Descriptive Geometry 2

Transcription

1 By Pál Ledneczki Ph.D Spring Semester 1

2 Conic Sections Ellipse Parabola Hyperbola 2007 Spring Semester 2

3 Intersection of Cone and Plane: Ellipse The intersection of a cone of revolution and a plane is an ellipse if the plane (not passing through the vertex of the cone) intersects all generators. F 2 T 2 Dandelin spheres: spheres in a cone, tangent to the cone (along a circle) and also tangent to the plane of intersection. P T 1 F 1 Foci of ellipse: F 1 and F 2, points of contact of plane of intersection and the Dadelin spheres. P: piercing point of a generator, point of the curve of intersection. T 1 and T 2, points of contact of the generator and the Dandelin sphere. PF 1 = PT 1, PF 2 = PT 2 (tangents to sphere from an external point). PF 1 + PF 2 = PT 1 + PT 2 = T 1 T 2 = constant Spring Semester 3

4 Construction of Minor Axis a Let the plane of intersection a second projecting A plane that intersects all generators. b K L =C =D The endpoints of the major axis are A and B, the piercing points of the leftmost and rightmost generators respectively. B The midpoint L of AB is the centre of ellipse. C A K L B Horizontal auxiliary plane b passing through L intersects the cone in a circle with the centre of K. (K is a point of the axis of the cone). The endpoints of the minor axis C and D can be D found as the points of intersection of the circle in b and the reference line passing through L Spring Semester 4

5 Intersection of Cone and Plane: Parabola The intersection of a cone of revolution and a plane is a parabola if the plane (not passing through the vertex d of the cone) is parallel to one generator. Focus of parabola: F, point of contact of the plane of E F T T 2 intersection and the Dadelin sphere. Directrix of parabola: d, line of intersection of the plane of intersection and the plane of the circle on the Dandelin sphere. P: piercing point of a generator, point of the curve of intersection. T: point of contact of the generator and the P T 1 Dandeli sphere. PF = PT (tangents to sphere from an external point). PT =T 1 T 2 = PE, dist(p,f) = dist(p,d) Spring Semester 5

6 Construction of Point and Tangent a V Let the plane of intersection a second projecting plane be parallel to the rightmost generator. The vertex of the parabola is V. b Horizontal auxiliary plane b can be used to find P, P the second image of a point of the parabola. The tangent t at a point P is the line of intersection of the plane of intersection and the tangent plane of the surface at P. The first tracing point of the tangent N 1 is the point V of intersection of the first tracing line of the plane of intersection and the firs tracing line of the tangent t P n 11 n 12 plane at P, n 11 and n 12 respectively. t = N 1 P N Spring Semester 6

7 Intersection of Cone and Plane: Hyperbola The intersection of a cone of revolution and a T 2 plane is a hyperbola if the plane (not passing through the vertex of the cone) is parallel to two F 2 generators. Foci of hyperbola: F 1 and F 2, points of contact of plane of intersection and the Dadelin spheres. T 1 P F 1 P: piercing point of a generator, point of the curve of intersection. T 1 and T 2, points of contact of the generator and the Dandelin spheres. PF 1 = PT 1, PF 2 = PT 2 (tangents to sphere from an external point). PF 2 PF 1 = PT 2 - PT 1 = T 1 T 2 = constant Spring Semester 7

8 Construction of Asymptotes b K a A L B Let the plane of intersection a second projecting plane parallel to two generators, that means, parallel to the second projecting plane b through the vertex of the cone, which intersects the cone in two generators g 1 and g 2. g 1 = g 2 The endpoints of the traverse (real) axis are A and B, the piercing points of the two extreme generators. g 1 a 1 L B The midpoint L of AB is the centre of hyperbola. The asymptotic lines a 1 and a 2 are the lines of intersections of the tangent planes along the generators g 1 and g 2 and the plane of intersection a. g 2 a Spring Semester 8

9 Perspective Image of Circle horizon F C axis vanishing line 2007 Spring Semester 9

10 Construction of Perspective Image of Circle K The distance of the horizon horizon F and the (C) is equal to the distance of the axis and the vanishing line. axis (C) vanishing line The type of the perspective image of a circles depends on the number of common points with the vanishing line: no point in common; ellipse one point in common; parabola two points in common; hyperbola (K) The asymptotes of the hyperbola are the projections of the tangents at the vanishing points Spring Semester 10

