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

Straight lines located in a plane Before studying the special straight lines located in a plane, we search the relationship between the plane and the straight lines located through it, and the condition improved as the straight line located through the plane. This relationship determined by the following theory: Theory: If the straight line located in a plane , the horizontal traces of the straight line (F) will be located on the horizontal traces of the plane (T1) , the face traces of the straight line (W) will be located on the face traces of the plane (T2) and the side traces of the straight line (G) will be located on the side traces of the plane (T3).

Proofing: The horizontal traces of the straight line (F) is the point of intersection of the straight line with the horizontal plane of projection. So, point (F) is a point of the horizontal plane of projection. When the straight line located through a known plane, the point F considered to be on of this planes’ points in term of the plane’s definition : (it is a plane that when connecting any two points in it by a straight line, all the points of this straight line located on it). So, point F located on two levels: the first is the horizontal plane of projection and the second is the plane that the straight line located through it.

Point F located on the intersection line of them in according to the intersection line definition. While the intersection line of the plane and the horizontal plane of projection called the horizontal traces of the plane (T1) , the horizontal trace of the straight line (F) located on the horizontal traces of plane (T1). In the same way, we can proofing that the face traces of the straight line (W) located on the face traces of the plane (T2) , and the side traces (G) located on the side traces of the plane (T3).

Example: The plane (45⁰ , 120⁰ , 5) and the straight line AB located through this plane, it’s known that A (1, ? , 2) B ( 2 , ? ,3 ). Draw the horizontal and face projection of the straight line AB. Sol: The last theory say that point F located on T1 , so we can draw F in term of F and T1. Also the theory say that face traces of the straight line AB located on T2 . When we stretched the face projection of the straight line AB ,crossing T2 by W which is in the same time W . By considering W point of the face level ,so it’s horizontal projection will be on the earth line. The work: 1. Draw A B and connect between them to get the face projection of the straight line AB.

2. stretched the face projection A B to intersect the earth line by point F . 3. Draw a vertical line to the earth line from point F to cut T1 by F. 4. Stretched A B to cut T2 by W . 5.From W draw a line vertical to the earth line and cut it by W. 6. Connect F W. 7. From A , draw a line vertical to the earth line and find A on FW . Also draw another vertical line from B to find B. 8. AB is the horizontal projection of the straight line AB.

Horizontal straight line located in a plane: The horizontal projection of the horizontal straight line located in a plane will be parallel to the horizontal traces of the plane ,and the face projection will be parallel to the earth line.

Example 2: The plane (45⁰ , 135⁰ , 6) , draw a horizontal straight line located through the plane and away from the horizontal plane of projection by (3 units). Analysis : The face projection of the horizontal straight line located through the plane will be parallel to the earth line , and the horizontal projection will be parallel to the horizontal traces of the plane. The steps of solution: Represent the plane. Draw a horizontal straight line at a distance 3 unit below the earth line and parallel to the last. Stretched the face projection of the straight line to intersect the plane by a point W. Draw a vertical line from W to intersect the earth line by W.

5. From W draw a line parallel to the horizontal traces of the plane which represent the horizontal projection of the line . Show the figure below:

Example 3: The plane ABC is known by it’s horizontal and face projection as shown in figure below. Draw: Horizontal straight line located through the plane and passes through point B. Horizontal straight line crosses AC ,BC. Horizontal straight line crosses AC, AB. Sol: 1. From B draw a line parallel to the earth line and intersect AC by D , find D on AC , connect D and B , this will be the required horizontal straight line . 2. Draw a straight line parallel to the earth line and intersect AC ,BC by X Y . Find X , Y on AC , BC which be the required straight line .

3. Draw a straight line parallel to the earth line and intersect AC ,AB by O F . Find O , F on AC , AB , connect O F which be the required straight line .

Face straight line located in a plane: The face projection of the face straight line located in a plane will be parallel to the face traces of the plane ,and the horizontal projection will be parallel to the earth line.

Example 1: The plane (135⁰ , 45⁰ , -2) , draw a face straight line located through the plane and away from the face plane of projection by (3 units). Analysis : The horizontal projection of the face straight line located through the plane will be parallel to the earth line , and the face projection will be parallel to the face traces of the plane. The steps of solution: Represent the plane. Draw a horizontal straight line at a distance 3 unit above the earth line and parallel to the last. Stretched the horizontal projection of the straight line to intersect the plane by a point F. Draw a vertical line from F to intersect the earth line by F.

5. From F draw a line parallel to the face traces of the plane which represent the face projection of the line . Show the figure below:

Example 2: The triangle plane ABC is known by it’s horizontal and face projection as shown in figure below. Draw: Face straight line located through the plane and passes through point A. Face straight line crosses AB ,BC. Face straight line crosses AC, BC. Sol: 1. From A draw a line parallel to the earth line and intersect BC by D , find D on AC , connect D and A , this will be the required face straight line . 2. Draw a straight line ( X Y ) parallel to the earth line and find X , Y which be the required straight line . 3. Draw a straight line ( O F ) parallel to the earth line and find O , F , connect O F which be the required straight line .

Side straight line located in a plane: The face and horizontal projection of the side straight line located in a plane will be parallel to the vertical folding line , and the side projection will be parallel to the side traces of the plane located through it.

Thank you For Your Attention