Climbing
Cam Geometry
The
general profile for a climbing cam lobe is an exponential spiral, also
known as a logarithmic spiral. This curve has the unique property that
the angle (psi) between its radial and tangential vectors (lines) is a
constant everywhere along the curve. In other words, no matter how you
rotate the curve about its center point, it will always have the same
angular relationship to a given line. The only other curve with this
property is a circle, which can be considered a special case of the
exponential spiral, where the angle (psi) is 90 degrees, and the
equation
below
reduces to
r = constant.
The exponential spiral can be readily derived from differential
geometry using polar coordinates. Because the angle (psi) is a
constant, it is possible to separate variables and obtain a solution to
the differential equation, as shown below.
