University of Warith AL-Anbiya’a College of Engineering Air-condition & Refrigeration Department Desprictive geometry II First stage Asst.Lec. Aalaa Mohammed AL-Husseini

plane The plane or plane surface is the surface where any two points are connected in a straight line, all the points of the straight line fall into it or belong to it. The plane is determined by three points that are not on a single straightness or a straight line and a point that is out of it or in two parallel or intersecting lines. As shown below:

The Traces of the plane: Are straight lines intersected with the three main projection levels. Horizontal traces of the plane: The intersection line of the known plane with the horizontal plane of projection, Symbolized by the symbol T. 2.Face traces of the plane: The intersection line of the known plane with the face plane of projection, Symbolized by the symbol T. 3.Side traces of the plane: The intersection line of the known plane with the side plane of projection, Symbolized by the symbol T.

Note: The traces of plane are straight lines and symbols are just conventions and that the presence of openings should not lead to confusion and can be symbolized by the symbol T1 , T2, T3 . General Theory: Theory 1: If two parallel levels are cut by a third level, they are cut by two parallel straight line as shown below. Theory 2: If two levels are cross with a third level by two parallel straight lines , the intersection straight line is parallel to these straight lines as shown below:

3. Theory 3: If two cross levels are cut by a plane parallel to their intersection line, it’s cut them by two parallel straight lines ,and the intersection of the levels straight line parallel to these straight lines as shown below. 4. Theory 4: If we have three planes and this planes don’t have parallel planes and not parallel to intersection line of other planes , so it’s connected by one point as shown below:

5. Theory 5: the horizontal and face traces of the plane either be parallel and parallel to the earth line or intersected by a point on the earth line or coincide on the earth line 6. Theory 6: the face and side traces of the plane either be parallel and parallel to the vertical folding line or intersected by a point on the vertical folding line or coincide on the vertical folding line

Result 1: If the horizontal traces of two cross levels are parallel , the intersection line of them is a horizontal straight line. Result 2: If the face traces of two cross levels are parallel , the intersection line of them is a face straight line.

Idiomatic code: We will mark the known level by it’s traces using one of tow methods: The first method: [ (0,Y1,Z1) , (X2,0, Z2) , Z] That’s mean: 1. The plane horizontal traces passes through the point (0,Y1,Z1) (the horizontal projection of the point). 2. The plane face traces passes through the point (X2,0, Z2) (the face projection of the point). 3. The distance Z represent the intersection point of the two traces T1 ,T2 from the side plane of projection. If the intersection point on the left, the value will be positive and the value be negative if the intersection point to the right.

Example : Represent the plane [(0,2,3),(4,0,4),6]

The second method: [ D1, D2, Z] That’s mean: 1. D1 is the angle between the earth line and the horizontal traces (T1) 2. D2 is the angle between the earth line and the face traces (T2) 3. The distance Z represent the intersection point of the two traces T1 ,T2 from the side plane of projection. If the intersection point on the left, the value will be positive and the value be negative if the intersection point to the right.

T1 D2 D1 T2

T1 D1 D2 T2

Level representation: The level can be represented by it’s traces or by using the projection of three point not in one straightness or by using the projection of point and straight line located through it or by the projection of two straight line located through it . Method of drawing level traces: If the level is defined by two parallel or intersecting lines: Steps: 1. Find the horizontal traces of the first line (F1) and the horizontal traces of the second line (F2), link F1F2 will get T1 (the horizontal traces of the plane).

2. Find the face traces of the first line (W1) and the face traces of the second line (W2), link W1W2 will get T2 (the face traces of the plane). 3. Use theory 6 to draw T3 2. If the level is defined by a straight line and the point outside of it, link the point to the straight line and turn the case to having traces of the level using two cross lines. 3. . If the level is defined by three points located through it ,make two intersection straight line and complete the solution.

Example 1: the straight line AB where A (1,4,3) B (4,1,7) and the point C (5,1,1) , draw the traces of the level passing through the straight line and the point.

Example : known : the projection of two parallel straight lines AB, CD Required : represent the plane passing through the straight lines by it’s traces . A(1,3,1) , B(3,1,4) , C(1,3,3) , D (2,1,6)

Thank you For Your Attention