Parallel projection Special type of mapping of 3D object on a 2D medium (technical drawing, display)

Size: px

Start display at page:

Download "Parallel projection Special type of mapping of 3D object on a 2D medium (technical drawing, display)"

Roderick Barnett
6 years ago
Views:

1 Parallel projection Special type of mapping of 3D object on a 2D medium (technical drawing, display) Parallel projection 1. Direction of projection s, plane of projection p 1

2 Parallel projection (parallel view) 1. Direction of projection s, plane of projection p 2. Point A, A p Parallel projection (parallel view) a 1. Direction of projection s, plane of projection p 2. Point A, A p 3. Point A =a p 2

3 Parallel projection (parallel view) a 1. Direction of projection s, plane of projection p 2. Point A, A p 3. Point A =a p, A p, AA s 4. Triangle DABC Parallel projection (parallel view) a b c 1. Direction of projection s, plane of projection p 2. Point A, A p 3. Bod A =a p 4. DABC 5. DA B C 3

4 Central projection (perspective) 1. Centre of projection S, plane of projection p Central projection (perspective) 1. Centre of projection S, plane of projection p, S p 2. Point A, A p 4

5 Central projection (perspective) 1. Centre of projection S, plane of projection p, S p 2. Point A, A p 3. Intersection point A =SA p Central projection (perspective) 1. Centre of projection S, plane of projection p, S p 2. Point A, A p 3. Intersection point A =SA p 4. DABC 5

6 Central projection (perspective) 1. Centre of projection S, plane of projection p, S p 2. Point A, A p 3. Intersection point A =SA p 4. DABC 5. DA B C Parallel view of a line 1. Direction s, plane of projection p, s not parallel to p 2. q inclined straight line 6

7 Parallel view of a line 1. Direction s, plane of projection p, s not parallel to p 2. q inclined straight line 3. Intersection point P (P = q p) piercing point Точка P - cлед прямой q Parallel view of a line a 1. Direction s, plane of projection p, s not parallel to p 2. q inclined straight line 3. Intersection point P (P = q p) 4. Points A,B on the line q 5. A =a p 7

8 Parallel view of a line 1. Direction s, plane of projection p, s not parallel to p 2. q inclined straight line 3. Intersection point P (P = q p) 4. Points A,B on the line q 5. A =a p 6. B =b p Parallel view of a line 1. Direction s, plane of projection p, s not parallel to p 2. q inclined straight line 3. Intersection point P (P = q p) 4. Points A,B on the line q 5. A =a p 6. B =b p 7. q1=a p 8

9 Parallel view of a line 1. Direction s, plane of projection p, s not parallel to p 2. q inclined straight line 3. Intersection point P (P = q p) 4. Points A,B on the line q 5. A =a p 6. B =b p 7. q1=a p 8. q1=a B or q1=a P Paralel view of a straight line is straight line or point. Measurement is not invariant property of parallel projection. Parallel view of two parallel lines 1. p q 2. P piercing point p (P =p p) 3. Q piercing point q (Q =q p) 9

10 Parallel view of two parallel lines 1. p q 2. P piercing point p (P =p p) 3. Q piercing point q (Q =q p) 4. C =c p 5. p1=c P Parallel view of two parallel lines 1. p q 2. P piercing point p (P =p p) 3. Q piercing point q (Q =q p) 4. C =c p 5. p1=c P 6. A =a p 7. q1=a Q Parallelism is invariant property of parallel projection. The equal parallel segments are projected to the two equal parallel segments. 10

11 A line parallel to the projection plane 1. q p 2. Piercing point Q at infinity (ideal point, Бесконечно удалённая точка) A line parallel to the projection plane a b 1. q p 2. Q ideal piercing point 3. A =a p 4. B =b p 5. q1=a B A line parallel to the projection plane and it s projection are parallel q q1. 11

12 A plane (ПЛОСКОСТЬ) parallel to the projection plane 1. Plane p p A plane parallel to the projection plane U 1. Plane p p 2. DABC p 12

13 A plane parallel to the projection plane U 1. Plane p p 2. DABC p 3. DA B C p DABC DA B C U Shape U in a plane p p and it s parallel view U are congruent. Determination of a plane by contained lines s 1. a,p, p plane of projection a inclined plane 13

14 Determination of a plane by contained lines 1. a,p, p plane of projection a inclined plane 2. p a =a p trace of the plane a След плоскости Determination of a plane by contained lines 1. a,p, p plane of projection a inclined plane 2. p a =a p trace of the plane a 3. p i...planes parallel to p 14

15 Determination of a plane by contained lines 1. a,p, p plane of projection a inclined plane 2. p a =a p trace of the plane a 3. p i...planes parallel to p 4. p i a Determination of a plane by contained lines 1. a,p, p plane of projection a inclined plane 2. p a =a p trace of the plane a 3. p i...planes parallel to p 4. h a i=p i a...contour lines (h a p a ) Главные линии плоскости 15

