Euler spiral
clothoid

The first four lines express the Euler spiral component.

The shapes are segments of the Euler spiral or clothoid curve.

This thus confirm that the original and normalized Euler spirals are having geometric similarity.

Where the Euler spiral meets the circular curve, its curvature becomes equal to that of the latter.

This geometry is an Euler spiral.

The Euler spiral has two advantages.

The spiral is a small segment of the above double-end Euler spiral in the first quadrant.

Normalized Euler spiral can be expressed as:

Normalized Euler spiral has the following properties:

The following is Mathematica code for the Euler spiral component (it works directly in wolframalpha.com):

Euler spirals have applications to diffraction computations.

An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin.

Euler spirals are also commonly referred to as spiros, clothoids or Cornu spirals.

Because of the capabilities of personal computers it is now practical to employ spirals that have dynamics better than those of the Euler spiral.

Marie Alfred Cornu (and later some civil engineers) also solved the calculus of Euler spiral independently.

I now think that's overkill, and G2-continuous splines (the Euler spiral ones) are plenty, and could be done with fewer points."

Since then, "clothoid" is the most common name given the curve, even though the correct name (following standards of academic attribution) is "the Euler spiral".

Sections from Euler spirals are commonly incorporated into the shape of roller-coaster loops to make what are known as "clothoid loops".

The resulting shape matches a portion of an Euler spiral, which is also commonly referred to as a clothoid, and sometimes Cornu spiral.

The 200 meter course consisted of an Euler spiral loop built from rammed-earth with banked curves and a heavy timber over-under bridge intersection in the middle.

Euler spirals are now widely used in rail and highway engineering for providing a transition or an easement between a tangent and a horizontal circular curve.

It is also referred to as the Euler spiral, the Cornu spiral, a clothoid, or as a linear-curvature polynomial spiral.

The Euler spiral, also known as Cornu spiral or clothoid, is the curve generated by a parametric plot of S(t) against C(t).

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:

The graph on the right illustrates an Euler spiral used as an easement (transition) curve between two given curves, in this case a straight line (the negative x axis) and a circle.