The Main Menu اPrevious اPrevious Next 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 B1B1 B* 2 D* 2 A* 2 C* 2 x 12 B* 1 A* 1 D* 1 C* 1 C1C1 D1D1 A1A1 C2C2 B2B2 D2D2 A2A2

The Main Menu اPrevious اPrevious Next B1B1 C2C2 A2A2 D2D2 B* 2 D* 2 A* 2 C* 2 B* 1 A* 1 D* 1 C* 1 C1C1 D1D1 A 1 B2B2 x 12 x 13 C3C3 A3A3 D3D3 B3B3 C* 3 A* 3 B* 3 D* 3 x 35 C 5 = C* 5 A 5 = A* 5 B 5 = B* 5 D 5 = D* 5 // * *

The Main Menu اPrevious اPrevious A* 3 B* 3 D* 3 // C* 3 x 35 C 5 = C* 5 A 5 = A* 5 B 5 = B* 5 D 5 = D* 5 * * // x 13 C3C3 A3A3 D3D3 B3B3 B* B C* C D* D // * * A* A B*.. A* A

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

A AB A B C C D D S S AB is a dimeterdimeterdimeterdimeter diameter diameter Parallel Parallel tothe Plane of Projection.. CD is a diam. normal normal to AB. AB.

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

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. 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. S S r S x r

S S S S S r r r r x O

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. v h s s x S S

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

2.Parallelism : 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 3. Intersection The point of intersection of a straight line and a plane. The straight line of intersection of two different planes m // //

m m r M R R = m r = 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 In figure the straight line k is lying in the plane The straight line m is parallel to the straight line k

v h k m m m x k SECOND PROBLEM : Parallelism of two planes THEOREM : a b b a 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.

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 i) The plane is given by two intersecting str. Lines M M a a b b a and b a and b ii) The plane is given by two parallel str. Lines a &b

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

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

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

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

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

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

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

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. a b a b c d d c v v r r S S R R

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

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

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