GEOMETRY AND GRAPHICS EXAMPLES AND EXERCISES. Preface

Transcription

1 GEOMETRY AND GRAPHICS EXAMPLES AND EXERCISES Preface This textbook is intended for students of subjects Constructive Geometry and Computer Graphics at the Faculty of Mechanical Engineering at Czech Technical University in Prague. The textbook is organized in the following way: Part I Geometry consists of many examples on their graphic representation. Because Constructive Geometry makes great demands on the student s spatial imagination, the threedimensional models of important geometric problems, which are constructed in this textbook in Monge projection, are available on as Geometry and Graphics 3D Supplement. Geometry and Graphics 3D Supplement is considered to be as an inseparable part of this textbook. Part II Graphics is devoted to the mathematical modelling of free form curves and surfaces which are used in many CAD/CAM systems nowadays. This part can be useful when modelling free form curves and surfaces in Ferguson, Bézier, Coons, B-spline and NURBS representation. Prague 2013 doc. Ing. Ivana Linkeová, Ph.D. 3

2 GEOMETRY AND GRAPHICS EXAMPLES AND EXERCISES Part I GEOMETRY Contents 1 Monge projection Oblique projection Orthogonal axonometry, technical isometry Linear perspective Kinematic geometry Surfaces of revolution Helix Helicoidal surfaces Envelope surfaces Developable surfaces Transition developable surfaces Solutions

3 1 MONGE PROJECTION 1.1 Construct the true length of straight line segment AB. a) AB π b) AB ν z 2 z 2 A 2 A 2 B 2 B 2 B 1 A 1 B 1 A 1 y 1 1 y c) AB in general position d) AB in general position z 2 z 2 B 2 B 2 A 2 A 2 B 1 B 1 A 1 A 1 y 1 y 1 5

4 1 MONGE PROJECTION 1.2 Construct adjacent views of equilateral triangle ABC and square ABCD lying in projecting plane ρ. a) ABC ρ π b) ABC ρ ν n 2 ρ ρ 2 A 2 B 2 A 1 ρ 1 B 1 p 1 ρ c) ABCD ρ π n 2 ρ d) ABCD ρ ν ρ 2 A 2 B 2 A 1 ρ 1 B 1 p 1 ρ 6

5 1 MONGE PROJECTION 1.3 a) Construct adjacent views of a circle k inscribed into square ABCD lying in projecting plane ρ π. b) Construct adjacent views of a circle k circumscribed around regular hexagon ABCDEF lying in projecting plane ρ ν. ρ 1 A 2 a) B 2 n 2 ρ ρ 2 A 1 B E 1 b) ρ p 1 7

6 1 MONGE PROJECTION 1.4 Construct adjacent views of a solid, which consists of hemisphere Σ (S, r = SR) and cube ABCDA B C D. Axis o of the solid, centre S of the hemisphere, front view R 2 of the point R of the hemisphere and front view A 2 of vertex A of the cube are given. R 2 S 2 A 2 S 1 o 1 8

7 2 OBLIQUE PROJECTION 2.1 In oblique projection (ω = 135, q = 3:4), construct oblique view of the cube ABCDA B C D with the base ABCD in plane (x, y). The side AB is given. Three circles k, k and k are inscribed into faces BCC B, CDD C and A B C D, respectively. Construct oblique views of circles k, k and k. z O =A=A =A' 1 1 y x B=B =B' 1 1 9

8 2 OBLIQUE PROJECTION 2.2 In oblique projection (ω = 120, q = 2:3), construct oblique view of a cone of revolution with the base in the plane parallel with plane (x, y). Radius of the base (r = 45 mm), centre S of the base and vertex V = O of the cone are given. z S O =V=V =S 1 1 y x 10

9 2 OBLIQUE PROJECTION 2.3 In oblique projection (ω = 135, q = 1:2), construct oblique view of a solid. Front view and right side view of the solid are given. Measure the dimension, which you will need. z 2 z 3 x 2 y 2 y 3 x 3 z O y x 11

10 2 OBLIQUE PROJECTION 2.4 In military perspective, construct oblique view of a solid. Top view and front view of the solid are given. Measure the dimension, which you will need. z 2 x 2 y 2 x 1 z 1 y 1 z O x y 12

11 3 ORTHOGONAL AXONOMETRY, TECHNICAL ISOMETRY 3.1 In technical isometry, construct a sphere with the centre S = O, and the radius r = 50 mm. Construct the points of intersection K, L, M of the sphere with x, y and z axis. Construct the curve of intersection e of the sphere and a) horizontal plane of projection π. b) frontal plane of projection ν. a) z S = O b) z S = O 13

