GEOMETRICAL CONSTRUCTIONS

          In this chapter, we deal with problems on Geometrical construction, which are mostly based on plane geometry and which are very essential in the preparation of Engineering Drawings.

They are:

1) Bisecting a Line

2) To draw Perpendiculars

3) To draw Parallel Lines

4) To divide a Line

5) To divide a Circle

6) To Bisect an Angle or Arc

7) To Trisect an Angle

8) To find the centre of an Arc

9) To construct an Ogee (or) Reverse curve

10) To construct Equilateral triangles

11) To construct Squares

12) To construct Regular Polygons

13) Special method of drawing Regular Polygons

14) Regular polygons inscribed in circles

15) To draw regular figures using T-square and set-squares

16) To draw Tangents

17)  Lengths of Arcs

18) Circles and Lines in contact

19) Inscribed Circles.

TO DRAW PARALLEL LINE -- To draw a line through a point and parallel to a given straight line

A) To draw a line through a given point, parallel to a given straight line

1. Let AB be the given line and P be the given point at a distance.

2. With centre P any convenient radius, draw an arc CD cutting AB at E.

3. With the same radius, from point E as centre draw an arc cutting AB at F.

4. Point E as centre and radius equals to FP, draw an arc cutting CD at Q.

5. Draw a line connecting PQ, This is the required line parallel to AB.

 

B) To draw a line parallel to, and at a given distance from a straight line

1. Let AB be the given line and R is the given radius.

2. Mark points P and Q on line AB, as far apart as convenient.

3. By taking R as radius draw an arc C from given point P.

4. with same radius draw another arc D from point Q.

5. Draw the line CD, just touching the two arc's, now this becomes the parallel line to the given line AB.

TO DRAW A PERPENDICULAR TO A GIVEN LINE FROM A POINT OUTSIDE IT (AWAY FROM IT)

 (A) When the point is nearer the centre

(i) Let AB be the line and P be the point.

(ii) With centre P and any convenient radius draw an arc cutting AB at C and D.

(iii) Take any radius greater than half the length of CD in compass, and with centres C and D draw the arcs intersecting each other at E.

(iv) Draw a line connecting P and E, this line cuts the line AB at Q. 

(v) Then PQ is the required perpendicular to AB line.

(B) When the point is nearer to the end of line

(i) Let AB be the line and P be the point.

(ii) With centre A and radius equal to AP, draw an arc cutting AB at C.

(iii) with centre C and radius equal to CP, draw an arc cutting previously drawn arc at D.

(iv)  Draw a line joining P and D and intersecting AB at Q.

(v) then PQ is the required perpendicular.

 

TO DRAW PERPENDICULARS TO A LINE

  1) To draw a perpendicular to a given line from a point within it.

Method 1:- 

(A) When a point P is near the middle of the line.

(i) Let us consider AB be the given line and P the point on it.

(ii) with P as centre and any convenient radius R1 draw arcs cutting AB at point C and D.

(iii) with any radius R2 greater than R1 and centres C and D, draw arcs intersecting each other at O.

(iv) by using scale draw a line connecting points P and O. Then this is the required perpendicular to the given line.

 

Method 2:- 

(B) When a point P is near the end of the line.

(i) Let us consider AB be the given line and P the point on it.

(ii) Mark a point O, with O as centre OP as radius, draw an arc greater than the semi-circle cutting AB at C.

(iii) Draw a line joining C and O, and extend it up to arc to cut it at Q,

(iv) Draw the line joining P and Q. Then PQ is the required perpendicular to the given line.

 

Method 3:- 

(C) When a point P is near the end of the line.

(i) Let us consider AB be the given line and P the point on it.

(ii)  with P as centre and any convenient radius, draw an arc greater than the semi-circle cutting AB at C.

(iii) With the same radius cut the arc into two equal divisions CD and DE.

(iv)  With the same radius drawn an arc from D and cut this arc by drawing same radius arc from E.

(v) Mark this intersecting point as Q. 

(vi) Draw a line joining P and Q, This is the perpendicular to the line AB.

Representing scale in engineering drawing

          The proportion between the drawing and the object can be represented by two ways as follows: 

a) Scale:- 1cm = 1m (or) 1cm = 100cm (or) 1:100

b) Representative Fraction:- (RF) = 1/100 

"The ratio of the length of the object represented on drawing to the actual length of the object represented is called the Representative Fraction (i.e., RF)".

There are three types of scales depending upon the proportion it indicates as 

1. Full Scale: Some times the actual dimensions of the object will be adopted on the drawing then in that case it is represented by the scale and RF as 

Scale:- 1cm = 1cm (or) 1:1 and by R.F =1/1 (equal to one)

2. Reducing scale: When the dimensions on the drawing are smaller than the actual dimensions of the object. It is represented by the scale and RF as 

Scale:- 1cm = 100cm (or) 1:100 and by R.F=1/100 (Less than one)

3. Enlarging scale: In some cases when the objects are very small like inside parts of a wrist watch, the dimensions adopted on the drawing will be bigger than the actual dimensions of the objects then in that case it is represented by scale and RF as 

Scale:- 10cm = 1cm (or) 10:1 and by R.F =10/1 (Greater than one)

Note: The scale or R.F of a drawing is given usually below the drawing. If the scale adopted is common for all figures under the title of the sheet.

Scales in drawing

          Usually the word scale is used for an instrument used for drawing straight lines, but in Engineering language scale means '' The proportion  (or) ratio between the linear dimensions adopted for the drawing to the actual dimensions of the object". 

It can be indicated in three different ways.

1) For example a 75mm long pencil may be shown by a drawing of 75mm length, drawings drawn of same size of the objects, are called full-size drawings.

2) For example the actual dimensions of the room say 10m X 8m cannot be adopted on the drawing. In suitable proportion the dimensions should be reduced in order to adopt conveniently on the drawing sheet. If the room is represented by a rectangle of 10cm X 8cm size on the drawing sheet that means the actual size is reduced by 100 times. This is called reduced scale.

3) For example if we draw a small size object to an increased size of our convenience, then this is called enlarging scale.