It seems that timing and cycles are important for humans to locate themselves in their time direction. As such, I should design for myself a clock that suits my orientation in time and is attractive given my interests.

Design

It should be a 24-hour clock, with noon at the top. The existing 12-hour clocks make fitting blocks of time into a day difficult by rolling over during times when I am normally or often active. A 24-hour clock will better allow me to approximate my time planning and usage day-to-day by simple visual inspection.

Notes for day-planning could be taken around the periphery of the clock. Tasks could be dragged onto one or more hours. A checkbox by each task would help identifying what is left to do.

Years pass by unnoticed; this has to do with the lack of intuition in our linguistic notation for dates. My clock should note the time of the year in a purely visual way: using a pointer travelling around a cycle. The pre-existing art here is to use an ecliptic, as in an astrolabe. This has the added benefit of my being able to visually estimate sunrise/sunset; note that daylight hours are an important consideration for some tasks. I am interested in astronomical phenomena, so this astronomy addition is quite pleasing. The clock will therefore require my latitude and information about daylight savings.

Knowing the weather is important on a daily basis to determine my carrying an umbrella and wearing coats. If I know temperature, precipitation, wind, and cloud cover, I should be pretty good. I haven’t decided on actual temperature or adjusted for wind chill &c.

Knowing where the sunlight hours are could come in handy when dealing with daylight-based scheduling constraints. Although sunrise/sunset should be estimable from the ecliptic and altitude lines, it’d be much easier to just read it off from some lines.

I’ve always wanted to learn stars but I don’t have a good method. Integrating a planisphere into the clock should do the trick. If I’m ending up with a computer-clock, then I could also mark the planets, something which would be extreemly difficult in a mechanical clock. Knowing the phase of the moon would also be cool.

The day of the week should be given, but I am undecided as to how to accomplish this visually instead of using day-of-week names.

In addition to visual estimation, the clock should allow detailed times to be known so as to better coordinate with other’s or fixed schedules. I have not explored methods for this, other than to integrate a standard clock as a smaller component. If ever I end up collaborating across time zones, I’d like the ability to have more than one standard clock, each set to a different timezone.

There may also be some benefit in including a Baha’i calendar so I’m not so surprised by events.

Notes

what is local sky projection?

Astrolabe Manual

Dark Sky API looks like a good place to get some weather info

Stereographic Projection

A.k.a. planispheric projection?

An important property we’ll use is that stereographic projections preserve circles (also angles, though we won’t use that). The projection is defined from point $P$ on a sphere to point $P'$ on the plane through the sphere’s equator.

Let $R$ be the radius of the circle $O$ and $\theta = \angle IOP$ Given $\theta$, solve for $r = \lVert\overline{OP'}\rVert$.

Note that $\triangle O'AP$ is similar to $\triangle O'OP'$ so that ${\lVert\overline{OP'}\rVert \over \lVert\overline{O'O}\rVert} = {\lVert\overline{AP}\rVert \over \lVert\overline{O'A}\rVert}$. Using trigonometric definitions, ${r \over R} = {R\cos\theta \over R + R\sin\theta}$. With a touch of simplification, we find $r = {R\cos\theta \over 1 + \sin\theta}$.

Ecliptic

In the left of the figure, a cross-section of the Earth is given, with the equator in red, and Tropics of Cancer and Capricorn in blue and green respectively. The ecliptic is defined¹ as a (particular) great circle osculating the two tropics.

On the right of the figure is the stereographic projection. Recall that stereographic projections preserve circles, therefore the ecliptic is still a circle after projection. Projective geometry also preserves tangency, and so the projected circle will also osculate the tropics in projection.

Given two circles of radii $R_1$, $R_2$ centered on the origin, the radius $r$ and magnitude of displacement from the origin $c$ of an osculating circle are given by:

• $r = {R_1 + R_2 \over 2}$
• $c = {R_2 - R_1 \over 2}$

An astrolabe’s (or astronomical clock’s) perimeter is given by a tropic (which one depends on hemisphere).

Grading the Ecliptic

Given $R = \lVert\overline{OP}\rVert$, $\Delta c = \lVert\overline{OO'}\rVert$, and $\phi = \angle IO'P$, find $\theta = \angle IOP$. From Solution of triangles, I see that I have a side-side-angle situation². Let $\alpha = \angle O'OP$ and $\gamma = \angle O'PO$.

Read off that $\sin\gamma = {\Delta c \over R} \sin\phi$ and $\alpha = {\tau \over 2} - \phi - \gamma = {\tau \over 2} - \theta$. With some rearrangement, we get $\theta = \phi + \arcsin({\Delta c \over R}\sin\phi)$.