Descriptive Geometry 1 - epab.bme.hu

1 Descriptive Geometry 1 Descriptive Geometry 1by P l Ledneczki of contents1 Multi-view of the purposes of studying Descriptive Geometry : 1. Methodsand means for solving 3D geometrical construction problems. In this sense Descriptive Geometryis a branch of 2D representation of 3D technical object, basics of Technical Drawing, instrument in technical is Descriptive Geometry ? What is Descriptive Geometry ? One simply takes two planes at right angles to each other, one vertical and the other horizontal then projects the figure to be represented orthogonally on these planes, the projections of all edges and vertices being clearly indicated. The projection on the vertical plane is known as the elevation , the other projection is called the plan.

2 Finally, the vertical plane is folded te ee ato, t e ot e p oject o s ca ed tepaay,te etcapaesodedabout the line of intersection of the two planes until it also is horizontal. This puts on one flat sheet of paper what we ordinarily visualize in 3D . (A History of Mathematics by Carl B. Boyer, John Wiley & Sons, New York, 1991) Gaspard Monge(1746 1818)1746 1818)was sworn not to divulge the above method and for 15 years it Gaspard Monge(1746 1818)1746 1818)was sworn not to divulge the above method and for 15 years, it was a jealously guarded military secret. Only in 1794, he was allowed to teach it in public at the Ecole Normale, Paris where Lagrange was among the auditors. With his application of analysis to Geometry , this devil of a man will make himself immortal , exclaimed Lagrange.

3 R Parthasarathyh//iki di/ iki/Gd MDescriptive Geometry Descriptive Geometry 1 MethodologyMulti-view representation, auxiliary projectionsMultiview representation, auxiliary projectionsAxonometyPerspectivepTypes of problems Incidence and intersection problems shadow constructionsIncidence and intersection problems, shadow constructionsMetrical constructionsRepresentation of spatial elements, polyhedrons, circle, sphere, Representation of spatial elements, polyhedrons, circle, sphere, cylinder and coneDescriptive Geometry 13 IntroductionIn Descriptive Geometry 1We shall studyrepresentation of spatial elements and analyze their mutual positionsdetermine their angles and distancesrepresent pyramids, prisms, regular polyhedrons,construct the intersection of polyhedrons with line and plane, intersection of two polyhedronsconstruct shadowsconstruct shadowscast shadow, self-shadow, projected shadowthe principles of representation and solution of 3D geometrical pppgproblems in 2 DDescriptive Geometry 14 IntroductionSpatial elements, relations, notation2parallel non parallelz, perpendicularRelationspair of points: determine a distancepoint and line.

4 Lying onDlying on, passingthrough, coincidentpygnot lying on plane, distancepair of lines: coplanarintersecting anglesCparallel distancenon coplanarskew angle and distanceABsegmentA,BlABkpoint and plane: lying onnot lying on distanceline and plane: parallel distance|AB|= lline A,BB = k 1lintersection = [BCD]plane B,C,Dlintersecting anglepair of planes: parallel distanceintersecting angleDescriptive Geometry 15A not lying onIntroductionRepresentation of point 2P P P 21stquadrant2ndquadrantxd2 PxP 13rdquadrantx1,2d1d1 1P x1,23rdquadrant4thquadrantd2In which quadrant or image plane is the point located, why is it special?A B B C N N2d1X X D F G H =J K L A BC N1N1 N2 d2d1d1= d2X =X D E =E K H L x1,2 Descriptive Geometry 16112F =G KJ Multi-view representationAuxiliary projectionsP P d2x2,3 Side view, third imageChain of transformationsP BVd2x1,2 PVA =B C =D QVAVCVP Fourth image, linked to the first imagex1,2x4,5Q B C DVx1,2P PIVd1d1P Q A D CIVQIVDIVP x1,4x1,4 PIVAD AIVBIVD escriptive Geometry 17 AMulti-view representationRepresentation of Straight Lines, Relative PositionsC A B v P Q A B =ABh first principal line second principal line profile line first proj.

5 Line second proj. linep1 p2 =M =N L first coincidingpointssecond coincidingpointsD AA B v x1,2P Q x1,2P Q =PQhh D p1 =K =L p2 K M ABQ PQ=PQC p2 Intersecting parallel skew intersecting parallelN a b c d e g k l m =n a b c d e g k =l m n first coinciding second coinciding Descriptive Geometry 18dglinesglinesMulti-view representationPoint and Line 2P R l NN M p 2P SS l KNN K pPx1,2Q QRllML Px1,2Q QllKL 1P Q R l LN L M 1P Q S l LN L K N K N P R P S M L N L N Q l Q l p p S l K L M N*M*L K P Q R l P Q S l p p Descriptive Geometry 19S l K pR l M pMulti-view representationTracing Points of a Line 2l NN1x1,2l N2 N2N1 QllN2Nl N2 l N1 N1 Problems.

