John
Sharp20 The Glebe,Watford Herts WD2 6LR, England PART 1: TYPES OF SPIRALS
Figure 1 The rules that relate the movement from the pole relative to the rotation affect the shape of the spiral. Many errors that arise in relation to descriptions of the spiral arise because of lack of appreciation of this fact. Throughout this article
Figure 2 It can be thought of as being based on addition or subtraction since the spiral is formed from the rule that for a given rotation angle (like one revolution) the distance from the pole has a fixed amount added. The relationship between the movement and rotation is that
the movement is directly proportional to the angle of rotation.
Thus the dependence of r = aq,where Different values of the constant Figure 3 If you magnify one of these spirals the correct amount, it
will fit directly on the other one
There is only one Archimedean
spiral.
Figure 4 The relationship between the movement and rotation is more
complicated than the Archimedean spiral. As the line rotates
so the distance of the point increases exponentially (due to
repeated multiplication instead of repeated addition). In this
case, the dependence of r = ab^{}q,with q measured in revolutions, or alternatively r = ae^{}q cot a,with q measured in radians, where
the constant The reason to make a explicit is the following important property of the logarithmic spiral: a line from the pole of a logarithmic spiral makes a constant angle with the tangent to the spiral, namely a. Hence it is often called the equiangular spiral (Figure 5). Figure 5 The mathematical relationships between a
and the aforementioned multiplication factor will be further
examined and exploited in Part
3 and Part 4. For now,
suffice it to note that different values of the constant By considering the spiral in terms of the tangent angle a, two curves can be seen as part of the family of logarithmic spirals. When a is 0° or 180° the spiral becomes a straight line, and when it is 90° it becomes a circle. This latter result is important when considering the Golden Section spiral produced approximately with arcs of circles. In characterising various Golden Section spirals, the tangent angle will be used to compare spirals produced in different ways. If you simply magnify a logarithmic spiral, it will not fit
on top of itself,
For curves made up of portions of other curves to 'flow' without kinks or cusps, the tangents of two curves at the intersection point must be the same line. For two circles to join as if they are one curve, the intersection point must be on a line through their centres, so two arcs can be made to appear as if they are one continuous line as follows (Figure 6). Figure 6 Such a 'piecewise' curve cannot be described by a single mathematical equation, and, as we shall see when we draw a Golden Section logarithmic spiral using this technique, it can lead to properties which are different from the those of single-equation curves. Note that the circles do not have to be the same size. As long as the line joining the centres goes though the intersection point, the curve will appear smooth.
Figure 7 Note that although this uses rectangles, it has nothing to do with the Golden Section, although the techniques described in Parts 2 and 3 for producing a Golden Section spiral use a similar drawing. There are two types of rectangles in Figure 7, with sides in the ratio 4:3 and 12:7 (the latter being the ones containing the flies).
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