12 3 ORTHOGONAL AXONOMETRY, TECHNICAL ISOMETRY 3.2 In technical isometry, construct a cube ABCDA B C D with the base ABCD in the plane (x, y). Vertex A = O is given. Vertex B lies on positive part of x axis, length of the cube side is 80 mm. Three circles k, k and k are inscribed into faces A B C D, BCC B and CDD C, respectively. Construct technical isometry of circles k, k and k. z O =A 14

13 3 ORTHOGONAL AXONOMETRY, TECHNICAL ISOMETRY 3.3 In technical isometry, construct a detail which is given by technical drawing. Axis of the detail is identical with x axis, centre S of the sphere lies at original O of coordinate system. 60 S z O = S 15

14 3 ORTHOGONAL AXONOMETRY, TECHNICAL ISOMETRY 3.4 In technical isometry, construct a detail which is given by technical drawing. Axis of the detail is identical with y axis, centre S of the sphere lies at original O of coordinate system S 50 SPHERE R S 30 z O x y 16

15 4 LINEAR PERSPECTIVE 4.1 In linear perspective (h, z, H, D d ), construct a squared mesh lying in the ground plane. The rotated top view of the squared mesh is given. a) A" 1 B" 1 C" 1 D" 1 H h A' 1 B' 1 C' 1 D' 1 A 1 B 1 C 1 D 1 z b) A" 1 D d B" 1 A' 1 C" 1 B' 1 H h A 1 C' 1 B 1 z C 1 D d 17

16 4 LINEAR PERSPECTIVE 4.2 In linear perspective (h, z, H, D d ), construct a right squared prism ABCDA B C D. The base ABCD of the prism lies in the ground plane. The prism is placed behind the perspective plane of projection. The rotated top view of the base ABCD and altitude of the prism v = 120 mm are given. C 1 H D 1 B 1 A 1 D d h z 18

17 4 LINEAR PERSPECTIVE 4.3 In linear perspective (h, z, H, D d ), construct a right squared pyramid ABCDV. The base ABCD of the pyramid lies in the ground plane. Pyramid is placed behind the perspective plane of projection. The rotated top view of the base ABCD and altitude of the pyramid v = 120 mm are given. C 1 H D 1 B 1 A 1 D d h z 19

18 5 KINEMATIC GEOMETRY 5.1 The motion is given by trajectories A, B of points A, B. a) Construct new positions of given points C, D, E. Draw the trajectory C, D, E. b) Construct new positions of given point C. Construct the tangent lines to the trajectory C at all new positions of point C. Draw C. a) τ B A 0 τ A C 0 D 0 E 0 B 0 b) C 0 τ B A 0 τ A B 0 20

19 5 KINEMATIC GEOMETRY 5.2 The motion is given by trajectories A, B of points A, B. a) Construct new positions of the given straight line segment AB. Construct points of contact of the straight line segment AB and its envelope (AB) at all new positions. Draw the envelope (AB). b) Construct fixed centrode p and moving centrode h at the given instant. a) τ B A 0 τ A B 0 b) τ B A 0 τ A B 0 21

20 5 KINEMATIC GEOMETRY 5.3 The motion is given by trajectory A of point A and point envelope (b) of straight line b. Construct new positions of the given circle k with centre C. Construct the tangent lines to the trajectory C at all new positions of point C. Draw the trajectory C. Construct the points of contact of the circle k and its envelope (k) at all new positions of circle k. Draw the envelope (k). Construct fixed centrode p. S τa C 0 A 0 (b) k 0 τ A 22

21 5 KINEMATIC GEOMETRY 5.4 The motion is given by envelopes (a), (b) of straight lines a, b. Construct new positions of the given point C. Construct the tangent lines to the trajectory C at all new positions of the given point C. Draw the trajectory C of point C. Construct fixed centrode p and moving centrode h at the given instant. C 0 (a) P 0 (b) 23

22 5 KINEMATIC GEOMETRY 5.5 Cycloidal motion is given by centrodes p, h. a) Construct new positions of the given point A. Construct the tangent lines to the trajectory A at all new positions of the given point A. Draw the trajectory A of the point A. b) Construct new positions of the given straight line segment a. Construct the points of contact of the straight line segment a and its envelope (a) at all new positions of straight line a. Draw the envelope (a). a) b) p p h 0 h 0 a 0 A 0 24

