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
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