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# Ta101 2014 l8

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### Transcript of "Ta101 2014 l8"

1. 1. TA 101 Think and Analyze Anupam Saxena Associate Professor Mechanical Engineering Compliant and Robotic Systems Lab Indian Institute of Technology Kanpur
2. 2. Axonometric Views Isometric views
3. 3. Organization of Lectures and Laboratory Assignments Topic Week (No. of Lectures) Lab Intro and Basic Constructions Week 1 (2) Orthographic Projections Week 2 (2) Lab 1 Orthographic Projections Week 3 (2) Lab 2 Isometric Projections Week 4 (2) Lab 3 Missing Views Week 5 (2) Lab 4 Sectional and Assembly Week 6 (2) Lab 5 Oblique Projections Week 7 (2) Lab 6 Perspective Projections Week 8 (2) Lab 7 Lines and Planes Week 9 (2) Lab 8 Lines and Planes Week 10 (2) Lab 9 Auxiliary Projections Week 11 (2) Lab 10 Intersection of lines/planes/solids Week 12 (2) Lab 11 Intersection and Development Week 13 (2) Lab 12 TOTAL 26 12
4. 4. ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION Unit cube x z   O (1, 0, 0) P R (0, 0, 1) Q(0, 1, 0) Rotation about ‘z’, then about ‘x’ and then projection on the y-z plane G = PRxRz = cos(𝛼) −sin(𝛼) 0 0 0 0 sin 𝛼 sin(𝛽) cos 𝛼 sin(𝛽) cos 𝛽 |OP|= cos(𝛼) 2 + 𝑠𝑖𝑛2(𝛼)𝑠𝑖𝑛2(𝛽) |OQ|= 𝑠𝑖𝑛2 𝛼 + 𝑐𝑜𝑠2(𝛼)𝑠𝑖𝑛2(𝛽) |OR|= cos(𝛽) 2 𝑐𝑜𝑠2 𝛼 𝑐𝑜𝑠2 𝛽 + 𝑐𝑜𝑠2 (𝛽) = 1 −𝑐𝑜𝑠2 𝛼 𝑐𝑜𝑠2 𝛽 + 2𝑐𝑜𝑠2 (𝛽) = 1 𝑐𝑜𝑠2(𝛽) = 2/3 𝛽 =  35.26 𝛼 =  45 |OR| = |OQ| and |OR| = |OP| gives
5. 5. Unit cube x z   O (1, 0, 0) P R (0, 0, 1) Q(0, 1, 0) Rotation about ‘z’, then about ‘x’ and then projection on the y-z plane G = PRxRz = 1 2 − 1 2 0 0 0 0 1 6 1 6 2 3 𝑐𝑜𝑠2(𝛽) = 2/3 𝛽 =  35.26 𝛼 =  45 XY Z O 30 deg30 deg foreshortening happens by cos(𝛽) = 2 3 ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION
6. 6. Unit cube x z   O (1, 0, 0) P R (0, 0, 1) Q(0, 1, 0) Rotation about ‘z’, then about ‘x’ and then projection on the y-z plane G = PRxRz = 1 2 − 1 2 0 0 0 0 1 6 1 6 2 3 𝑐𝑜𝑠2(𝛽) = 2/3 𝛽 =  35.26 𝛼 =  45 XY Z O 30 deg30 deg foreshortening happens by cos(𝛽) = 2 3 ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION
7. 7. ANUPAM SAXENA TA101 LECTURE VII ISOMETRIC VIEW 20 40 30 45 40 10
8. 8. ANUPAM SAXENA TA101 LECTURE VII ISOMETRIC VIEW: SCALE 1:0.67 20 40 30 45 40 10 Not Unique: Choose the orientation NO HIDDEN LINES: ALL LINES ARE SOLID BE CAREFUL WITH VISIBLE/INVISIBLE LINES ISOMETRIC VIEWS ARE “REALISTIC” WE SHOW WHAT WE SEE !