11 Tangent Planes, Surface Normals n n A P A P n P A n A P n P A 2007 Spring Semester 11

12 Intersection of Cone and Cylinder Spring Semester 12

13 Intersection of Cone and Cylinder Spring Semester 13

14 Intersection of Cone and Cylinder Spring Semester 14

15 Methods for Construction of Point 1 Auxiliary plane: plane passing through the vertex of the cone and parallel to the axis of cylinder 2007 Spring Semester 15

16 Methods for Construction of Point 2 Auxiliary plane: first principal plane 2007 Spring Semester 16

17 Principal Points S 1 K 1 K 5 K 2 K 7 D S 1 K 1,2 S 2 K 6 S 2 K 8 K 4 K 1 K 3 K 2 D S 1 S 2 Double point: D Points in the plane of symmetry: S 1, S 2 Points On the outline of cylinder: K 1, K 2, K 3, K 4 Points On the outline of cone: K 5, K 6, K 7, K Spring Semester 17

18 Surfaces of Revolution: Ellipsoid Prolate ellipsoid Oblate ellipsoid Affinity Axis plane of affinity Capitol 2007 Spring Semester 18

19 Ellipsoid of Revolution in Orthogonal Axonometry Z Z IV anti-circle E F 1 O O iv Y X X IV =Y IV F Spring Semester 19

20 Intersection of Ellipsoid and Plane Spring Semester 20

21 Surfaces of Revolution: Paraboloid 2007 Spring Semester 21

22 Paraboloid; Shadows Find - the focus of parabola - tangent parallel to f - self-shadow - cast shadow - projected shadow inside 2007 Spring Semester 22

23 Torus z (axis of rotation) equatorial circle meridian circle throat circle y x 2007 Spring Semester 23

24 Classification of Toruses Spring Semester 24

25 Torus as Envelope of Spheres 2007 Spring Semester 25

26 Outline of Torus as Envelope of Circles 2007 Spring Semester 26

27 Outline of Torus 2007 Spring Semester 27

28 Classification of Points of Surface hyperbolical (yellow) elliptical (blue) parabolical (two circles) 2007 Spring Semester 28

29 Tangent plane at Hyperbolic Point 2007 Spring Semester 29

30 Villarceau Circles Antoine Joseph François Yvon Villarceau ( ) 2007 Spring Semester 30

31 Construction of Contour and Shadow 2007 Spring Semester 31

32 Ruled Surface led.html htm 2007 Spring Semester 32

33 Hyperboloid of One Sheet 03d984a4e4803ded Spring Semester 33

34 Hyperboloid of One Sheet 2007 Spring Semester 34

35 Hyperboloid of One Sheet, Surface of Revolution 2007 Spring Semester 35

36 Shadows on Hyperboloid of one Sheet 2007 Spring Semester 36

37 Hyperboloid of One Sheet, Shadow 1 f e The self-shadow outline is the hyperbola h in the plane of the selfshadow generators of asymptotic cone. h f e h The cast-shadow on the hyperboloid itself is an arc of ellipse e inside of the surface Spring Semester 37

38 Hyperboloid of One Sheet, Shadow 2 f The self-shadow outline is the ellipse e 1 inside of the surface. e 1 The cast-shadow on the hyperboloid itself is an arc of ellipse e 2 on the outher side of the surface. e 2 f e 1 e Spring Semester 38

39 Hyperboloid of One Sheet, Shadow 3 f O 2 The outline of the self_shadow is a pair of parallel generators g 1 and g 2. g 1 = g 2 O 2 ** f g 1 O 2 * g Spring Semester 39

40 Hyperboloid of One Sheet in Perspective 2007 Spring Semester 40

41 Hyperboloid in Military Axonometry 2007 Spring Semester 41

42 Construction of Self-shadow an Cast Shadow f V * : shadow of the center V that is the vetrex of the asymptotic cone T 2 t: tangent to the throat circle, chord of the base circle d: parallel and equal to t through the center of the base circle, diameter of the asymptotic cone f T 1 V A 2 B 2 Tangents to the base circle of the asymptotic cone from V* with the points of contact A 1 and A 2 : asymptotes of the castshadow outline hyperbola d t Line A 1 A 2 : first tracing line of the plane of self-shadow hyperbola; V A 1 and VA 2 : asymptotes of the self-shadow outline hyperbola T 2 * A 1 V * B 1 T 1 * 2007 Spring Semester 42