23 5 KINEMATIC GEOMETRY 5.6 Involute motion is given by centrodes p, h. Construct new positions of the given point C. Construct the tangent lines to the trajectory C at all new positions of given point C. Draw the trajectory C of point C. Construct new positions of the given straight line a. Construct the points of contact of the straight line a and its envelope (a) at all new positions of straight line a. Draw the envelope (a). p h 0 a 0 C 0 25

24 5 KINEMATIC GEOMETRY 5.7 Hypocycloidal motion is given by centrodes p, h (r p : r h = 2:1) a) Construct new positions of the given point C. Construct the tangent lines at all new positions of point C. Draw the trajectory C. b) Construct new positions of the given straight line AB. Construct the points of contact of the straight line AB and its envelope (AB) at all new positions of straight line AB. Draw the envelope (AB). a) p C 0 A 0 h 0 B 0 b) p A 0 h 0 B 0 26

25 5 KINEMATIC GEOMETRY 5.8 Epicycloidal motion is given by centrodes p, h (r p : r h = 3:1). Construct new positions of the given points A, B, C. Draw the trajectories A, B, C. A 0 h 0 R15 B 0 C 0 p R45 27

26 6 SURFACES OF REVOLUTION 6.1 A surface of revolution κ is given by its generating curve k and axis of revolution o. In Monge projection, construct the missing view of the point A lying on the surface κ. At the point A construct the tangent plane τ of the surface κ and the normal line n to the surface κ. a) k 2 k 1 o 1 A 2 b) S 2 k 2 k 1 S 1 o 1 A 1 28

27 6 SURFACES OF REVOLUTION 6.2 The surface of revolution κ is given by its generating curve k and axis of revolution o. In the given half-plane, construct the principal half-meridian m of the surface κ. Use Monge projection. k 2 k 1 o 1 σ 1 29

28 6 SURFACES OF REVOLUTION 6.3 Two surfaces of revolution are given by their half-meridian m, m and axis of revolution o, o. Construct the intersection of these surfaces. Indicate the visibility. Use Monge projection. a) m' 2 m' 1 o' 2 o' 1 o 1 m 2 m 1 b) m' 2 m' 1 o' 2 o' 1 o 1 m 2 m 1 30

29 6 SURFACES OF REVOLUTION 6.4 A conical surface of revolution (principal half-meridian m, axis o ) and a torus (principal half-meridian m, axis o ) are given. In Monge projection, construct the intersection of the torus and the conical surface of revolution. Indicate the visibility. o' 2 m 2 m' 2 m 1 o 1 o' 1 m' 1 31

30 6 SURFACES OF REVOLUTION 6.5 A conical surfaces of revolution κ and a cylindrical surface of revolution κ are given. In Monge projection, construct the intersection of the surfaces κ and κ. Indicate the visibility. κ 2 κ' 2 o' 2 κ 1 κ' 1 o 1 o' 1 32

31 6 SURFACES OF REVOLUTION 6.6 A conical surface of revolution κ and a cylindrical surface of revolution κ are given. In Monge projection, construct the intersection of the surfaces κ and κ. Indicate the visibility. a) κ'2 κ 2 o' 2 κ 1 κ' 1 o 1 o' 1 b) κ'2 κ 2 o' 2 κ 1 κ' 1 o' 1 o 1 33

32 7 HELIX 7.1 A cylinder of revolution κ (bases k, k', axis o is identical with z-axis) and the point A lying on the base k are given. In military perspective, construct one and half thread of right-handed helix h generated by screw motion of the given point A. The axis of the screw motion is identical with the axis o of cylinder of revolution κ. The lead is to be equal to 120 mm. k' z = o κ o 1 k A = A 1 x y 34

33 7 HELIX 7.2 In Monge projection, construct one thread of right-handed helix h generated by screw motion of the given point A. Construct point of intersection B of the helix h and plane σ and point of intersection C of the helix h and plane. Determine how many solutions the problem has. Axis of the screw motion o π, the lead of screw motion v = 120 mm, plane σ π and plane ν are given. ρ 2 A 2 A 1 o 1 σ 1 35

34 7 HELIX 7.3 A cylinder of revolution κ (bases k, k', axis o is identical with z-axis) and the point A lying on the base k are given. In military perspective, construct one thread of left-handed helix h (axis o π, left-handed, generating point A, parameter of screw motion v 0 ) and one thread of the surface generated by tangent lines of the helix h. Construct only the part of surface between the helix and horizontal plane of projection. k' z = o κ o 1 v 0 A = A 1 k y x 36

