logarithmic spiral
The equation of the logarithmic spiral in polar coordinates is
(1) |
where and are constants (). Thus the position vector of the point of this curve as the coordinate vector is written as
which is a parametric form of the curve.
Perhaps the most known of the logarithmic spiral is that any line emanating from the origin the curve under a constant angle . This is seen e.g. by using the vector and its derivative , the latter of which gives the direction of the tangent line (see vector-valued function):
One obtains
whence
It follows that . The angle is called the polar tangential angle.
The logarithmic spiral (1) goes infinitely many times round the origin without to reach it; in the case one may state that
The arc length of the logarithmic spiral is expressible in closed form; if we take it for the interval , we can calculate in the case that
thus
Letting one sees that the arc length from the origin to a point of the spiral is finite.
Other properties
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Any curve with constant polar tangential angle is a logarithmic spiral.
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All logarithmic spirals with equal polar tangential angle are similar.
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A logarithmic spiral rotated about the origin is a spiral homothetic to the original one.
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The inversion causes for the logarithmic spiral a reflexion against the imaginary axis and a rotation around the origin, but the image is congruent to the original one.
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The evolute of the logarithmic spiral is a congruent logarithmic spiral.
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The catacaustic of the logarithmic spiral is a logarithmic spiral.
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The families and are orthogonal curves to each other.
Title | logarithmic spiral |
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Canonical name | LogarithmicSpiral |
Date of creation | 2013-03-22 19:02:26 |
Last modified on | 2013-03-22 19:02:26 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 26 |
Author | pahio (2872) |
Entry type | Topic |
Classification | msc 14H45 |
Synonym | Bernoulli spiral |
Related topic | AngleBetweenTwoCurves |
Related topic | EvoluteOfCycloid |
Related topic | PolarTangentialAngle2 |
Related topic | AngleBetweenTwoLines |