GEOMETRIC CONSTRUCTION: POLYGONS AND TANGENCY

ENGINEERING DRAWING: LECTURE 3

TOPICS/SCHEDULE

• Introduction
• Introduction to drawing instruments and their proper use
• Use of scales, line work, lettering
• Dimensioning
• Polygons, Tangency, Loci: paths of points of simple mechanism (Covered in this lecture)
• Loci: Cam profile, ellipse, hyperbola, parabola, involute, cycloids
• Orthographic projection in 1st and 3rd Angle
• Isometric projection and oblique projection
• Isometric projections from orthographic projection

Polygons

Polygons are fundamental plane figures in engineering drawing.

1. Plane figures bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit.
2. Segments are called its edges or sides.
3. Points where two edges meet are the polygon's vertices or corners.

Construction of Regular Hexagon: Given The Distance Across Flats

1. Draw a circle of diameter equal to the given distance.
2. Draw vertical and horizontal lines across the diameter.
3. With the Tee-square, draw tangents to the circle at the top and below the circle.
4. With the 60º set square correctly placed on the Tee-square draw tangents to the circles.
5. Join the tangents, as required, to produce the given hexagon.

Note: A regular octagon can be produced with the procedures above using 45º set squares.

Construction of Regular Hexagon: Given the Distance Across Corners

1. Draw a circle of diameter equal to the given distance.
2. Draw vertical and horizontal lines across the diameter to intercept at O.
3. Using the ends of the horizontal line as starting point step off the radius of the circle round it.
4. Join the points to produce the hexagon.

Construction of Regular Octagon: Given the Distance Across Corners

1. Draw a circle of diameter equal to the given distance.
2. Draw vertical and horizontal lines through centre O to intercept the circle at A, B, C & D.
3. With the 45° set square, draw two cross lines through centre O to intersect the circle at E, F, G & H.
4. Join all the interceptions to obtain the octagon (A-E-D-H-B-F-C-G-A).

Construction of Regular Pentagon (Given a side AB)

1. Draw the given line AB.
2. With A as centre draw a circle of radius AB.
3. With B as centre draw a circle of radius AB to intersect the previous circle at C and D.
4. Draw a line through C and D.
5. With centre D draw a circle of radius AB to intersect the previous circles at E and F and line CD at G.
6. Draw a line through E and G to intersect the second circle at H.
7. Draw a line through F and G to intersect the first circle at J.
8. With centres H and J and radius AB, draw arcs to meet at K.
9. Join points AJKHB to complete the pentagon.

Construction of General Regular Polygons (Given side AB)

1. Draw the given side AB and bisect it.
2. Draw a line through A inclined at 45° using 45° set square to intersect the bisection line at 4.
3. Draw a line through B inclined at 60° using 60° set square to intersect the bisection line at 6.
4. Bisect 4 and 6 to obtain 5.
5. Step off arcs 4-5 starting from 6 to produce 7, 8, 9, ..., n (where n is the number of sides).
6. To draw a heptagon, use point 7 as centre to draw a circle passing through A and B.
7. Starting from A, step off arcs on the circle with radius AB.
8. Join the intersections together to complete a regular heptagon (or any n-gon).

PRINCIPLES OF TANGENCY

What is a Tangent?

• A tangent to a curve is a straight line that touches the circle at only one point.
• A tangent is perpendicular to the radius of a circle drawn at the point where the tangent meets the circle.
• Tangents can be drawn to meet any curve (e.g., finding slope/gradient of quadratic/polynomial curve at a given point).
• Tangents are common in engineering drawing.

Construction Methods for Tangency

1. To Draw Tangent to a Circle from any Point on the Circumference

1. Draw the radius of the circle to the given point.
2. Construct an angle 90° from the point where the radius crosses the circumference. This perpendicular line is the tangent.

2. To Draw a Tangent from Any Point P to a Circle (P is outside the circle)

1. Draw the given circle and locate point P.
2. Join OP (from center O to point P).
3. Bisect OP.
4. Draw a Semi-circle with the bisected point as center and half of OP as radius, to cut the given circle at A.
5. Join AP.
6. AP is the required tangent.

3. To Construct a Common Tangent to Two Equal Circles (Direct Tangent)

1. Join the centres of the two circles.
2. From each centre, construct lines at 90° to the centre line, passing through the circle circumference.
3. Join the points of intersection on the circumference to give the tangent.

4. To Construct a Common Cross Tangent to Two Equal Circles

1. Join centres of circle (O-O1).
2. Bisect O-O1 to give point A.
3. Bisect OA to give point B.
4. Draw a semi-circle with radius OB and centre B to cut the first circle at C.
5. Locate D on the second circle such that AC=AD (or by drawing a line from O1 parallel to OB to find the tangent point).
6. Join C-D. This is the common cross tangent.

5. To Construct a Common Tangent to Two UNEqual Circles (Direct Tangent)

1. Join centres of circles (O-O1).
2. Bisect O-O1 to give A.
3. Draw a semi-circle with radius OA and centre A.
4. Draw a circle centre O with radius (R-r) (difference of radii) to cut the semi-circle at B.
5. Draw a line through OB to cut the first circle at C.
6. Draw line O1D parallel to OC.
7. Join CD. This is the common direct tangent.