9. 9. TA: What happens to the shape of a circle on the x-y plane ? G = PRxRz = 1 2 − 1 2 0 0 0 0 1 6 1 6 2 3 x y X = a + rcos Y = b + rsin  Z = 0 𝑥 𝑛 𝑦 𝑛 𝑧 𝑛 = 1 2 − 1 2 0 0 0 0 1 6 1 6 2 3 𝑎 + 𝑟𝑐𝑜𝑠𝜃 𝑏 + 𝑟𝑠𝑖𝑛𝜃 0 2𝑥 𝑛 − 𝑎 + 𝑏 = 𝑟𝑐𝑜𝑠 𝜃 − 𝑟𝑠𝑖𝑛 𝜃 6𝑧 𝑛 − 𝑎 − 𝑏 = 𝑟𝑐𝑜𝑠 𝜃 + 𝑟𝑠𝑖𝑛 𝜃 2𝑥 𝑛 − 𝑎 + 𝑏 2 + 6𝑧 𝑛 − 𝑎 − 𝑏 2 = 2𝑟2 A𝑥2 + 2𝐻𝑥𝑧 + 𝐵𝑧2 + 2𝐹𝑥 + 2Gz + C = 0 A= 2; 𝐻 = 0; 𝐵 = 6; 𝐻2 − 𝐴𝐵 = −12 < 0 Ellipse on the ‘isometric Xn-Zn’ plane ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION x z O P R Q
10. 10. TA: What happens to the shape of a circle on the y-z plane ? G = PRxRz = 1 2 − 1 2 0 0 0 0 1 6 1 6 2 3 y z X = 0 Y = a + rcos Z = b + rsin  𝑥 𝑛 𝑦 𝑛 𝑧 𝑛 = 1 2 − 1 2 0 0 0 0 1 6 1 6 2 3 0 𝑎 + 𝑟𝑐𝑜𝑠𝜃 𝑏 + 𝑟𝑠𝑖𝑛𝜃 − 2𝑥 𝑛 − 𝑎 = 𝑟𝑐𝑜𝑠 𝜃 6𝑧 𝑛 − 𝑎 − 2𝑏 = 𝑟𝑐𝑜𝑠 𝜃 + 2𝑟𝑠𝑖𝑛 𝜃 2𝑥 𝑛 + 𝑎 2 + 3 2 𝑧 𝑛 − 𝑏 + 𝑥 𝑛 2 2 = 𝑟2 A𝑥2 + 2𝐻𝑥𝑧 + 𝐵𝑧2 + 2𝐹𝑥 + 2Gz + C = 0 A= 2 + 1 2 ; 𝐻 = 3 2 ; 𝐵 = 3 2 ; 𝐻2 − 𝐴𝐵 = 3 4 − 5 2 3 2 = −3 < 0 Ellipse on the ‘isometric Xn-Zn’ plane ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION x z O P R Q
11. 11. TA: What happens to the shape of a circle on the x-z plane ? G = PRxRz = 1 2 − 1 2 0 0 0 0 1 6 1 6 2 3 x z X = a + rcos Y = 0 Z = b + rsin  𝑥 𝑛 𝑦 𝑛 𝑧 𝑛 = 1 2 − 1 2 0 0 0 0 1 6 1 6 2 3 𝑎 + 𝑟𝑐𝑜𝑠𝜃 0 𝑏 + 𝑟𝑠𝑖𝑛𝜃 2𝑥 𝑛 − 𝑎 = 𝑟𝑐𝑜𝑠 𝜃 6𝑧 𝑛 − 𝑎 − 2𝑏 = 𝑟𝑐𝑜𝑠 𝜃 + 2𝑟𝑠𝑖𝑛 𝜃 2𝑥 𝑛 + 𝑎 2 + 3 2 𝑧 𝑛 − 𝑏 − 𝑥 𝑛 2 2 = 𝑟2 A𝑥2 + 2𝐻𝑥𝑧 + 𝐵𝑧2 + 2𝐹𝑥 + 2Gz + C = 0 A= 2 + 1 2 ; 𝐻 = − 3 2 ; 𝐵 = 3 2 ; 𝐻2 − 𝐴𝐵 = 3 4 − 5 2 3 2 = −3 < 0 Ellipse on the ‘isometric Xn-Zn’ plane ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION x z O P R Q
12. 12. 60 60 40 10 40 30 40 30 ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION: EXAMPLE WITH ELLIPSES
13. 13. ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION: SCALE 1:1 60 60 40 10 40 30 40 30 10 X Y Z Z Y X
14. 14. ANUPAM SAXENA TA101 LECTURE VIII ISOMETRIC PROJECTION: SCALE 1:1 60 60 40 10 40 30 40 30 10 TA: TRICK ! SAVES A LOT OF TIME X Y Z Z Y X
15. 15. keep Thinking and Analyzing Until next time...
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