The Geometry of Euler’s Formula – part 3
by greg ehmka
Euler’s Formula as a Helix
If we take Euler’s formula:
and, as before, graph the output on the y and iz axes, but this time, rather than the animation variable θ, have the input variable be the linear x axis, we then have:
which gives the helix:
and, with a frequency coefficient a:
and then, as in part 2, additional terms will give the various orbits in helix form; for example, with:
a “triangular helix” results:
And, the two-dimensional projections: side (RSIP) in blue, and front (FRP) in violet:
A “Loops and More Loops” Helix is generated with:
The second graph above is of the front (violet), top (red) and side (blue) views, two-dimensional projections as they naturally occur orthogonally to one another.
Next, in section 3.5 of the eBook, there are examples of how curves in polar coordinates can be easily converted to ‘3Di coordinates’. It essentially amounts to: a) replacing θ by x, and r by r(x); and b) inserting r(x) as a coefficient to the helix. Using Fermat’s Spiral as an example:
The positive square root (aqua) and the negative square root (yellow) give the two helixes. The side view is also shown:
And last, if one of the exponential coefficients is replaced with an animation variable:
and the base is varied to a different real number (the helix base and Euler’s Formula itself may be positive, negative, real, imaginary or complex), for example, the black helix animation below is:
then, the helix will continuously morph through an array of curves.
The blue object below is the side view (RSIP) of the helix. And, the red object (above the blue object) is the side view of the resulting helix after performing the following integral operation on the helix:
Notice in general, that the blue object has loops whereas the red object has cusps. Also notice, at various places the associated curves (evolute, pedal, etc.), show relationships between the two. For example, the one shown here of the Tricuspoid and Trifolium for
In section 6.32 of the eBook, and in part 6 of this series of blogs, Euler’s Formula is upgraded to include any base: positive, negative, real, imaginary or complex.
And more happy news to come!
In blog #1 Euler’s formula is used as a rotation of the exponential graph though an imaginary circle.
In blog #2 Euler’s formula is used as an ‘orbit operator’.
In #3 as a helix with both fixed three-dimensional geometry, and in four dimensions with an animation variable.
In #4 we show that Euler’s identity actually maps a point on an ‘exponential/natural log surface’!
In #5 a ‘circular helicoid standing wave’ is shown in which Euler’s imaginary exponential is used no less than five times in the equation: as a surface operator, an orbit operator, a synchronizing rotator, a spin coefficient, and as a reciprocal spin coefficient!
In #6 we present the equation that updates Euler’s formula for all bases: positive, negative, real, imaginary and complex.