Geometric Construction

Geometric Construction This Lecture explains some of the essential techniques to draw specific engineering operations for students and drafters .. It will include: Drawing principle (continues) Engineering Operations on a line Engineering Operations on an arc Engineering Operations on a Circle Drawing a polygon Drawing of Ellipses

Drawing Principles Line Types

Drawing Principles Line Types

Drawing Principles Drawing Tables - Workshop orders need more information than just the drawing (eg. Name of the designer, name of component, date of design, material, number of parts …) General Information Table Specific Information Table

Drawing Principles 1. Name of Institution (University) 2. Name of branch (College) 3. Name of Factory (Faculty) 4. Name of department (Specialty) 5. Name of Designer (Student) 6. Group Symbol (Group Number) 7. Date of Design 8. Name of Design 9. Plate Number 10. Drawing Scale 11. Drawing Symbol

Drawing Principles

Drawing Principles Drawing Scale Sometimes we need to enlarge to clarify Sometimes we need to minimize to fit paper Eg. Draw 2000mm length  minimize to 200mm  minimized by 10 times  Scale is 1:10 Original dimensions should be written

Drawing Principles Drawing Scale Types Full Scale 1:1 Reduced Scale 1:2, 1:5, 1:100 …. Enlarged Scale 2:1, 5:1, 25:1 …. Choosing the right SCALE depends mostly on: - dimensions of part - dimensions of drawing paper

Engineering Operations On a straight line Bisecting a line Using a compass, open it more than half the length of the line you want to bisect. Position the compass on one edge (eg. Point A) and draw an arc on the two sides of the line. Do the same from the other end on the line (eg. Point B) Notice that the arcs crosses in two points. Drawing a line from the intersection of the two arcs gives a line that intersect the original line at point E. Point E is in the middle of the line AB. The line CD is also perpendicular to AB.

B. Drawing a perpendicular line on a certain point. Open the compass and draw from the point (0) two arcs that crosses the original line (AB) at (D) and (E). From each point D&E, draw arcs above and below the line. (similar to bisecting a line). Drawing a line from either (F) or (L) to the original line (AB) will be perpendicular

C. Drawing a perpendicular line from an outer point. Using the compass from point C and an opening bigger than the line expected. Draw two arcs that crosses the line AB at D&E. From the points D&E draw arcs in the opposite side of the line AB. The line from points C&L is perpendicular to AB.

D. Drawing a parallel line. We want to draw a parallel line from point C to the line (AB). Chose a point at the line AB (eg. Point D). Using the compass draw an arc from point D that crosses C and the line AB. Draw an arc from point C (eg. arc R). Draw an arc from point E that crosses the arc R at point F. The line CF is parallel to AB. Note that the opening of the compass is constant.

E. Bisecting an angle. We have the known angle (A) With an (X) opening of a compass, draw two arcs that crosses the two lines of the angle at points D&E. Draw an arc with the same opening from point D (eg. Arc R). Draw an arc from point E that crosses the arc R at point F. The line AF is bisecting the angle A.

F. Dividing a line equally we want to divide the line AB by 6 equal pieces Draw a line AL that has a sharp angle. Draw an arc from point A on AL (eg. Point D) with an opening (X). With the same opening, draw an arc from point D at AL. Continue doing that to have a total of 6 arcs at the points (E, F, J, H, C). Draw a line from point C to B. Draw lines from each point toward AB that are parallel to CB as HH1, JJ1, FF1, EE1, DD1. The points D1, E1, J1, H1, divide the line AB by 6 equal pieces.

2. Engineering Operations on an Arc A. Finding the center of an arc or a circle We have the circle (a) and arc (b) that is missing a center. Draw any two lines in the circle as (AB & CD). Draw a perpendicular lines from the center of these lines as stated before. The intersection of these two lines is the center of the circle.

B. Drawing a tangent arc to two lines We have the two lines AB and CD,(on the right) and want to draw an arc that is tangent to these two lines with a known radius (R). Draw a line parallel to AB and with a distance R. Do the same for the line CD. The intersection (point E) is the center of the arc. Draw the arc from point E with the opening (R).

C. Arc Tangent to a Circle and a Line The arc has a known radius (R), and should be tangent to both the line AB and the circle. Draw a line CD that is parallel to AB and with a distance R. open the compass to an opening of (R+r) and draw an arc on the line CD from the point (0). The arc’s intersection (point E) is the center of the arc we will draw.

