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Square-ness, Example #1 (solution) || Steep-ness, Example #2 (solution) Compact-ness, Example #3 (solution) || Crowded-ness, Example #4 (solution) Awkward-ness, Example #5 (solution) || Sharp-ness, Example #6 (solution)
Malcolm Swan
Jim Ridgway
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Initially it makes sense to estimate the difficulty of a bend by the angle it turns through ( ![]()
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At this stage, most students will probably try to answer the question by eye. However, (anticipating question 4), if you try to draw the largest circular arc through each bend, you find that bends A and B will both allow arcs of radius 3.4 cm. Bend C will permit an arc of 6.8 (double the radius for bends A and B) and bend D will permit an arc of radius 13.1 cm.
Thus, the order of difficulty of the bends is now:
It may be seen that one possible measure for the 'difficulty' of a bend is radius of the largest circular arc that may be inscribed within the bend, that is tangential to the outer edges of the road.
The only drawback is that the larger this radius, r, the easier the bend. A measure which takes larger values for more difficult bends would be 1/r. This would take the'difficulty' value 0 for a straight road. The most difficult turn would be a U turn (when Alternatively, one could define a measure using two variables, by reasoning in the following way:
Thus we could define difficulty by a proportional model such as: ![]()
See answer to question 2.
Suppose the road is of width w, and turns through an angle ![]() ![]()
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OP = r - w OQ = r
< POQ =
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Notice that r is directly proportional to w. (This may also be seen, for example, by considering road C to be an enlargement of road A by scale factor 2, thus enlarging the radius by the same scale factor).
As
Compact-ness, Example #3 (solution) || Crowded-ness, Example #4 (solution) Awkward-ness, Example #5 (solution) || Sharp-ness, Example #6 (solution)
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