Tag Archives: TPE

Plan C: How To Plan a Time Sequence Shot

If you missed the last total lunar eclipse, don’t worry. You’ll have another chance in October, 2014. For that, I’m grateful since as you can see I had some problems with my apparatus (the CamRanger). The battery failed after the 7th shot of the moon you see below, and then it stopped working again after 3 more shots, and needed to be slayed and restarted just as the moon was transitioning to fully eclipsed.

But this column is not about our troubles, it is about how I planned for the lunar eclipse shot you see below.

Plan C: San Jose City Hall Eclipse Sequence


The planning began with a list of possible foreground subjects. The San Jose City Hall Rotunda was “Plan C” and the least well researched of my plans. What were plan A and B? Those were one of my favorite lighthouses and a favorite landmark in San Francisco, California. For each arrangement I had to:

  1. Calculate where to stand to make sure the moon would be in an interesting phase above the object. The plan required solving these problems
    1. Determine how high in the sky the moon would be (to know what viewing angle was best)
    2. Determine which DIRECTION I needed to face to capture the moon.
    3. Determine how “wide” a lens I needed to get the sequence I wanted.
  2. Monitor the weather at each location.

After planning all that was left was to make a last-minute decision where the most likely target would have favorable conditions and make any final on-site adjustments.  I had a Plan D, too… but it was also in San Jose so it would have only been chosen had I found some serious obstacle at the City Hall rotunda.

San Jose City Hall Panorama

Calculating the Angles

Determining the angles needed is pretty simple. I used The Photographer’s Ephemeris including all the nifty tricks we teach in our Catching the Moon Webinar. Below you can see a screen shot from the Photographer’s Ephemeris which shows the moon altitude and direction at the beginning of the eclipse. I also moved the time ahead to show the same for the middle of the eclipse.  The moon’s altitude angle (32 to 41 degrees) gave me an idea how close to be to the rotunda to get the moon overhead.  Lower angles allow me to get farther away which allows me to photograph the moon larger relative to the foreground object. This eclipse, however, and the one in October will have the moon high overhead.

Coming up with a Foreground

There is no good substitute for knowing what interesting foregrounds are possible. And also knowing which direction(s) you should be facing.  I knew that the San Jose City Hall Rotunda was generally easterly because I had watched a sun rise through it. I also knew that the eclipse would be at maximum when the moon was in the southern sky so I knew that the range was SE to S directionally.  You can see a diagram from The Photographer’s Ephemeris below for more complete planning.

Calculating Where to Stand

I had to know approximately how tall the foreground object is. For the San Jose City Hall I flat-out guessed.  I found the overall height of the building through Google, and I guess the Rotunda was 60 to 80 feet tall.   My original calculations had me much closer to the building… it was only when I got on site that I realized that there were adaptations that needed to be made.

Watching the Weather

Remember that the Rotunda was plan C.  I kept a close eye on the weather for each of the planned sites.  My favorite weather app is provided by weather.gov – in particular the hourly graphs. We talked about this tool in detail in a prior column.  Why do I like it so much? Because it gives me numbers instead of “partly cloudy”.  It was pretty obvious that the coastal region for Plan A, and the San Francisco Landmark (plan B) were likely to have bad weather – both fog and clouds. Indeed my friends who headed those directions were frustrated by poor visibility.  We had clouds passing through San Jose, but as the weather predictions had read: it got clearest right near totality, and overall was not a hindrance.

Last Minute Adaptation

When I first got to the site, I realized that the Rotunda was taller than I thought. I set up across the street in order to be able to have the moon over the Rotunda… but there were other problems, too. One of the problems is the floodlight on the top of the building. Another was a street light just to the right of where the red marker is in the graph below. These are problems that would only reveal themselves if you visit at night!

And then there are all of those flag posts.  My original guess at the Rotunda Height would have allowed me to stand between the fountain (brown area) and the building… but that clearly didn’t work as the rotunda was too high.  Setting up across the street (and very low) also had its challenges… namely buses and cars that came regularly.  I also realized that I had miscalculated the eclipse time by an hour (forgot it was now daylight savings time).  The miscalculation turned out to be a good thing as it left plenty of time to move around.  It would seem the ideal spot was in the MIDDLE of Santa Clara Street, but that wouldn’t have worked, of course.  Eventually I picked the spot with the red marker as a compromise between altitude of the moon above the structure, removing the glare from the tower lights, the wash-out of the street light, and the many flag poles in the way.