D. Arc tangent to two circles We need to draw an arc that is tangent to the two circles and with a radius (R). (right picture) Draw an arc with an opening (R+R1) from point (01) Draw an arc with an opening (R+R2) from point (02) The intersection is the center of the arc we will draw at point (M) The left picture is the same with different compass opening as (R-R1) & (R-R2)

E. A Line tangent to a Circle We want to draw a line that is tangent to a circle and crosses a known point (M). Draw a line from the point M to the center (0). Bisect the line M0 at E. Draw an arc from point E with a radius of (M0/2). The arc will intersect with the circle at point A, which will be the tangent point. Draw the tangent line from point M to A

3. Engineering Operations on a Circle A. Dividing a Circle into Three Equal Parts. Can be done by two methods First Method (right): Draw the line AB. Draw an arc with a radius R from point A that crosses the circle at points (1&2). The points (1,2,B) divide the circle into three equal parts. Second Method (left): Draw the line AB. Then use the triangle. With the triangle perpendicular to AB, move it until it touches (0) and draw point 1. do the same in the other side and draw 2

B. Dividing the Circle into four parts First Method (right) Draw the line AB. Similar to the Second method (previous slide) but with using the 45 degree triangle. Second Method (left): Draw a line AB, then draw two arcs (S1, S2) from A with a radius bigger than R. Do the Same at point B and draw (C1, D1). The line that crosses the intersections (CD) makes four equal parts (A,B,C,D)

C. Dividing a circle into any number of equal parts. Dividing into 5,6,7 parts is in the book. Lets divide into 16 parts. Draw the lines AB and CD. Draw 2 arcs that crosses CD at C1 & D1 from point A with a radius 2R. Divide the line (1-9) by half the number desired for the circle dividing. Draw lines from each point (C1 & D1) that crosses through the numbers (2,3,4,5,6,7,8). The intersections will give the 16 equal parts.

4. Drawing a Polygon Can be done by multiple methods. The first method is by drawing a circle that the polygon will be inside touching the circle. (in the book) The second: lets draw a polygon with 10 known sides. (AB) Draw the side AB. Then with a compass draw the arc BM. Divide the arc into the same number of sides needed with the point (1,2,3,4,5,6,7,8,9) Draw lines from point A to the points (1,2,3,4,5,6,7,8,9) Using the compass with radius (AB) draw an arc that intersect with the line A1 from the point B, to create point C Do the same from point C to create point D, and so on.

5. Drawing an Ellipse There are some approximate methods (1st & 2nd methods - read book) and some more accurate methods. 3rd method (accurate): With knowing the 2 diameters AB and CD. Draw them and extend CD. Draw two circles (M1,M2) with diameters AB, and CD Draw a line from point 0 that crosses the two circles at point 1&2. Draw a parallel line to AB from point 1, and draw a parallel line to CD from point 2. the intersection is point E, which is a point on the conic section required. Repeat the same to have as much points as possible and then connect them. The more points, the better accuracy.

4th method (used to draw ellipses in Isometric (3D) objects): With knowing one of the ellipse’s diameters. The two diameters of the ellipse incline on each other the same inclination of the isometric object. If the long diameter (AB) is known … 1. Draw the diamond (FECD) around (AB) where the lengths are equal to (AB) and the inclination is the same as the isometric object. 2. From (F) draw a perpendicular line to (CD) and (DE) which will intersect on (H) and (G). 3. From (D) draw a perpendicular line to (CF) and (FE) which will intersect on (K) and (L). Specify points (P) and (Q). Open the compass size (LD) center at (D) and draw arc (KL) also center at (F) and draw arc (GH) Open the compass size (QH) center at (Q) and draw arc (LH) also center at (P) and draw arc (KG)

Class Exercise 1: Draw the following with a scale of (1:1):

Class Exercise 2: Draw the following with a scale of (3:1): Draw a six sided polygon where each side measures 40mm Draw an eight sided polygon where each side measures 30mm

Summary and Exercises ==================================== Next Geometric Construction Engineering Operations on a line Engineering Operations o n an arc Engineering Operations o n a Circle Drawing a polygon Drawing of Ellipses ==================================== Next First Angle and Third Angle Projection Homework: Finish your Exercises

End of Lecture 2 Prepared with the assistance of Mechanical Engineering Department at College of Engineering & Islamic Architecture