Euler spiral (Cornu spirals)
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids or Cornu spirals.
Euler spirals have applications to diffraction computations. They are also widely used as transition curve in railroad engineering/highway engineering for connecting and transiting the geometry between a tangent and a circular curve. The principle of linear variation of the curvature of the transition curve between a tangent and a circular curve defines the geometry of the Euler spiral:
Its curvature begins with zero at the straight section (the tangent) and increases linearly with its curve length.
Where the Euler spiral meets the circular curve, its curvature becomes equal to that of the latter. (For more images)
See more(formulations & description) at: Euler spiral on Wikipedia - Weisstein, Eric W. “Cornu Spiral.” From MathWorld. & Fresnel integral.
Figure 1: Euler spiral (x, y) = (C(t), S(t)). The spiral converges to the centre of the holes in the image as t tends to positive or negative infinity.
Figure 2: Plot the Cornu parametric spiral on Mathematica.stackexchange.
Figure 3: Glacier Empress by Rhätischen Bahn - For more: Spiral railway near Brusio, Switzerland.
Figure 4,5,6: Cornu’s spiral tooth gear. A Cornu’s spiral is applied to a tooth profile of a gear. In accordance with the application of the gear, the Cornu’s spiral may be modified two-dimensionally by some constant proportion. Since Cornu’s spirals have similarity are similar, when a pressure angle is determined, the tooth profile can be standardized by a module, so that it is highly versatile. By replacing the tip of a tooth and the bottom of a tooth by another curve, such as an arc, the range of application can be increased. Since working is always by flank contact between a protrusion and a recess, contact surface stress is low. In addition, the radius of curvature of a dedendum is large. Therefore, tooth bending stress is low, and fatigue strength is high. Even if the number of teeth is small, an undercut is not produced. Even if there is no backlash, a smooth working is achieved, so that the tooth profile is suitable for devices requiring precise alignment - Inventors: Tsutomu Miyaoku, Jiro Kanehiro, Hidekazu Sasaki.