35 7 HELIX 7.4 In Monge projection, construct point of intersection B of helix h (generating point A, axis o π, parameter v 0 ) and plane σ. Construct tangent line of helix h at the given point A and at the point of intersection B both. a) Orientation of the screw motion is left-handed. b) Orientation of the screw motion is right-handed. a) o 1 v 0 σ 1 A 2 A 1 b) o 1 v 0 σ 2 A 2 A 1 37

36 8 HELICOIDAL SURFACES 8.1 In Monge projection, construct the missing view of a point A lying on helicoidal surface κ (generating curve k, axis o π, parameter of screw motion v 0 ) and tangent plane τ of helicoidal surface κ at the point A. a) Orientation of the screw motion is left-handed. b) Orientation of the screw motion is right-handed. Choose one solution only. a) k 1 A 1 v 0 o 1 k2 b) k2 S2 v 0 k 1 S1 o 1 A 2 38

37 8 HELICOIDAL SURFACES 8.2 In Monge projection, construct the missing view of point A lying on generating curve k (circle with centre S) of helicoidal surface κ (generating curve k, axis o π, parameter of right-handed screw motion v 0 ). Construct normal section k of helicoidal surface κ by plane σ ν. Construct point A lying on normal section k corresponding to the given point A. Construct the tangent plane τ of helicoidal surface κ at point A. σ 2 v 0 S 2 k 2 o 1 A 1 S 1 k 1 39

38 8 HELICOIDAL SURFACES 8.3 In Monge projection, construct the normal section k of helicoidal surface κ (generating curve k, axis o π, parameter of left-handed screw motion v 0 ). A 2 S 2 C 2 k 2 B 2 v 0 σ 1 o 1 A 1 k 1 B = S 1 1 C 1 40

39 8 HELICOIDAL SURFACES 8.4 In Monge projection, construct the principle half-meridian of helicoidal surface κ (generating curve k, axis o π, parameter of right-handed screw motion v 0 ) in given half-plane σ. v 0 S 2 k 2 o 1 σ 1 S 1 k 1 41

40 8 HELICOIDAL SURFACES 8.5 In Monge projection, construct the normal section k of helicoidal surface κ (generating curve k, axis o π, parameter of left-handed screw motion v 0 ). A 2 S 2 C 2 k 2 σ 2 B 2 v 0 k 1 o 1 A 1 B = S 1 1 C 1 42

41 8 HELICOIDAL SURFACES 8.6 In Monge projection, construct the principal meridian k, k of helicoidal surface κ (generating curve k, axis o π, parameter of right-handed screw motion v 0 ) in given plane σ. v 0 k 2 Q 2 P 2 P 1 k 1 Q 1 o 1 σ 1 43

42 8 HELICOIDAL SURFACES 8.7 In Monge projection, construct the normal section k of helicoidal surface κ (generating curve k, axis o π, parameter of right-handed screw motion v 0 ). P 2 σ2 k 2 Q 2 v 0 Q 1 o 1 P 1 44

43 9 ENVELOPE SURFACES 9.1 In Monge projection, construct the characteristic curve k of envelope surface κ which is generated by rotation of plane σ around axis o π. Construct top view and front view of envelope surface κ. a) Rotated plane σ π (given by its top view σ 1 and frontal trace n 2 σ ) is parallel to axis o. b) Rotated plane σ (given by triangle ABC) intersects axis o. a) o 1 σ 1 n 2 σ b) A 2 A1 C 2 C 1 o 1 B 2 B 1 45

44 9 ENVELOPE SURFACES 9.2 In Monge projection, construct the characteristic curve k of envelope surface κ generated by translation of sphere Σ along path p. Construct top view and front view of envelope surface κ between horizontal plane of projection π and frontal plane of projection ν. Σ 2 S 2 p 2 p 1 Σ 1 S 1 46

45 9 ENVELOPE SURFACES 9.3 In Monge projection, construct the characteristic curve k of envelope surface κ generated by rotating sphere Σ around the axis o π. Construct principle half-meridian m in given plane σ. Construct top view and front view of envelope surface κ. Σ 2 S 2 o 1 σ 1 Σ 1 S 1 47

46 9 ENVELOPE SURFACES 9.4 In Monge projection, construct the characteristic curve k of envelope surface κ generated by a screw motion of the plane σ. Screw motion is given by axis o π and parameter v 0. a) Plane σ π (given by its top view σ 1 and frontal trace n 2 σ ) is parallel to axis o. Orientation of the screw motion is left-handed. b) Plane σ (given by its from view σ 2 and frontal trace p 1 σ ) intersects axis o. Orientation of the screw motion is right-handed. a) σ 1 o 1 v 0 n 2 σ b) σ 2 o 1 v 0 p 1 σ 48

