If a disc stands on its edge on a flat surface it will touch the surface at one point. This point is known as the point of tangency, as shown in Fig. 9.12 and the straight line which represents the flat plane is known as a tangent. A line drawn from the point of tangency to the centre of the disc is called a normal, and the tangent makes an angle of 90° with the normal.
The following constructions show the methods of drawing tangents in various circumstances.
Point of tangency
Point of tangency
To draw a tangent to a point A on the circumference of a circle, centre O (Fig. 9.13)
Join OA and extend the line for a short distance. Erect a perpendicular at point A by the method shown.
To draw a tangent to a circle from any given point A outside the circle (Fig. 9.14)
Join A to the centre of the circle O. Bisect line AO so that point B is the mid-point of AO. With centre B, draw a semi-circle to intersect the given circle at point C. Line AC is the required tangent.
To draw an external tangent to two circles (Fig. 9.15)
Join the centres of the circles by line AB, bisect AB, and draw a semi-circle. Position point E so that DE is equal to the radius of the smaller circle. Draw radius AE to cut the semi-circle at point G. Draw line AGH so that H lies on the circumference of the larger circle. Note that angle AGB lies in a semi-circle and will be 90°. Draw line HJ parallel to BG. Line HJ will be tangential to the two circles and lines BJ and AGH are the normals.
To draw an internal tangent to two circles (Fig. 9.16)
Join the centres of the circles by line AB, bisect AB and draw a semi-circle. Position point E so that DE is equal to the radius of the smaller circle BC. Draw radius AE to cut the semi-circle in H. Join AH; this line crosses the larger circle circumference at J. Draw line BH. From J draw a line parallel to BH to touch the smaller circle at K. Line JK is the required tangent. Note that angle AHB lies in a semi-circle and will therefore be 90°. AJ and BK are normals.
To draw internal and external tangents to two circles of equal diameter (Fig. 9.17)
Join the centres of both circles by line AB. Erect perpendiculars at points A and B to touch the circumferences of the circles at points C and D. Line CD will be the external tangent. Bisect line AB to give point E, then bisect BE to give point G. With radius BG, describe a semi-circle to cut the circumference of one of the given circles at H. Join HE and extend it to touch the circumference of the other circle at J. Line HEJ is the required tangent. Note that again the angle in the semi-circle, BHE, will be 90°, and hence BH and AJ are normals.
To draw a curve of given radius to touch two circles when the circles are outside the radius (Fig. 9.18)
Assume that the radii of the given circles are 20 and 25 mm, spaced 85 mm apart, and that the radius to touch them is 40 mm.
With centre A. describe an arc equal to 20 + 40 = 60 mm.
With centre B, describe an arc equal to 25 + 40 = 65 mm.
The above arcs intersect at point C. With a radius of 40 mm, describe an arc from point C as shown, and note that the points of tangency between the arcs lie along the lines joining the centres AC and BC. It is particularly important to note the position of the points of tangency before lining in engineering drawings, so that the exact length of an arc can be established.
To draw a curve of given radius to touch two circles when the circles are inside the radius (Fig. 9.19)
Assume that the radii of the given circles are 22 and 26 mm, spaced 86 mm apart, and that the radius to touch them is 100 mm.
With centre A, describe an arc equal to 100 - 22 = 78 mm.
With centre B, describe an arc equal to 100 - 26 = 74 mm.
The above arcs intersect at point C. With a radius of 100 mm, describe an arc from point C, and note that in this case the points of tangency lie along line CA extended to D and along line CB extended to E.
To draw a radius to join a straight line and a given circle (Fig. 9.20)
Assume that the radius of the given circle is 20 mm and that the joining radius is 22 mm.
With centre A, describe an arc equal to 20 + 22 = 42 mm.
Draw a line parallel to the given straight line and at a perpendicular distance of 22 mm from it, to intersect the arc at point B.
With centre B, describe the required radius of 22 mm, and note that one point of tangency lies on the line AB at C; the other lies at point D such that BD is at 90° to the straight line.
To draw a radius which is tangential to given straight lines (Fig. 9.21)
Assume that a radius of 25 mm is required to touch the lines shown in the figures. Draw lines parallel to the given straight lines and at a perpendicular distance of 25 mm from them to intersect at points A. As above, note that the points of tangency are obtained by drawing perpendiculars through the point A to the straight lines
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