A circle unrolled into a sine wave
circle-to-sine is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
A circular sector of area 0.60 and a hyperbolic sector of area 0.60
Equal-area slices of the hyperbola, 5 of them
The circular and hyperbolic functions on one set of axes
A hanging chain and a parabola through the same three points
The Lissajous figure with frequencies 1 and 2
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the 1:1 figure crosses itself 2pq − p − q times ×11
- at phase 1.178 the 1:2 figure has the generic crossing count ×7
- a coprime pair up to five ×1
- a generic Lissajous figure crosses itself 2pq − p − q times ×1
- a step up to 1 and two to six sectors ×1
- and they stay on the hyperbola ×1
- between three and seven offsets ×1
- cosh and sinh are the even and odd halves of e^t ×1
- every slice between neighbouring points has the same area ×1
- the catenary's parameter is between 0.2 and 2 ×1
- the circle's parameter is between 0 and 2π ×1
- the circular sector's area is half its parameter ×1
- the folding phases really fold ×1
- the grid runs to between 2 and 5 ×1
- the hyperbola's parameter is between 0 and 2.2 ×1
- the hyperbolic sector's area is half its parameter ×1
- the offset is generic — the curve does not fold back on itself ×1
- the parabola and the catenary really differ ×1
- the point is on the circle ×1
- the point is on the hyperbola ×1
- the point's height is the wave's height there ×1
- the range is between 1 and 3.2 ×1
- the ratio is between one and three ×1
- the row shows both a folded and an open figure ×1
- the time spent in each cell is close to the product of two arcsine densities ×1
- the two coordinates are a point of the unit circle ×1
- the view is one of sector, squeeze, waves, catenary, lissajous, lissgrid, lissphase, lissdense ×1
- two whole frequencies up to six with no common factor ×1
- x + y is e to the parameter ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A sine wave is a circle seen from the side
Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.
AnalysisThe angle that is really an area
On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.
DynamicsThe histogram an orbit leaves
When no single step of an orbit is worth reporting, what is left is where it spends its time. That distribution is not uniform, it does not depend on where the orbit started, and it can be computed in closed form.
AnalysisWhen two circular motions come home
Drive a point across with one sine wave and up and down with another. If the two frequencies are in a whole-number ratio the point retraces a closed figure whose crossings can be counted in advance — 2pq − p − q of them — and if they are not, it never comes back and fills the square, spending twenty times longer in the corners than in the middle.