6 X1,2 11) find the tracing points of principal /profile lines2) determine lines by means of tracing points Descriptive Geometry 110 Multi-view representationRepresentation of PlanePair of intersecting lines a c d m =n Pair of parallel lines b c k l x1,2a b d k =l m n A Plane figures2 3 Ti liA B C 1 24 n2 Tracing linesx12n2A B C 1 2 3 4 n1spanned plane1,2n1 Descriptive Geometry 111A spanned planespanned planeslanted planeslanted planeMulti-view representationLine and Plane2 3 lying on (incident)l 2 3 P intersectingl [1234]1 2 3 4 l =g P g 1[1234]=P[]a 1 4 ll ll 1 2 4 g ab parallela 1 3 P a2[ab]a b a 14 l Descriptive Geometry 112b Multi-view representationIntersection of Two PlanesA 2 3 P 23A 2 3 4 P1 P2 P1= |12|1[ABC]2 AAB B C 1 2 4P2= |AC|1[1234]B P1P2C 3 P1 P2 BC4B 1 A 1 4 211 2 3 4 A B C Descriptive Geometry 11341A Multi-view representationTl f P i Of Sk Li Pi Thh Transversal of a Pair Of Skew Lines Passing Through a Given PointSketch and algorithm Solution in Monge sYour solutionAa P A abtBa P B Pb b a x1,2P B = [Pa] 1bt= |PB|orP a a b B t= [Pa] 1[Pb]b b2 [P] []a A Descriptive Geometry 114a2a , [Pa]= [aa]

7 Multi-view representationTransversal of a Pair Of Skew Lines Parallel to a Given Transversal of a Pair Of Skew Lines Parallel to a Given Direction Sketch and algorithm Solution in Monge sYour solutionAt A X a abtBXda AB d d* d d*b x1,2b X ab d d* a d* X, d*2dB = b1[ad*]tB t 2d*a t A B X b d Descriptive Geometry 115tB, t 2dMulti-view representationAuxiliary Projections on Special Purposes 1 True length of a segmentDistance of a pair of skew linesA D xB A B C CVx1,2B x1,2B C D DVdAV=BVx14 BIVA BIVA Cx4,5x1,4 AIVx1,4 AIVDIVCIVD escriptive Geometry 116 Multi-view representationAuxiliary Projections on Special Purposes 2 Application: find the distance dof the point Pand the plane [ABC].

8 Edge view of a plane: transformation of a plane in projecting plane 4 h, 4 1 4A B h P 4 2h hA AIVx1,2 BIVC 1Ax3,4 CIVPIVB h d 1h A Ax1,2x1,4P A C Descriptive Geometry 117 Multi-view representationAuxiliary Projections on Special Purposes 3B Construction of the true shape of a figure lying in a general planeGeneral plane 6fourth projecting plane 6fifth principal plane 4A C h 4 2h hA AIVx4,5x1,2 BIVX4,5 1AX1,4 AVCIVB h AIVBV 1h A Ax1,2 5x1,4A C AVCVD escriptive Geometry 118 Multi-view representationCast Shadow, Self-shadow, Projected Shadow Descriptive Geometry 119 Shadow constructionsShadow in Traditional Descriptive GeometryRiess, C.: Grundz ge der darstellenden Geometrie(Stuttgart : Verl.)

9 J. B. Metzler schen Buchhandlung, 1871)Application of Descriptive Geometry for Construction of Projected Shadow (plate X.) Romsauer Lajos: br zol geometria (Budapest : Franklin-T rsulat, 1929)htt //3 h /kti / d tb i / Descriptive Geometry 120 constructionsShadow in VisualisationDescriptive Geometry 121 Shadow constructionsShadows - BasicsAB NA*ff f ABA B**zBB**N2 Afff fAA*xAB**B*fA yA*ff Descriptive Geometry 122N1 Shadow constructionsShadow Properties1)Our constructions are restricted to parallel )We do not represent transition between dark and light )We usually construct three types of shadow: cast shadow on the ground or on the image 3)We usually construct three types of shadow: cast shadow on the ground or on the image planes, self-shadow (shade) and projected )Shadow of a point.

10 Piercing point of the ray of light passing through the point, in the surface (on ground plane, picture plane etc.)5)Shadow of a straight line: intersection of the plane passing through the line, parallel to the direction of lighting and the surface (screen).6)Shadow of a curve: the intersection of cylinder (whose generatrix is the curve, the generators are rays of light) withthe surface (screen)are rays of light) withthe surface (screen).7)Shadow-coinciding points: pair of distinct points, whose shadows )Alongside cast shadow the surface is in )In case of equal orientation of a triangle and its shadow, the face of triangle is )The cast shadow outline is the shadow of the self-shadow Geometry 123 Shadow constructionsCast Shadow, Projected Shadow323**AAshadow coinciding pointsAC11*44*CC* Descriptive Geometry 124 Shadow constructionsIntersection of Pyramid and Lineauxiliary intersectionparallel and similar to the baseFind the intersection of line and pyramidhauxiliary intersectionpassing through the apexlapexn1 Descriptive Geometry 125 intersection problemsIntersection of Polyhedron and Projecting Planedhf ld lhd1 C B D A Find the intersection of plane and polyhedron3 4 M 1 3 C D A 2 3 M Descriptive Geometry 126B intersection problemsIntersection of Polyhedron and Plane (auxiliary projection)