16 Determination of a plane by contained lines 1. a,p, p plane of projection a inclined plane 2. p a =a p trace of the plane a 3. p i...planes parallel to p 4. h a i=p i a...contour lines (h a p a ) 5. s a steepest line in a (s a p a ) Линии наибольшего наклона Dividing ratio of three different collinear points + B C (ABC )= AC BC A Internal division of a line segment (ABP )<0 Point P lies inside the segment AB. External division of a line segment (ABP )>0 Point P lies outside segment AB. 16

17 Dividing ratio of three different collinear points Dividing ratio of three different collinear points 17

18 Dividing ratio of three different collinear points ( ABP) AP BP (A B P )=? Parallel projection preserves dividing ratio. (ABP)=(A B P ) Orthogonal projection - s p 1. s direction of projection, p plane of projection s p 2. q inclined line in general position 3. P = q p 4. line segment A,B on q 5. A =a p 6. B =b p 7. q1=a p 8. q1=a B and also q1=a P Edges that are inclined to a plane of projection appear as foreshortened lines. A B = AB cos a 18

19 Orthogonal projection of right angle p q Orthogonal projection of right angle p q, p p 1. p p 19

20 Orthogonal projection of right angle p q, p p 1. p p 2. q a p 3. <(p,q )=? Orthogonal projection of right angle p q, p p 1. p p 2. q a p 3. <(p,q )=? 4. p a?? 5. p s, p q p a 6. p p, p a q p 20

21 Orthogonal projection of right angle p q, p p Let perpendicular straight lines q, p are not projection lines and p is parallel to projection plane. Than the projections p, q are perpendicular. Only in that case Parallel projection preserves the right angle. Image of the circle 21

22 Parallel View of a Circle 1. Projectors of all points on the circle form a cylinder. 2. Circle which is not in edge view is projected as an ellipse intersection curve of the projecting cylindrical surface and drawing plane p. How to determine the ellipse? Conjugate diameters of an ellipse Ellipse is given by the major axis AB and minor axis CD. 1. Ellipse A,B,C,D 2. Line through points 3. Diameter MN 22

23 Conjugate diameters of an ellipse 1. Foci F1,F2 2. Lines F1N,F2N 3. Normal line n angle bisector of F1N, F2N 4. Tangent line t n Conjugate diameters of an ellipse Tangent lines at points M, n are parallel. 23

24 Conjugate diameters of an ellipse Diameter PQ t Conjugate diameters MN, PQ of an ellipse In the case of a circle, all diameters have the same length, tangent parallelogram becomes tangent square and conjugated diameters are always perpendicular. 24

25 Parallel View of a Circle 1. k a...circle Parallel View of a Circle 1. k a...circle 2. PQ, MN conjugated diameters 25

26 Parallel View of a Circle 1. k a...circle 2. PQ, MN conjugated diameters 3. M N,P Q projections of conjugated diameters MN,PQ Parallel View of a Circle 1. k a...circle 2. PQ, MN conjugated diameters 3. M N,P Q projections of conjugated diameters MN,PQ 4. ABCD ellipse Two mutually perpendicular diameters MN PQ of the circle are projected as conjugated diameters M N, P Q. 26

27 Exercises> point construction, tangent line and normal ELLIPSE, parallelogram method 27

28 Rytz s axis construction it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters MN, PQ. Rytz s axis construction it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters MN, PQ. 1. M S MS, M S = MS 28

29 Rytz s axis construction it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters MN, PQ. 1. M S MS, M S = MS 2. Point O is midpoint of M P Rytz s axis construction it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters MN, PQ. 1. M S MS, M S = MS 2. Point O is midpoint M P 3. k (O, r = OS ) 29

30 Rytz s axis construction it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters MN, PQ. 1. M S MS, M S = MS 2. Point O is midpoint of M P 3. k (O, r = OS ) 4. X,Y k M P Rytz s axis construction it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters MN, PQ. 1. M S MS, M S = MS 2. Point O is midpoint of M P 3. k (O, r = OS ) 4. X,Y k M P 5. Line of major axis XS Line of minor axisys 30

Rytz s axis construction it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters MN, PQ. 1. M S MS, M S = MS 2.

31 Rytz s axis construction it is possible to find the major and minor axis and the vertices of an ellipse, starting from two conjugated diameters MN, PQ. 1. M S MS, M S = MS 2. Point O is midpoint of M P 3. k (O, r = OS ) 4. X,Y k M P 5. Line of major axis XS Line of minor axisys 6. M X = PY = SA = a M Y = PX = SC = b 7. Ellipse ABCD Approximation of an ellipse by means of osculation circles 31

32 Approximation of a hyperbola by means of osculation circles Approximation of a parabola by means of osculation circles 32

ME 111: Engineering Drawing. Geometric Constructions

ME 111: Engineering Drawing. Geometric Constructions ME 111: Engineering Drawing Lecture 2 01-08-2011 Geometric Constructions Indian Institute of Technology Guwahati Guwahati 781039 Geometric Construction Construction of primitive geometric forms (points,