding that the theorem A is the consequence, through a chain of intermediaries, of a theorem Z of which the establishment is easy. This geometrical analysis is not the subject of the present article; but in the reasoning from form to form of an equation or systemof equations, with the object of basing the algebraical proof of a geometrical fact on other facts of a more obvious character, the same logic is utilized, and the name “analytical geometry” is thus in part explained.
4. In algebra real positive number was alone at first dealt with, and in geometry actual signless distance. But in algebra it became of importance to say that every equation of the first degree has a root, and the notion of negative number was introduced. The negative unit had to be defined as what can be added to the positive unit and produce the sum zero. The corresponding notion was readily at hand in geometry, where it was clear that a unit distance can be measured to the left or down from the farther end of a unit distance already measured to the right or up from a point O, with the result of reaching O again. Thus, to give full interpretation in geometry to the algebraically negative, it was only necessary to associate distinctness of sign with oppositeness of direction. Later it was discovered that algebraical reasoning would be much facilitated, and that conclusions as to the real would retain all their soundness, if a pair of imaginary units ±√−1 of what might be called number were allowed to be contemplated, the pair being defined, though not separately, by the two properties of having the real sum 0 and the real product 1. Only in these two real combinations do they enter in conclusions as to the real. An advantage gained was that every quadratic equation, and not some quadratics only, could be spoken of as having two roots. These admissions of new units into algebra were final, as it admitted of proof that all equations of degrees higher than two have the full numbers of roots possible for their respective degrees in any case, and that every root has a value included in the form a + b √−1, with a, b, real. The corresponding enrichment could be given to geometry, with corresponding advantages and the same absence of danger, and this was done. On a line of measurement of distance we contemplate as existing, not only an infinite continuum of points at real distances from an origin of measurement O, but a doubly infinite continuum of points, all but the singly infinite continuum of real ones imaginary, and imaginary in conjugate pairs, a conjugate pair being at imaginary distances from O, which have a real arithmetic and a real geometric mean. To geometry enriched with this conception all algebra has its application.
5. Actual geometry is one, two or three-dimensional,i.e.lineal, plane or solid. In one-dimensional geometry positions and measurements in a single line only are admitted. Now descriptive constructions for points in a line are impossible without going out of the line. It has therefore been held that there is a sense in which no science of geometry strictly confined to one dimension exists. But an algebra of one variable can be applied to the study of distances along a line measured from a chosen point on it, so that the idea of construction as distinct from measurement is not essential to a one-dimensional geometry aided by algebra. In geometry of two dimensions, the flat of the land-measurer, the passage from one point O to any other point, can be effected by two successive marches, one east or west and one north or south, and, as will be seen, an algebra of two variables suffices for geometrical exploitation. In geometry of three dimensions, that of space, any point can be reached from a chosen one by three marches, one east or west, one north or south, and one up or down; and we shall see that an algebra of three variables is all that is necessary. With three dimensions actual geometry stops; but algebra can supply any number of variables. Four or more variables have been used in ways analogous to those in which one, two and three variables are used for the purposes of one, two and three-dimensional geometry, and the results have been expressed in quasi-geometrical language on the supposition that a higher space can be conceived of, though not realized, in which four independent directions exist, such that no succession of marches along three of them can effect the same displacement of a point as a march along the fourth; and similarly for higher numbers than four. Thus analytical, though not actual, geometries exist for four and more dimensions. They are in fact algebras furnished with nomenclature of a geometrical cast, suggested by convenient forms of expression which actual geometry has, in return for benefits received, conferred on algebras of one, two and three variables.
We will confine ourselves to the dimensions of actual geometry, and will devote no space to the one-dimensional, except incidentally as existing within the two-dimensional. The analytical method will now be explained for the cases of two and three dimensions in succession. The form of it originated by Descartes, and thence known as Cartesian, will alone be considered in much detail.