Planning Moonrise

If only my CamRanger had cooperated, I’d have had a continuous sequence of shots of the moon passing over the Rotunda.  There is always October… and maybe Plan A will work for that!

Of course that’s not ALL that was required to get the shot. I also had to composite each of the moon shots into their proper locations. I did that by first taking a panorama of the area, then making sure that when the exposures began I had a piece of the rotunda in each shot so I could properly align the moon over its actual location.  The creation of the image used the Easy HDR method we have previously described.

Stratospheric Exercise for Moonatics

The moon setting behind the US Capitol Building, Washington, DC

If you’re going to chase the moon (or the sun), there are problems that you need to solve.  Here are some exercises to hone your sun and moon chasing skills so you can turn the chasing into catching.  The questions get progressively harder.  Those who have taken our Catching the Moon Webinar will find the answers and much more detail on the private course materials page.  The tools you will need to solve the problems include

You might also want to read some of our past articles on the topic, especially part 1 and part 2.

The Stratosphere Tower, Las Vegas, Nevada

I’ve picked a place that I hope few people are intimately familiar with.  Many of use have been to Las Vegas, Nevada and know that there is one of the worlds tallest towers there. I’ve even had the thrill of hopping on the “Big Shot” ride – the tallest ride in the world.  The Stratosphere Tower is second in the Americas in height only to the CN Tower in Toronto.  The Stratophere’s height above relatively flat surroundings makes it an easier target for catching a sun or moon set or rise from a distance far enough to make the tower seem small.  Even though the Stratosphere is tall, there are complications – including surrounding buildings and surrounding mountains. The farther you move away from the tower, the more significant those obstructions and potential obstructions become.

If you have never seen the tower, above is a relatively close up shot captured from Google Street View. Take note of the height of the mast above the “bulge” in the tower – that’s where you find the ride “Big Shot.” To my thinking it would not be terribly interesting to get an alignment with the sun or moon behind the mast of the tower. On the other hand, if the moon/sun diameter is not as large or larger than the bulge, the shot may not be all that interesting either.

On to the questions, starting from the basic data you need to collect, and on through to solving a “real life” alignment problem.

  1. What is is the correct GPS location for the Stratosphere Tower in Las Vegas, Nevada?
  2. The base of the Stratosphere Tower is at what elevation?
  3. Looking west from the base of the Stratosphere Tower, what is the azimuth and altitude of the tallest natural obstruction in the range of West, south west, to west north west (235 to 295 degrees)?
  4. From the tower at ground level: sunset on Tuesday, August 28, 2012 occurs in line with which of these natural features:
    1. La Madre Mountain
    2. Griffith Peak
    3. Lone Mountain
    4. Frenchman Mountain
    5. Mt Charleston
  5. On what day in August, 2012 will the sun appear to set on (not behind) La Madre Mountain peak?
  6. How far is the summit of La Madre Mountain from the base of the Stratosphere Tower?
  7. Can the Stratosphere Tower be seen from the intersection of Boulder Hwy (Nevada Route 582,aka Fremont Street) and East Sahara Avenue?
  8. If the tower is visible from the above intersection, which part of the intersection provides the least obstructed view?
    1. East
    2. North
    3. South
    4. West
  9. How tall is the Stratosphere Tower (excluding the antenna/mast on top)?
  10. How far is the Stratosphere Tower from the Fremont Street/East Sahara avenue intersection?
  11. What is the difference in altitude between ground level at the Stratosphere, and the ground level at the intersection?
    1. The intersection is 279 feet lower
    2. The Stratosphere is 279 feet lower
    3. No change
    4. The Stratosphere is 1,402 feet higher
  12. What is the altitude (angle above ground) from the intersection to the tip of the mast of the Stratosphere?
  13. On Wednesday, December 19, 2012 from the intersection, the moon will pass closest to the Stratosphere tower at what time:
  14. From the intersection the apparent moon size is about:
    1. Equal to the tower height, excluding the mast
    2. Half the height of the tower, excluding the mast
    3. 1/6 the height of the tower, excluding the mast
    4. Twice the height of the tower, including the mast
  15. On Wednesday, December 19, 2012 at the time calculated in question 13 the moon will:
    1. Pass just under the bulge in the tower
    2. Pass just over the bulge in the tower
    3. Pass behind the bulge of the tower
    4. Pass through the mast of the tower
  16. As seen from the intersection: what is the first day after June 13, 2012 when a nearly full moon (at least 95% illuminated) will appear to set behind the Stratosphere Tower?
  17. What is the NEXT day after the date found in the previous calculation that a nearly full moon will appear to set behind the Stratosphere Tower? (Hint: it’s more than a year later than the previous event).
  18. You want to catch Venus crossing the face of the sun as the sun sets behind the Stratosphere tower on June 5, 2012. In what publicly accessible location would you stand, and at what time so that:
    1. The sun is as large as possible relative to the tower (i.e. you’re standing as far away as practical).
    2. You are confident there is a visible line of sight to the tower.
    3. There are as few obstructions as possible in your line of sight.
    4. There is no mountain, hill or other building behind the tower along the sightline.
    5. You have at least a little bit of room to move to correct for misalignments in your calculations (e.g. standing on a manhole cover in the middle of the freeway is not advisable!)

