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2 Tangent Constructions Session 1 Drawing and designing logos Many symbols are constructed using geometric shapes. The following section explains common geometrical constructions. Common geometrical constructions As part of the construction of symbols it is often necessary to make geometrical constructions in order to draw a symbol, part of a symbol or a border. The following basic constructions are only some of the many available. Other geometrical constructions can be found in graphics textbooks. Quickly construct (not draw) each tangent example as you read, especially the unfamiliar ones. An arc tangential to two straight lines An arc that is tangential to two straight lines can be found by constructing two lines parallel to the given lines. The constructed line should be a distance (C) equal to the radius of the tangential arc away from the given lines. Theoretically we should find the exact points of tangency of each of the four construction arcs but realistically this isn't necessary. Points of tangency are defined by constructing a 2 4 mm medium line across the point where the tangent occurs. Page 1 of 7

A tangent to a circle from a point outside the circle To construct a tangent on a circle (or arc) from a point outside the circumference, first name the centre of the circle ('O' in this case) and

3 A tangent to a circle from a point outside the circle To construct a tangent on a circle (or arc) from a point outside the circumference, first name the centre of the circle ('O' in this case) and the point outside the circle (P), then: STEP 1 Join OP and bisect the line OP to find its centre X. STEP 2 Draw a semicircle (radius XP) to find the point of tangency T on the circumference as shown. STEP 3 Draw the tangent from point P through point T. Note: The tangent PT will touch the circumference at only one point. This point must be a tangent since PTO is a right angle. Page 2 of 7

4 An arc tangential to two arcs or circles The tangential arc to be constructed in this situation is best thought of in terms of the position of its centre in relation to the centres of the two given circles or arcs. Internal Follow these steps to find the centre of an arc tangential to two internal arcs. STEP 1 Draw a measuring line anywhere on the page. STEP 2 Mark CD, the radius of the tangential arc, onto the measuring line. Note: All measurements should be transferred from the measuring line and not from a ruler. From D mark the radius of the large circle A. Also from D, mark B, the radius of the small circle, onto the line CD, as shown in the illustration on the next page. STEP 3 Scribe an arc, radius CA, from the centre of the larger circle in the vicinity of where the centre of the tangential arc will be. Note: CA is obtained by subtracting the radius of the large circle from the radius of the tangential arc. STEP 4 Similarly, take the distance CB and scribe an arc from centre B to intersect the arc drawn in Step 3. Note: CB is obtained by subtracting the radius of the small circle from the radius of the tangential arc. STEP 5 The point where the two arcs intersect will be the centre of the tangential arc, radius CD. STEP 6 The points of tangency are found by drawing a line from the tangential arc's centre through the centres of both circles and then on to the furthest part of the circumferences of each circle. STEP 7 Before drawing the tangential arc, check with a compass that the distances from the points of tangency to the centre of the tangential arc are the same. External A similar process to the one just outlined is used to find the centre of an arc tangential to two external arcs. The only difference is that the radii of both the large and small circles need to be added to the radius of the tangential arc, rather than subtracted as in the internal method. STEP 1 Draw a measuring line and mark the radius of the tangential arc CD onto it. From D add A and also from D add B, the radii of both circles as shown in the illustration on the next page. Page 3 of 7

5 STEP 2 Scribe an arc CA from the centre of the large circle and an arc CB from the centre of the smaller circle in the vicinity of where the centre of the tangential arc will be. STEP 3 The point where the two arcs intersect will be the centre of the tangential arc, radius CD. STEP 4 The points of tangency are determined by a similar method to Step 7 on the previous page. The difference is that the lines from the tangential arc's centre don't need to be extended past the centre of each circle to find the points of tangency. An arc tangential to a line and another arc or circle To find a tangential arc to an arc (or circle) and a line, you need to locate the centre of the tangential arc. To do this, use a combination of the two methods discussed previously. STEP 1 To find a tangential arc to a line and an arc: Draw a line parallel to the given line which is the radius of the tangential arc (B) away from it. (See the tangential arc to two given lines example on page 17.) Scribe an arc to locate the centre of the tangential arc. This is done by adding the radii of the tangential arc and the given circle (or arc) together. The distance AB is scribed from the arc centre (in our case) to intersect the parallel line constructed previously. STEP 2 A tangential arc can now be drawn using the intersection of the parallel line and constructed arc as its centre. Page 4 of 7

6 A direct common tangent To construct a direct common tangent between two circles: STEP 1 Join the centres of the two circles A and B to form line AB. STEP 2 Bisect the line AB to find its centre X. STEP 3 With centre X draw a semicircle, radius XA. STEP 4 From the centre of the given large circle (centre B), draw a circle with a radius equal to the difference between the two given circles, that is BR Ar. (Take the smaller radius away from the larger radius). STEP 5 Draw a line from the centre of the large circle B through point C, the point of intersection between the circle drawn in Step 4 and the semicircle drawn in Step 3. Extend this line to the circumference which will become the point of tangency D. STEP 6 Draw a line from the centre of the given small circle A to point C. Transfer distance AC from tangent point D onto the circumference of the given small circle to locate point E (use compasses or dividers). This point is the second point of tangency. STEP 7 Draw the tangent DE. Page 5 of 7

7 A transverse common tangent Transverse common tangents are also known as indirect common tangents. To draw a transverse common tangent between two circles: STEP 1 Join the centres of the two given circles A and B, bisect the line AB and draw a semicircle radius XA (or XB) as in Steps 1 3 of the previous example. STEP 2 From the centre of the given smaller circle B, draw a circle with radius BR, equal to the sum of the two given circles. This can be achieved by adding the small and large radii and transferring them from a line as discussed previously. STEP 3 Draw a line from the centre of the small circle B to where the circle constructed in Step 2 intersects the semicircle constructed in Step 1. Name this point C. The point where the line BC intersects the circumference of the given small circle will become the point of tangency D. STEP 4 Join point C to the centre of the given larger circle A. Transfer distance CA from tangent point D onto the circumference of the given larger circle to locate point E. This point is the second point of tangency. STEP 5 Line in the tangent DE. Remember to mark the points of tangency. Page 6 of 7

8 How to construct a pentagon Some other geometric constructions are often required when producing logos. Below is a popular technique for pentagon construction. Refer to the Resources for other useful techniques. Bisect AB at C and extend the bisector. Measure AB (using dividers) and mark this distance on the extended bisector at D, from C (vertically). Join AD and extend. Mark distance DE equal to AC from D. With centre A and radius AE, scribe an arc to cut the bisector at F. This will become a corner of the pentagon. From points A, B and F, scribe intersecting arcs, radius AB (the length of a side of the pentagon) to locate the two missing corners X and Y. Join these to form the pentagon. Always check your accuracy by making sure the length of each diagonal is equal (especially, AY BX KY). Page 7 of 7

ME 111: Engineering Drawing. Geometric Constructions

ME 111: Engineering Drawing. Geometric Constructions ME 111: Engineering Drawing Lecture 2 01-08-2011 Geometric Constructions Indian Institute of Technology Guwahati Guwahati 781039 Geometric Construction Construction of primitive geometric forms (points,