43 Construction of Projected Shadow f H The cast-shadow of H on the ground plane: H* Draw a generator g 1 passing through H Find B 1, the pedal point of the generator g 1 f g 1 g 2 H Find n 1 = B 1 H* first tracing line of the auxiliary plane HB 1 H* The point of intersection of the base circle and n 1 is B 2 B 1 B 2 n 1 l l H* The second generator g 2 lying in the plane HB 1 H* intersects the ray of light l at H, the lowest point of the ellipse, i.e. the outline of the projected shadow inside Spring Semester 43

44 Hyperboloid of One Sheet with Horizontal Axis 2007 Spring Semester 44

45 Hyperboloid of One Sheet, Intersection with Sphere 2007 Spring Semester 45

46 Ruled Surface: Hyperbolic Paraboloid Spring Semester 46

47 Hyperbolic Paraboloid: Construction Spring Semester 47

48 Saddle Surface Spring Semester 48

49 Axonometry and Perspective 2007 Spring Semester 49

50 Saddle Point an Contour A C A C K 1 K 2 S S D D B B 2007 Spring Semester 50

51 Shadow at Parallel Lighting A C D B C* A* 2007 Spring Semester 51

52 Intersection with Cylinder 2007 Spring Semester 52

53 Composite Surface 2007 Spring Semester 53

54 Intersection with Plane 2007 Spring Semester 54

55 Conoid Conoid Studio, Interior. Photo by Ezra Stoller (c)esto Courtesy of John Nakashima /Nakashima.htm Sagrada Familia Parish School. Despite it was merely a provisional building destined to be a school for the sons of the bricklayers working in the temple, it is regarded as one of the chief Gaudinian architectural works Spring Semester 55

56 Conoid Definition: ruled surface, set of lines (rulings), which are transversals of a straight line (directrix) and a curve (base curve), parallel to a plane (director plane). Parabola-conoid (axonometric sketch) Right circular conoid (perspective) Eric W. Weisstein. "Right Conoid." From MathWorld--A Wolfram Web Resource Spring Semester 56

57 Tangent Plane of the Right Circular Conoid at a Point r The intersections with a plane parallel to the base plane is ellipse (except the directrix). The tangent plane is determined by the ruling and the tangent of ellipse passing through the point. The tangent of ellipse is constructible in the projection, by means of affinity {a, P ] (P)} P Q e P r s l T t Q e n Spring Semester 57

58 Contour of Conoid in Axonometry Find contour point of a ruling r = e = n 1 Method: at a contour point, the tangent plane of the surface is a projecting plane, i. e. the ruling r, the tangent of ellipse e and the tracing line n 1 coincide: r = e = n 1 1. Chose a ruling r 2. Construct the tangent t of the base circle at the pedal point T of the ruling r K 3. Through the point of intersection of s and t, Q draw e parallel to e 4. The point of intersection of r and e, K is the projection of the contour point K r K T t Q e s 5. Elevate the point K to get K 2007 Spring Semester 58

59 Contour of Conoid in Perspective r = e = n 1 Find contour point of a ruling V F h Method: at a contour point, the tangent plane of the surface is a projecting plane, i. e. the ruling r, the tangent of ellipse e and the tracing line n 1 coincide: r = e = n 1 (t) (T) T t Q K e K s r a 1. Chose a ruling r 2. Construct the tangent t of the base circle at the pedal point T of the ruling r 3. Through the point of intersection of s and t, Q draw e parallel to e ( e 1 e = V h) 4. The point of intersection of r and e, K is the projection of the contour point K 5. Elevate the point K to get K C 2007 Spring Semester 59

60 Shadow of Conoid r Find shadow-contour point of a ruling Method: at a shadow-contour point, the tangent plane of the surface is a shadow-projecting plane, i. e. the shadow of ruling r*, the shadow of tangent of ellipse e* and the tracing line n 1 coincide: r* = e* = n 1 1. Chose a ruling r f 2. Construct the tangent t of the base circle at the pedal point T of the ruling r f K 3. Through the point of intersection of s and t, Q draw e parallel to e* e r K* Q K t T s 4. The point of intersection of r and e, K is the projection of the contour point K, a point of the self-shadow outline 5. Elevate the point K to get K 6. Project K to get K* r* = e* = n Spring Semester 60

