Hi Tom,
Your indoor sundial is a great idea. I’m now searching the internet for convex
mirrors. I note that you have used a convex ball mirror, I guess that’s needed
to cover 6am to 6pm? I’ve always wondered how to do that.
Thanks for a great idea,
Roderick Wall.
From: Tom Egan
Sent: Thursday, February 16, 2012 12:54 PM
To: sundial mailing list
Subject: A compact indoor sundial using a convex mirror on the north sideof the
roof
Dave Bell, in his 2/12/2012 response (appended) to Ruben's invitation to us to
view his cosmic room, discusses ways to use flat mirrors -- one being
effectively a reflective pinhole aperture -- to obtain a noon line throughout
the year in a living room that is well north of the equator (at 37.3 deg.).
I've developed an indoor sundial that uses a large convex mirror to capture the
sun's position throughout the day and year. It allows a compact dial face, if
desired, that can fit on a table-top. (For development purposes, I'm using a
target the size of a piece of letter paper.)
I presented a slide show on it at the 2011 NASS annual conference in Seattle
WA. I'll summarize the description from an article that's in preparation:
A sundial system based on a three-dimensional convex mirror can overcome many
of the limitations of a plane mirror. The convexity provides a wide field of
view, reducing the apparent arc of the sun’s travel throughout the day and year
to no more than the width of the mirror. This, in turn, permits the dial face
to be compact enough to fit on a table-top.
A convex mirror, while creating a compact view of the world, is a virtual
view. That is, additional components are required to turn the sun’s disk into
a real image of a sun spot. This is accomplished by incorporating a pinhole
“camera obscura” that projects the sun’s image onto the dial face, as for
example, a piece of paper with drawn hour lines and seasonal markings.
When the convex mirror is mounted above a window on any side of the building
– even the north side – and above the roof line, the system can deliver the sun
spot to the dial face inside the building. Being above the roof line provides
the mirror with a view of the sun’s travels that is least shadowed by nearby
structures and vegetation.
Two flat mirrors at 45 degree angles are used to change the path of light
that is reflected from the convex mirror from vertical to horizontal --
allowing perpendicular entry into the building through a window to avoid
distortion -- and then again re-orienting the beam to vertical to allow a
convenient and easy-to-read horizontal dial face.
Figure 1 (attached) illustrates the concept. The photo shows a side view of
the gazebo I'm using to develop the system. The convex mirror reflects the
sunlight down to a flat mirror tilted approximately 45 degrees just outside the
window. This flat mirror turns the sun beam from vertical to horizontal so it
can penetrate the window glass. On the inside of the window, a mask with
pinhole is affixed, creating the camera obscura effect. The brightest object
in the obscura image is the sun, which casts a bright spot onto the dial face
after being verticalized by another 45 deg. flat mirror. This bright image of
the sun indicates both time of day and season of the year, since it acts as a
nodus.
To lay out the hour and month lines on the dial face, I developed a math model
of the system using vector notation. The smooth mirror is conceptualized as a
large disco ball covered in infinitesimally small flat mirrors. This permits
the use of simple relations suitable for plane mirrors.
The equations turned out to be implicit ones, so I used the Solver feature of
the Excel spreadsheet to perform a global step search to converge to a solution.
Bob Kellogg subsequently devised an elegant least squares optimizer programmed
in DeltaCad to solve the implicit equations.
We're in the process of documenting this for the NASS Compendium.
On 2/12/2012 5:42 PM, Dave Bell wrote:
Beautiful rooms and concepts, Ruben! A wonderful teaching facility.
Your low latitude is certainly a help in fitting these dial displays to an
interior.
After viewing your pages, I once again looked at my living room with an eye
towards a noon line, as I have a roof window well situated near the South
wall, but due to its height (just over 10 feet) and my latitude (37.3°), the
resulting analemma doesn't fit the living room at all. It's too far from the
South wall in Summer, and extends well past the open floor space in Winter.
However, it did get me to thinking again, and doing some calculations.
If I designed the line for a more southern location, say 9° North, the total
length of the analemma was much shorter, and the Summer end would extend
South of the ceiling hole, quite close to my South wall.
Moving my home 28° to the South isn't very practical, but it occurred to me
that I could introduce a deviation in the incoming rays, and accomplish the
same result! I thought of a prism, but the geometry isn't very good; finding
the right prism (apex angle and refractive index) would be difficult and/or
expensive, and I would have to deal with chromatic dispersion, as well. But
a pair of mirrors, even first-surface mirrors, is much easier. If the first
mirror (to the north of center) was blacked-out, with a small, "pinhole"
reflective aperture, inclined vertically or slightly to the North, its
reflected beam could fall on the second, larger, flat a few inches to the
south, inclined 14° more to the north. The 14° apex angle of the pair would
introduce a 28° deviation towards the south, independent of the exact
inclination of the pair. (They would have to be accurately aligned
East-West, of course.)
Any inaccuracies in mirror angles and orientation would be hard to allow for
in plotting the Noon analemma, but I could always fall back on the ancient
empirical method of driving a tack at 12:00 local solar time, once a week!
Something a little like the ASCII sketch below.
Assume the light enters from above and to the right...
|
| \
| \
\
\
Dave
-----Original Message-----
On Behalf Of ruben nohuitol
Hi friends, please look my cosmic room,
my page is www.ruben.mx
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