Good luck!

PS If you’re stumped, I recommend our Catching the Moon (and Sun) Webinar.

NOTE: You are free to ask or answer any of the questions in comments, but those comments will remain private so that those who come along later won’t be tempted to cheat!

Alignment (Part 2 of 2)

If you are finding these articles useful, please spread the word. Share us, Tweet Us, Digg us. Like us on Facebook. And if you would like one on one instruction please consider a Star Circle Academy Workshop. Now back to your regularly scheduled program.

In a previous column: Alignment Part 1 of 2, I touched upon the many elements that complicate capturing the moon near an object on the horizon. Here they are again for consideration:

  • The amount of moon illumination changes daily.
  • The moon’s rising and setting location must be accurately calculated – and it changes daily.
  • Exposures to capture moon detail require the right amount of foreground illumination.
  • The site chosen must have an unobstructed view of the sky in the desired direction.
  • To get a “big moon” it is necessary to get far enough away from the foreground.  If too close, depth of field problems may arise.
  • A well supported telephoto lens is required.
  • Capturing a shot of the moon near the horizon means the atmosphere must be relatively clear of clouds, dust and haze and when very low in the horizon there is more atmospheric distortion.

Figuring out how to tackle the moon location is computationally challenging. Fortunately with the internet there are many free resources to aid in this endeavor. And more fortunately, there is one tool which is almost ideal for the task: The Photographer’s Ephemeris.

We will address the problems step by step.

  1. Obtain the appropriate camera gear.
  2. Identify a suitable target.
  3. Calculate how far away we want to be from the target.
  4. Identify possible vantage points to shoot that target.
  5. Verify (visually, if possible), that the target is viewable from the vantage point and that there is sky behind our target.
  6. Verify that the moon will pass near our target and at an opportune time of day.
  7. Determine how high in the sky the moon should be.
  8. Fine tune the location to be sure the geometry is correct.
  9. Pray for good weather!

The camera gear element of the puzzle is easy: get the longest telephoto lens available. 2,400 mm will work great with a 35 mm (full frame) camera. I do not have anything that big (or expensive), so I use a 70-200 mm lens with a 1.4x extender on a 1.6 crop factor camera.  That effectively gets me 200 \times 1.4 \times 1.6 = 448 mm focal length. The “short” focal length of 448 millimeters means I can not fill my frame with the moon – it would take 32 moons laid out in a grid. Getting more foreground in the shot creates more opportunity for an arresting image however. Besides, those really big lenses are not only expensive, but unwieldy. In fact, they call them telescopes! Working with a crop camera in this scenario is a benefit.

No telephoto? Well then I probably would not bother – at least I would not bother trying to capture moon DETAIL.

Picking a Target

The moon is obviously one of our targets, but we want something interesting in the foreground to pair the moon with. Ideally we want a target that clearly stands above the surroundings and preferably one that allows us to get the proper distance away to maximize the “big moon phenomenon”. How far away?  Here is an easy formula: multiply the height of the object by 114.6.  If the object is 100 feet tall, the proper distance is 1,114.6 feet away.  If the object is 20 meters tall, the distance is 2,292 meters.  If 6 inches, then a distance of 687.6 inches is about right.

For the curious, the number 114.6 corresponds to 1 \div {\tan{(0.5)}}, where 0.5 is the number of degrees of the angular size of the moon from anywhere on earth. If shooting from somewhere else in space more advanced trigonometry may be needed.

