Basic Geometric Constructions

In engineering drawing, frequently problems have to be solved which call for the knowledge of the method, or systematic procedure, to be used for preparing basic constructions. At the same time, the performance of such tasks develops the skills in handling drawing tools (compasses and dividers, triangles, rulers, templates) and promotes logical thinking.

1.7.1. Basic Problems

Problem 1:

Fig. 1.36. Bisecting a line segment Solution:

Draw arcs using B as the centre with the compass set so that its opening is practically . Draw arcs using B as the centre with the compass set as above.

Connect the intersection (C and D) by a line. The connecting line bisects AB in point M. Problem 2:

Fig. 1.37. Bisecting an angle Solution:

Draw an arc with S as the centre, thus A and B are obtained. Draw any arcs with A and B as the centres which intersect in C The connecting line C − S bisects the angle.

Problem 3:

Draw a line perpendicular to the line AB in A!

Fig. 1.38. Drawing a line perpendicular to a given line Solution:

With the compass set to any radius draw an arc about A as centre, thus C,

an arc about C as centre, thus D, an arc about D as centre, thus E, an arc about E as centre, thus F,

connecting line F to A is the perpendicular. Problem 4:

A given line is to be divided into 5 equal parts!

Fig. 1.39. Dividing a line into equal parts Solution:

From A of this line draw a second line at any convenient angle and lay off 5 equal spaces (points C − G). Connect G with B; Draw parallel lines to GB.

Problem 5:

Fig. 1.40. Determining the centre of a circle Solution:

Draw any two chords AB and CD.

Draw lines perpendicular to the two chords.

The intersection of these two perpendiculars is the centre (M) of the circle.

1.7.2. Regular Polygons

Problem 6: Hexagon

Fig. 1.41. Drawing a regular hexagon Solution:

Draw the vertical and horizontal centre lines.

Draw circular arc with M as the centre and the compass set to . With the compass set to draw arcs intersecting the circle. Problem 7: Pentagon

Fig. 1.42. Drawing a regular pentagon Solution:

Draw the vertical and horizontal centre lines.

Draw circular arc with M as the centre (thus obtaining A, B, C, D).

Bisect line segment MA (thus obtaining E); Draw arc with the radius EB and with E as the centre (thus obtaining F).

BF is the length of one side of the pentagon. Problem 8:

Fig. 1.43. Drawing a square standing on a corner Solution:

Draw the vertical and horizontal centre lines.

Draw circle with M as the centre, thus obtaining A, B, C and D. Connect the intersections.

Problem 9:

Fig. 1.44. Drawing a square standing on a flat Solution:

Draw the vertical and horizontal centre lines.

Draw circle with M as the centre, thus obtaining A, B, C and D. Bisect the angle AMB, thus obtaining E.

Bisect the angle CMB, thus obtaining F. EF is a flat side of the square.

Connect the points and mark off the distances on the circle. Problem 10:

Ellipse

Fig. 1.45. Drawing an ellipse Solution:

Draw the major axis and the minor axis, thus also obtaining, A, B, C, D and M. Draw a circle with M as the centre and the radius equal to the major axis.

Draw a circle with M as the centre and the radius equal to the minor axis. Draw any number of radii from M. Mark off the points on the outer circle (F, H, K, N).

Mark off the points on the inner circle (E, G, I, L). Draw further radii (without letter in the illustration).

Draw lines parallel to major axis through points provided by the intersections. Draw lines parallel to the minor axis through points provided by intersections. The intersections of the vertical and horizontal lines provide points of the ellipse.

1.7.3. Circular Arc Connections

Problem 11:

Construct the tangents to a circle having a diameter of 40 mm! The tangents have to pass through the common intersection A. The distance AM is 55 mm.

Fig. 1.46. Tangents to a circle Solution:

Find the centre M.

Draw circle with M as the centre and a radius of 20 mm. Find A (AM = 55 mm).

AM Bisect AM, thus obtaining M' (AM' = ).

Draw arcs with M' as the centre and AM' as the radius, thus obtaining B and C. Problem 12:

The two legs of a right angle have to be connected by a circular arc with a radius of 30 mm.

Solution:

Draw a right angle, thus obtaining S.

Draw two parallels within the angle at a distance of 30 mm in any case. The intersection of the parallels is the centre M.

The perpendiculars (AM and BM) are the points of connection. Problem 13:

The two legs of an angle of 120° have to be connected by a circular arc with a radius of 36 mm.

Fig. 1.48. The two legs of any angle are connected by a circular arc Solution:

Draw the angle (90° + 30°!), thus obtaining S.

Draw two parallels at a distance of 36 mm each, thus obtaining M. The perpendiculars (AM and BM) are the points of connection. Problem 14:

Two adjacent circles have the following diameters: d₁ = 50 mm

d₂ = 30 mm

The distance between their centres is 70 mm. The two circles have to be connected with a transition radius (R_tr) of 25 mm!

Fig. 1.49. Transition radius, two adjacent circles Solution:

Draw the two circles, thus obtaining M₁ and M₂.

Draw a circular arc with M₁ as the centre and the radius of R₁ + R_tr (30 mm + 25 mm). Draw a circular arc with M₂ as the centre and the radius of R₂ + R_tr (15 mm + 25 mm). The intersection is M.

Connect M with M₁ and M₂, thus obtaining A and B. A and B are the points of connection.

Problem 15:

Two circles, an inner circle and an outer circle, have to be connected by a circular arc. The circles have the following diameters:

Fig. 1.50. Transition radius, two circles one inside the other Solution:

Draw the two circles, thus obtaining M₁ and M₂.

Connect the centres and extend to the circles, thus obtaining A and B. Bisect AB, thus obtaining M as the centre for the circular arc.

Problem 16:

Two adjacent circles have to be connected by a tangent. Their diameters are:

d₁ = 80 mm d₂ = 34 mm

Fig. 1.51. Tangents to two circles Solution:

Draw the two circles, thus obtaining M₁ and M₂ (M₁ M₂ = 85 mm).

Draw circle with the centre M₁ and the radius of R₁ − R₂ (40 mm − 17 mm). Bisect M₁ M₂. Thus obtaining M₃.

Draw a circle with M₃ as the centre and M₁M₃ as the radius, thus obtaining A. Extend M₁A over A, thus obtaining B.

Draw a parallel through M₂ to M₁B, thus obtaining C BC is the tangent.