61 Intersection of Conoid and Plane P P P 2007 Spring Semester 61

62 Intersection of Conoid and Tangent Plane 2007 Spring Semester 62

63 Developable Surfaces Developable surfaces can be unfolded onto the plane without stretching or tearing. This property makes them important for several applications in manufacturing. martaherford - Frank Gehry C:\Documents and Settings\Pali\Dokumentumok\palidok\Tematika\Tematikus\Developable\Rhino3DE Developable Surfaces.htm 2007 Spring Semester 63

64 Developable Surface Find the proper plug that fits into the three plug-holes. The conoid is non-developable, the cylinder and the cone are developable Spring Semester 64

65 Developable Surface 1. Divide one of the circles into equal parts z Find the corresponding points of the other circle such that the tangent plane should be the same along the generator O Use triangulation method for the approximate polyhedron O M 2 x 4. Develop the triangles one by one O 2 2 (2) T y 1 y 9 10 (O 2 ) (T y ) (10) 2007 Spring Semester 65

66 Helix Helical motion: rotation + translation Fold a right triangle around a cylinder 2007 Spring Semester 66

67 Left-handed, Right-handed Staircases While elevating, the rotation about the axis is clockwise: left-handed While elevating, the rotation about the axis is counterclockwise: right-handed x(t) = a sin(t) y(t) = a cos(t) z(t) = c t c > 0: right-handed c < 0: left-hande 2007 Spring Semester 67

68 Classification of Images of Helix Sine curve, circle Curve with cusp Stretched curve Curve with loop 2007 Spring Semester 68

69 Helix, Tangent, Director Cone P: half of the perimeter p: pitch a: radius of the cylinder g M t P c: height of director cone = parameter of helical motion M: vertex of director cone g M P g: generator of director cone t P t: tangent of helix M g P c p = 2 cπ t a P = 2 aπ 2007 Spring Semester 69

70 Helix with Cuspidal Point in Perspective T = t = N 1 =V t T: point of contact t: tangent at T N 1 : tracing point V t : vanishing point The tangent of the helix at cuspidal point is perspective projecting line: T = t = N 1 = V t Since t lies in a tangent plane of the cylinder of the helix, it lies on a contour generator of the cylinder (leftmost or rightmost) 2007 Spring Semester 70

71 Construction Helix with Cusp in Perspective 1 (N 1 ) Let the perspective system {a, h, ( C )} and the base circle of the helix be given. A right-handed helix starts from the rightmost point of the base circle. Find the parameter c (height of the director cone) such that the perspective image of the helix should have cusp in the first turning. F h (t ) N 1 =T t (O) (P o ) 1) T is the pedal point of the (T ) contour generator on the left. T O P o a ( C ) 2) The rotated (N 1 ) can be found on the tangent of the circle at (T ): dist((n 1 ), (T )) = arc((p o ), (T )). 3) T can be found by projecting (N 1 ) through (C), because, T = N 1. ( C ) 2007 Spring Semester 71

72 Construction Helix with Cusp in Perspective 2 V t axis g 2 t, g = V t O G: pedal point of the generator g, the point of intersection of g and the circle (ellipse in perspective) F h t (t ) N 1 =V t =T (O) g g (P o ) G M T O P o a g 2 t, g = V t O M: vertex of director cone c = dist(m,o) 2007 Spring Semester 72

73 Helicoid Definition: A ruled surface, which may be generated by a straight line moving such that every point of the line shall have a uniform motion in the direction of another fixed straight line (axis), and at the same time a uniform angular motion about it. Eric W. Weisstein. "Helicoid." From MathWorld--A Wolfram Web Resource Spring Semester 73

74 Tangent Plane of Helicoid The tangent plane is determined by the ruling and the tangent of helix passing through the point Spring Semester 74

75 Contour of Helicoid T Find contour point of a ruling Method: at a contour point, the tangent plane of the surface is a projecting plane, i. e. the ruling r and the tangent of helix e coincide: r = e 1. Chose a ruling r 2. Construct the tangent t of the helix at the endpoint T of the ruling segment r t N 1 K r = e t s T r K O Q e 3. Connect the tracing point N 1 of t and the origin O with the line s 4. Through the point of intersection of s and r = e, Q draw e parallel to t 5. The point of intersection of r and e, K is the projection of the contour point K 2007 Spring Semester 75