6. To Construct a Cross Tangent to Two UNEqual Circles

1. Join centres of circle (O-O1).
2. Bisect O-O1 to give A.
3. Draw a semi-circle with radius OA.
4. Draw a circle centre O with radius (R+r) (sum of radii) to cut the semi-circle at B.
5. Draw OB to cut the larger circle at C.
6. Draw O1D parallel to OB.
7. Join CD. This is the common cross tangent.

7. To Construct a Curve Tangential to two Perpendicular lines (Arc of given radius R)

1. With the intersection point (A) of the two perpendicular lines as centre and radius R, draw arcs to cut the lines at B and C.
2. With centres B and C, and radius R, draw arcs to intersect at O.
3. With O as centre, draw the required curve (arc of radius R).

8. To Construct a Curve Tangential to two Lines (at an angle, arc of given radius r)

1. Construct lines parallel to the given lines at a distance r away. These parallel lines will intersect at O.
2. Draw the required curve (arc) with centre O and radius r, touching both lines.

9. To Construct a Curve OF radius R Tangential to a Line and a Circle of radius r

1. Construct a line parallel to the given line, at a distance R away from it.
2. From the centre of the circle (B), draw an arc with radius (R+r) to intersect the parallel line at O.
3. Draw the required curve from O with radius R. This curve will be tangential to both the line and the circle.

10. To Construct a Curve Tangential to Two Circles (Externally)

1. Draw the two given circles of radius r1 and r2, with centres A and B respectively.
2. With centre A and radius (R+r1), draw an arc.
3. With centre B and radius (R+r2), draw an arc to intersect the first arc at O.
4. With centre O and radius R, draw the required tangential curve.

11. To Construct a Curve Tangential to Two Circles (Internally)

1. Draw the two given circles of radius r1 and r2, with centres A and B respectively.
2. With centre A and radius (R-r1), draw an arc.
3. With centre B and radius (R-r2), draw an arc to intersect the first arc at O.
4. With centre O and radius R, draw the required tangential curve.

Additional Principles of Tangency (Illustrated Diagrams)

The document includes diagrams (FIG.1, FIG.2, FIG.3) illustrating:

• Drawing an arc of given radius to touch a given straight line in various orientations.
• Drawing an arc of given radius to touch a given arc externally.
• Drawing an arc of given radius to touch a given arc internally.

TANGENCY PROBLEMS

A series of tangency problems (Figures 1-16) are provided for practice, showing various complex shapes constructed using tangency principles and dimensions.

SELF PRACTICE (SP 002)

Submission Deadline: Tuesday 19th March by 12:00 NOON

Q1(a): Construct a nonagon of sides AB = 50 mm

(Refers to the general polygon construction method on page 8, adapted for a 9-sided polygon.)

Q1(b):

• Construct a pentagon of sides 50 mm.
• Construct an hexagon A/C (Across Corners) of diameter 60 mm.
• Construct an hexagon A/F (Across Flats) of diameter 60 mm.

Q1(C):

• Construct a scalene triangle of any convenient sides and inscribe a circle in it.
• Construct a scalene triangle of any convenient sides and circumscribe a circle in it.
• Draw a line of any convenient length and bisect it.
• Draw a line of any convenient length and divide it in the ratio 3:5.
• Draw an acute angle and bisect into two.

Q2: Draw A Tangent to a Circle OF RADIUS 30 mm from any Point on the Circumference

(Follows the method on page 11)

Q3: Draw a Tangent from Any Point P to a Circle of radius 50 mm. P IS OUTSIDE THE CIRCLE.

(Follows the method on page 12)

Q4: Construct a Common Tangent to Two Equal Circles OF RADIUS 40 mm AND 110 mm APART.

(Follows the method on page 13)

Q5: Construct a Common Cross Tangent to Two Equal Circles OF RADIUS 40 mm AND 110 mm APART.

(Follows the method on page 14)

Q6: Construct a Common Tangent to Two Unequal Circles OF RADIUS 50 mm & 30 mm RESPECTIVELY & 120 mm APART.

(Follows the method on page 15)

Q7: Construct a Cross Tangent to Two UNEqual Circles OF 50 mm & 30 mm RESPECTIVELY & 120 mm APART.

(Follows the method on page 16)

Q8: Construct a Curve OF RADIUS 30 mm Tangential to two Perpendicular lines

(Follows the method on page 17)

Q9: Construct a Curve OF RADIUS 20 mm Tangential to two Lines WHICH SUBTENDS ANGLE 30° WITH EACH OTHER.

(Follows the method on page 18)

Q10: Construct a Curve OF RADIUS 30 mm Tangential to two Lines WHICH SUBTENDS ANGLE 120° WITH EACH OTHER.

(Follows the method on page 18)

Q11: Construct a Curve OF radius R= 50 mm Tangential to a Line and a Circle radius r = 30 mm. THE LINE IS 50 mm AWAY FROM THE CENTRE OF THE CIRCLE.

(Follows the method on page 19)

Q12 - Q15: REPRODUCE THE FIGURE BELOW

These questions require the reproduction of complex engineering drawing figures using the learned tangency and polygon construction principles, adhering to the given dimensions and radii.