47 9 ENVELOPE SURFACES 9.5 In Monge projection, construct the characteristic curve k of envelope surface κ generated by screwing sphere Σ. Screw motion is given by axis o π and parameter v 0 of right handed screw motion. Generating sphere Σ is placed in basic position (S 2 ). Σ 2 S 2 v 0 Σ 1 S 1 o 1 σ 1 49

48 9 ENVELOPE SURFACES 9.6 In Monge projection, construct the characteristic curve k of envelope surface κ generated by rotation of surface of revolution Σ around axis o π. Construct principal half-meridian m in the given plane σ. Σ 2 S 2 S' 2 o 1 σ 1 S 1 S' 1 Σ 1 50

49 9 ENVELOPE SURFACES 9.7 In Monge projection, construct the characteristic curve k of envelope surface κ generated by rotation of cylindrical surface of revolution Σ around axis o π. Construct the principal half-meridian m in the given plane σ. S' 2 Σ 2 S 2 o 1 σ 1 Σ 1 S 1 S' 1 51

50 10 DEVELOPABLE SURFACES 10.1 The right circular cylinder κ (directrix k, axis o π) and section plane σ ν are given. Develop the surface of cylinder κ and curve of intersection e = κ σ. k' 2 κ 2 e 2 k 2 κ 1 k 1 =k'=e 1 1 o 1 σ 2 σ p 1 52

51 10 DEVELOPABLE SURFACES 10.2 The intersection of two right circular cylinders Σ (directrix k, axis o π) and Σ (directrix k, axis o ν) degenerates. Develop the surfaces of cylinders Σ and Σ. Σ' 2 k' 2 S2 k 2 C 2 Σ 2 k ' 2 D 2 o' 2 S' 2 A 2 o' 1 o 1 =S S' 1 S ' 1 1 S ' 2 B 2 A 0 B 0 C 0 D 0 53

52 10 DEVELOPABLE SURFACES 10.3 A right circular cone κ (directrix k, vertex V) and section plane σ ν are given. Develop the surface of cylinder κ and curve of intersection e = κ σ. κ 1 k 1 k 2 κ 2 e 2 V 2 V 1 σ 2 σ p 1 54

53 10 DEVELOPABLE SURFACES 10.4 An oblique circular cylinder κ (diretrix k, centre line ST) and point A are given.. Construct top and front views of the cylinder κ. Construct the normal section e of the cylinder κ with the plane which passes through point A. Develop the surface of cylinder κ and normal section e. B 2 B 2 k 1 S 2 A =A 1 2 T 2 S 1 T 1 B 0 55

54 10 DEVELOPABLE SURFACES 10.5 An oblique circular cone κ (diretrix k, vertex V) is given. Construct top view and front view of the cone κ. Develop the surface of the cone κ. V 2 k 2 A 0 S 2 A 2 S 1 A1 V1 k 1 V 0 56

55 11 TRANSITION DEVELOPABLE SURFACES 11.1 A polyline ABC and circle k are given. In Monge projection, construct the smooth developable transition surface between polyline ABC and circle k. Develop the transition surface. A 2 B 2 k 2 S 2 k 1 S 1 A 1 B 1 A 0 B 0 57

56 11 TRANSITION DEVELOPABLE SURFACES 11.2 A polyline ABCD and circle k are given. In Monge projection, construct the smooth developable transition surface between polyline ABCD and circle k. Develop the transition surface. In military perspective, construct oblique view of transition the surface. k 2 S 2=E 2 z 2 A =B 2 2 C =D 2 2 S 1 A 1 D 1 k 1 E 1 B 1 C 1 y 1 A 0 B 0 z O x y 58

57 11 TRANSITION DEVELOPABLE SURFACES 11.3 A rectangle ABCD and circle k are given. In Monge projection, construct the smooth developable transition surface between rectangle ABCD and circle k. Develop the transition surface. E 2 S 2 k 2 z 2 A 2=B 2 C =D A 1 k 1 D E 1 S 1 A 0 B 1 C 1 y 1 B 0 59

58 11 TRANSITION DEVELOPABLE SURFACES 11.4 A hexagon m and circle k are given. In Monge projection, construct the smooth developable transition surface between hexagon m and circle k. Develop the transition surface. In military perspective, construct oblique view of the transition surface. S 2= C2 k 2 z 2 m 2 A 2 B 2 k 1 S 1 m 1 C 2 y 1 A 1 B 1 z O A 0 B 0 x y 60