It might be tempting to start with something short and nearby, like a golf ball. But getting a good depth of field is going to be difficult.

Let’s get started on the target, shall we? Fire up The Photographer’s Ephemeris (TPE) and follow along with me.  Switch to Ephemeris Mode (it is the first selection in the upper left). In the search bar (lower left), enter “Pioneer Park, San Francisco, CA“.

Now would be a good time to make the TPE window as large as possible, and select the “Satellite” mode in the map.

Figure 1: Pioneer Park Coordinates and Elevation according to TPE

Right above the upper right corner of the map you should notice two things: an elevation (here shown as +190 ft), and the GPS coordinates (37.8…blahblahblah).  If you prefer metric (or it shows metric and your prefer feet, you can change that using “Configure”).

Looking at the zoomed in map, put the cursor over the map near the bottom and click and drag upward. The map should move and soon you should see a conical shape casting a long shadow. Hooray. You found the Coit Tower. Double click in the center of the structure and it should look about like this.

Figure 2: Coit Tower in Pioneer Square, San Francisco, CA

Here I cheated and moved the elevation (+266 ft) and the GPS coordinates on to the image from above the map from the bar above.  I also zoomed out a bit so you can see the parking lot that you first landed on.

Did you notice that the elevation moved up from 190 to 266 feet?  You gained 76 feet in just a few parking spaces! It is steep there, but that number is NOT a measurement of the height of the tower, my friend. That is the elevation of the BASE of the tower. Don’t believe me… click a few spots near, but not on the tower or the building.  Click things farther away if you like, I’ll wait.  As you can see from the image at left taken from the parking lot, there is clearly not a gain of 76 feet between the two places.  The elevation information comes from a variety of sources, mostly the United States Geological Services (USGS) data.

What you hopefully learned is not to COMPLETELY trust the elevation shown. The elevation does not include buildings or trees and is not that precise, but it will probably be good enough.

In a while you will need to know the height of the tower above the base. Guess where you can find that? Yep, Google. Did you find it yet? It’s 210 feet (65.4 meters) tall.

So doing the math: ideally we’d like to be 210 x 114.6 feet away (24,066 feet or 4.5 miles) to have the moon’s apparent size be as big as the tower. Unfortunately going to the east, our choices are mostly in the San Francisco Bay, farther away on the Oakland Shore (near the Bay Bridge), or closer. Treasure Island looks like a good spot. It’s 2.11 miles and there is a lot of flat, publicly accessible shoreline to move along to align the moon behind the Coit Tower.  And besides even though the Coit Tower sits up on a high hill, only about the top half of the tower is above the sky line. So 2.11 miles might work out very nicely.

Since we have chosen a site to the east of the Coit Tower when can the moon appear behind it?  Near moon SET of course.

If you want your diagram to look exactly like mine, change the calendar to June 15, 2011. And change the Ephemeris mode to “Detail” (use the D key, or click the box down near the calendar).

When you switch to Detail mode, a hollow little gray marker will appear. Usually to the right of the red marker near the right edge of the map. Don’t lose it – you’ll need it in a minute.

Calculate the Moon Location Near Moon Set

You may have noticed all those colored lines extending from the Coit Tower in Figure 2. Here is what they mean: the light yellow line is the direction of sunrise, the orange line is the direction of sunset. The light blue line is the direction of moon rise and the dark blue is the direction of moon set.  All by itself that won’t help much. To see the moon setting in the west behind the Coit Tower, you obviously must stand to the EAST. But where?

Zoom out your map until you can see the Coit Tower on the left, and Treasure Island on the right. Make sure you are in Detail Ephemeris mode (you’ll know when you see a graph like this:

Figure 3: Sun/Moon graph and time slider.

Your map will look something like this:

Figure 4: SF Bay Map with Coit (lower left) and Treasure Island (upper right)

I have stripped off all the stuff around it to focus your attention. You’re focused, right?

Now would be a good time to play with the time slider. Click and drag it. Whoa! Did you see the lines moving? The skinny ones, that is.  There is a lot going on here, but the one thing you’re not yet seeing is where you need to stand to see the moon behind the Coit.

Stephen Trainor, the author of TPE put a cool feature in this tool. He did so because I asked politely and I support him with donations – I urge you to do so too. Buy his iPhone/iPad version of the tool (or Android if that’s available) or make a donation if you’re using the desktop (free) version of the Ephemeris. It’s the right thing to do!

