Definition
In polar coordinates the logarithmic curve can be written as
or
with being the base of natural logarithms, and and being arbitrary positive real constants.
In parametric form, the curve is
with real numbers and .
The spiral has the property that the angle φ between the tangent and radial line at the point is constant. This property can be expressed in differential geometric terms as
The derivative of is proportional to the parameter . In other words, it controls how "tightly" and in which direction the spiral spirals. In the extreme case that the spiral becomes a circle of radius . Conversely, in the limit that approaches infinity (φ → 0) the spiral tends toward a straight half-line. The complement of φ is called the pitch.
Read more about this topic: Logarithmic Spiral
Famous quotes containing the word definition:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
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—Ana Castillo (b. 1953)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)