1. Lecture 2: The straight line By Dr. Samah Mohamed Mabrouk www.smmabrouk.faculty.zu.edu.eg By Dr. Samah Mohamed Mabrouk www.smmabrouk.faculty.zu.edu.eg

2. The straight line 1- Representation of a straight line. 2- Special positions of a straight line. 3- True length of a straight line. 6- Examples 4-The mutual position of two straight lines in the space 5-Traces of a str. Line

3. The straight line 1- Representation of a straight line. 1- Two points. 2- A point and direction C B d A m m

4. Represent the three projections of a straight line m{A(2,2,5)and B(6,5,1)} Z +, y - Z -, y + x +, y - x-, y + B1B1 A1A1 B2B2 A2A2 m3m3 B3B3 A3A3 m2m2 m1m1 m 1 is the horizontal projction m 2 is the vertical projction m 3 is the profile(side) projction

5. Find the missing projections Z +, y - Z -, y + x +, y - x-, y + B1B1 A1A1 B2B2 A2A2 m3m3 B3B3 A3A3 m2m2 m1m1 L : A 1 L : A 3 L : B 2 L : B 1 مساقط النقطه تقع على مساقط الخط

6. X +,y -- X --,y+ z +,y -- Z --,y + Special positions of a str. line  1- m //  1 (Horizontal line) m2m2 m3m3 m 1 = T.L الخط الموازى للمستوى مسقطه على هذا المستوى = طوله الحقيقى (True length)

7. X +,y -- X --,y+ z +,y -- Z --,y + 2- m //  2 frontal line m1m1 m3m3 m 2 = T.L    ( m,  1 )   ( m,  2 )   ( m,  3 )

8. X +,y -- X --,y+ z +,y -- Z --,y + 3- m //  3 profile line m2m2 m1m1 m 3 = T.L  

9. X +,y -- X --,y+ z +,y -- Z --,y + 4- m   1 m1m1 m 3 = T.L m 2 = T.L الخط العمودى على مستوى مسقطه على هذا المستوى = نقطة وعلى باقى المستويات = طول حقيقى

10. X +,y -- X --,y+ z +,y -- Z --,y + 5- m   2 m2m2 m 3 = T.L m 1 = T.L

11. X +,y -- X --,y+ z +,y -- Z --,y + 6- m   3 m3m3 m 2 = T.L m 1 = T.L

12. True length of a straight line Z +, y - Z -, y + x +, y - x-, y + B1B1 A1A1 B2B2 A2A2 m3m3 B3B3 A3A3 m2m2 m1m1 zBzB  z AB zAzA T.L. of m  B1B1  A1A1  z AB

13. True length of a straight line Z +, y - Z -, y + x +, y - x-, y + B1B1 A1A1 B2B2 A2A2 m3m3 B3B3 A3A3 m2m2 m1m1  y AB T.L. of m yAyA yByB B2B2  A2A2  y AB 

14. True length of a straight line Z +, y - Z -, y + x +, y - x-, y + B1B1 A1A1 B2B2 A2A2 m3m3 B3B3 A3A3 m2m2 m1m1  x AB T.L. of m xAxA xBxB B3B3  A3A3  x AB 

15. B1B1  T.L. of m A1A1  z AB B2B2  T.L. of m A2A2  y AB B3B3  T.L. of m A3A3  x AB Triangles of solution

16. Example(1): Find the true length of a straight line AB where A(2, 2, 5) and B(6, 3, 1) and then get the point C which at distance 2 from A Z +, y - Z -, y + x +, y - x-, y + B1B1 A1A1 B2B2 A2A2  y AB T.L. of AB 2 C2C2 [C] C1C1  y AB [B]

17. The mutual position of two straight lines in the space x 12 m1m1 n1n1 m2m2 n2n2 m  n  m 1  n 1, m 2  n 2, m 3  n 3 n2n2 m2m2 m1m1 n1n1 Q1Q1 Q2Q2 2- الخطان متقاطعان 1- الخطان متوازيان

18. The mutual position of two straight lines in the space x 12 n2n2 m2m2 m1m1 n1n1 Q1Q1 P2P2 2- الخطان متخالفان

19. Z +, y - Z -, y + x +, y - x-, y + H1H1 H2H2 m3m3 H3H3 m2m2 m1m1 Traces of a str. Line 1- Horizontal trace H are the points of intersection with principal plane H 2 and H 3  x-axis H   1 and H  m  z H =0, m   1 = H

20. Z +, y - Z -, y + x +, y - x-, y + V1V1 V2V2 m3m3 V3V3 m2m2 m1m1 Traces of a str. line L:V 2 L:V 3 2- Vertical trace V V 1  x, V 3  z V   2 and V  m  y V =0, m   2 = V

21. Z +, y - Z -, y + x +, y - x-, y + m3m3 L:S 3 m1m1 Traces of a str. line S1S1 S2S2 S3S3 3- Side trace S S 1 and S 2  z-axis  x S =0, m   3 = S S   3 and S  m m2m2

22. Z +, y - Z -, y + x +, y - x-, y + H1H1 V1V1 V2V2 H2H2 m3m3 H3H3 V3V3 m2m2 m1m1 Traces of a str. line S1S1 S2S2 S3S3

23. Example(2): Given A 2 B 2 the vertical projection of a segment AB and the horizontal projection A 1 of the point A and <(AB,  3 )=45 ,find the remaining projections of AB Z +, y - Z -, y + x +, y - x-, y + B1B1 A1A1 B2B2 A2A2  x AB yAyA A3A3 B3B3 T.L. of AB A3A3 B\3B\3 B3B3 yAyA L :B 3 L :A 3 L :B 1 45 

24. Example(3): Given the straight line m in general position and a point A lying on m required, find a point M on m at a distance 4cm from A x +, y - x-, y + B1B1 A1A1 B2B2 A2A2  z AB m2m2 [M] m1m1 4  z AB M1M1 T.L. m M2M2

25. Example(4): Represent the two projections of an equilateral triangle ABC if its side AB is given and C(?, 1, ?) x +, y - B1B1 A1A1 B2B2 A2A2  y AB C1C1  y BC L :C 1 1  z AB  y AC  z AB T.L. of AB = AC T.L. of AB = AC = BC C2C2  T.L. of AB =AC A2A2  y AC C2C2 T.L. of AB =AC B2B2  y BC L :C 2 C2C2

26. X 12 Example (5): A cube ABCDA \ B \ C \ D \ is given by it’s vertex A \, it’s base ABCD   1 and it’s vertex C =(?, 5, ?), represent the cube by it’s two projections A2\A2\ A 1 \ =A 1 L: A 2, B 2, C 2, D 2 A2A2 T.L of square T.L of AC L: C 1 B2B2 D2D2 C2C2 B2\B2\ D2\D2\ C2\C2\ C 1 \ =C 1 D 1 \ =D 1 B 1 \ =B 1 B\B\ C\C\ C A\A\ A D\D\ D B 11 A 1 B 1 C 1 D 1 T.S A 2 B 2 C 2 D 2  x- axis

27. X +,y -- X --,y+  h2h2 h 1 = T.L f1f1 f 2 = T.L  h //  1 f //  2