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Orthogonal Projection and Position Problems in Geometry

Learn about successive auxiliary projection, orthographic circles, and position problems in geometry.

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Orthogonal Projection and Position Problems in Geometry

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  1. A*2 B*2 D*2 C*2 D2 B2 C2 x12 B1 B*1 A*1 A1 C*1 C1 D1 D*1 Next The Main Menu اPrevious Example (6) Using the successive auxiliary projection, construct the development of the given regular oblique tetragonal prism ABCD A*B*C*D* whose base ABCD is a square A2

  2. A*2 B*2 D*2 C*2 B1 A2 D2 C2 B2 x12 B*1 A*1 A 1 C*1 x13 A3 C3 D3 B3 C1 x35 D1 D*1 A*3 B*3 D*3 C*3 C5 = C*5 // D5 = D*5 * * // A5 = A*5 Next The Main Menu اPrevious B5 = B*5

  3. A B // A* C * D B* // A * . C* . D* A* The Main Menu اPrevious A3 D3 B3 C3 x13 x35 D*3 B*3 C*3 A*3 C5 = C*5 // D5 = D*5 * * // A5 = A*5 B5 = B*5 B* A*

  4. THE CIRCLE • The orthogonal projection of a circle : B r S r C D A A D C S r s B

  5. AB is a diameter Parallel t the o C Plane of . dimeter Projection . S CD is a diam . A normal B A to AB . D D B A S C

  6. REMARK M { {{{{{{{{{{{{{{ b To find the length of the semi minor axis if the major axis and a point M on the ellipse are given a a r S = A = B r Example 1 x Represent a circle lying in a plane perpendicular with V.P. if its centre and its radius are given . find S S r

  7. EXA MPLE 2 Represent a circle lying in a plane perpendicular with S.P. ( i.e. parallel to the x-axis ) if its centre and its radius are given . r S S r x S

  8. r S S r r x O r S S S

  9. EXAMPLE 3 Construct a circle lying in a plane (-7,8,6) ,its centre s (1,4,?) and its rsdius is of length 3.5 cms . s S v x s S h

  10. CHAPTER 8 POSITION PROBLEMS The position problems deal with : 1 . INCIDENCE A point lying on a straight line. A point lying in a plane. A straight line lying ln a plane . M m m M M

  11. 2 .Parallelism : A straight liine is parallel to another straight line , a straight line is parallel to a plane, a plane is parallel to a given plane . m // m 3 . Intersection The point of intersection of a straight line and a plane . The straight line of intersection of two different planes

  12. m m r M R R = m r = FIRST PROBLEM : Parallelism of a straight line and a plane THEOREM: A straight line m is parallel to a given plane iff m is parallel to a straight line lying in the given plane. In figure the straight line k is lying in the plane The straight line m is parallel to the straight line k

  13. m k v x k m h SECOND PROBLEM : Parallelism of two planes a THEOREM : b A plane is said to be parallel to another plane iff the plane contains two intersecting straight lines a and b, each of them is parallel to the plane . b a m

  14. Given a plane and a point M out side it. It is required to construct a plane passing through M and parallel to the given plane The plane is given by two intersecting str. Lines i) M b a and b a a b M ii) The plane is given by two parallel str. Lines a &b

  15. M b a b c a M x x a c a b M b M iii) The plane is given by its traces v v v M M x x M h M h h M

  16. v M M v v x x h M h M h The plane is perpendicular with H.P. M v v v M x h h h M M The plane is perpendicular with V. P. v v v s v M M M x M h h M The plane is parallel to x-axis h

  17. THIRD PROBLEM: INTERSECTION OF TWO PLANES V V v v v v r r r r h h r H h h Some special cases H i - One of the two planes is vertical: v v r x h h = r

  18. Ii- One of the two planes is perpendicular with V . P. v v=r v x r h v v r x r h h iii- One of the two planes is parallel to x- axis

  19. Iv- The two planes are parallel to the x- axis v v r r s s x X o r h h v- One of the two planes is horizontal r v = v v r r h

  20. v Vi- One of the two planes is frontal r r v r r X h Vii- Two traces do not intersect h v R We use an auxiliary frontal or horizontal plane to find one point of intersection. v v v v v v v r R h h H

  21. R v v r v R v H S v v R r h r h r h h R H H viii) Both vertical and S h Horizontal traces do not intersect .

  22. EXAMPLE Construct the line of intersection of a plane given by two intersecting str. Lines a&b with a plane given by two parallel str. Lines c& d . d a c R v r v S b R b r S d c a

  23. m EXAMPLE x Given a straight line m m v m i. To pass a vertical plane through the straight line m x m v h ii. To pass a plane normal to V.P. through m m x h m

  24. FOURTH PROBLEM: POINT OF INTERSECTION OF A STRAIGHT LINE m AND A PLANE v m m i) The plane is in a special position : v R 1- The plane is horizontal R m m 2. The plane is frontal R mm x m h R

  25. R m v 3. The plane is vertical x m R h m v 4. The plane is normal to V.P. x m h

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