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9.0.0  APPENDIX A:  GRAPHIC CONSTRUCTIONS 

9.1.0  Dividing straight lines and spaces into an equal number of segments.
9.2.0  Dividing a circle into a number of pie segments (inscribing a regular polygon of any number of sides in a given circle)
9.3.0  Given one side of a regular polygon, construct the polygon

9.1.0  DIVIDING STRAIGHT LINES AND SPACES INTO AN EQUAL NUMBER OF SEGMENTS

Figure 9.1:  Dividing lines and spaces into equal segments.

9.1.1  When drawing repetitive lines that are precisely equidistant from each other, as may be the case with brick courses or floor boards, it is possible to lay out the lines without measuring from one to the next.

9.1.2 

1)  Draw a line AB perpendicular to the first of the set of parallel lines to be constructed (Figure 9.1);

2)  place a scale, ruler or meter stick so that 0 is on the base line, and the number corresponding to the number of increments on a point the required distance from A;

3)  with a 4x0 pen (if constructing the lines on vellum) mark each increment through which the lines will be drawn;

4)  draw a few lines through these points to check the distance between them.
 

Figure 9.2:  Dividing a circle into an equal number of segments.

9.2.1  The technique illustrated in Figure 9.2 may be used to locate regularly occurring features on the plan of a cylindrical structure; or to draw a polygonal structure.  For precision, it is recommended that a compass with a 4x0 pen be used.  NB:  This method will produce a geometrically correct polygon.  Do not use this method to idealize the structure, but draw it as measured. 

9.2.2  For a polygon with seven sides:

1)  Draw circle A7BC, with its center at O, and a radius of at least 3";

2)  draw the two diameters C7 and AB perpendicular to each other;

3)  using the technique for dividing lines in 9.1.1, divide the diameter C7 into seven equal parts--the number of required sides;

4)  prolong the diameter AB;

5)  with either C or 7 as a center, and C7 as a radius, describe an arc to intersect the vertical center line AB at A';

6)  through A' and 2, the second division from C on C7, draw the line A'D cutting the circumference at D;

7)  draw the chord CD; it is one side of the required polygon; 

8)  with the compass set to the length of CD, step off the remainder of the polygon on the circumference. 

9)  The resulting polygon can now be reduced photographically or by projection.

9.3.0  GIVEN ONE SIDE OF A REGULAR POLYGON, CONSTRUCT THE POLYGON

Figure 9.3:  Constructing a regular polygon, given one side.

9.3.1  As above, the following method may be used to lay out a plan or detail ONLY if the structure as measured is perfectly regular.  Also, it is preferable to use a 4x0 pen to plot the construction points.  The following example is for constructing an octagon (Figure 9.3).

1)  Set a compass to radius AC, draw semicircle A1234567B;

2)  divide the semicircle into as many parts as there are sides (in this case, eight);

3)  from point C, and through the second division from B (6), draw straight line C6;

4)  bisect line AC and C6 by perpendiculars intersecting at O;

5)  using O as a center, set the compass to radius OC, describe the circle AC6DEFGH;

6)  from C, and through points 1, 2, 3, 4, 5 in the semicircle, draw lines CH, CG, CF, CE, CD, meeting the circumference of the circle;

7)  joining points C and 6, 6 and D, D and E, etc. by straight lines will produce the required polygon.

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