Move your time slider to 5:13 as in Figure 3.  Now hold down the shift key. Did you see the thin blue line jump out? That blue line traces roughly where the shadow of the moon would appear. It can’t be completely accurate, however since the exact location would have to take into account topography, trees and man-made structures. We helped ourselves around that worry by choosing a flat shoreline where not much can get in our way.

Now would be a good time to find that hollow gray marker. Lost it? Click “D” then “D” again. It will appear near the right side of your map connected by a dim gray line to the red marker.

Hold down the shift key again, and drag and drop the gray marker on the Treasure Island shore DIRECTLY over the dark thin blue line.  Zoom in if you have to and get the marker EXACTLY on the line. And try not to stand behind a building or a palm tree.

You probably didn’t notice, but three things appeared at the bottom of your Ephemeris Graph in the box labeled Geodetics.  Those are: Apparent Altitude (which here will be negative), Change in Elevation (also negative), and Distance and Bearing.  Each time you move the gray or red marker it will recalculate the distance, altitudes and angle between gray and red.

One last little coup for now… notice next to the word Geodetics it has a little red and gray dot with an arrow over the top? Yeah, click that. The gray and red locations magically flip. Now all of your altitude and elevations will be positive. The calculations are FROM red TO gray. Since red is at sea level now, and gray up 266 feet on the top of Pioneer Hill the angle above the horizon toward the hill is  positive: specifically the base of the Coit tower is 1.1 degrees above the horizon. So can we conclude that the moon must be 1.1 degrees high in the sky?

NOPE. Sorry, we can’t. So close and yet SO far!

Q: What is wrong? Did you figure it out?
A: TPE has no idea how tall the Coit Tower is! (Stephen tells me one day he’s going to add the ability to specify the height at the red or the gray marker), but for now, YOU have to make that adjustment yourself. I’m afraid it’s going to involve some math. Trigonometry, actually.

What is the CORRECT Angle?

If you can answer this question, you’ll get the solution. “If an object at 2.12 miles away is 210 feet taller than the current 1.1 degree elevation, how many more degrees will that be?”

\tan^{-1}(Height / Distance) = altitude\ in\ degrees

Or in this case  InverseTangent( 210ft / 11311ft ) = 1.06 degrees.

So the CORRECT altitude is 1.06 + 1.1 or 2.16 degrees.

Hint Use the built in calculator in MS Windows in Scientific mode (Alt+2). Set the units to degrees. To get to the inverse tangent function (also called tan-1) use the “i” (inverse) key.

NOTE: If you do not want to do the trigonometry, there is another way to find the angle: use your camera.  Go to the desired site, take a picture with your telephoto lens aimed level with the horizon and with the top of the object visible. Determine the angular field of view of your lens/camera combination. Then measure the height of the target on the image and use the ratio of the height of the target to the field of view.  That sounds complicated, but it’s actually pretty easy. Using a 200mm lens, my angular field of view is 4.3 degrees. My photo shows that the tower spans 1000 of 1800 possible pixels. So the tower is 4.3^{\circ} \times (1000 / 1800) = 2.388^{\circ}

Now that we know the moon altitude must be 2.16 degrees we do not have to start over. Let us make sure the red maker is back on the tower and adjust our slider until the moon height is 2.16 degrees, then follow the line of the direction of the moon set to get our new location.

Of course if we move significantly higher, lower, nearer or farther away we must recheck the angle calculations.  In a hilly or mountainous location it is extremely non-trivial to get all the heights and angles just right. Using the “Terrain” mode of the map may help, but changes of a few dozen feet may make a big difference in the alignment.

Just remember the following things:

  1. The satellite maps may be out of date. A tree, building, crater, fence or obstacle might be in the location you want – or directly in front of it.
  2. There is no substitute for prechecking the line-of-sight BEFORE the event (see 1 above)
  3. Terrain maps are not visible when zoomed in.
  4. Elevations of the terrain are ROUGH.
  5. Moving 10 feet to the left or right may make or break the shot.
  6. I am NOT available to solve your trigonometry problems! Ok, I am but there will be a fee!

But wait, there’s more!

Getting the Ideal Exposure

To get the ideal scenario for moon details AND foreground light, it helps that the sun is on the opposite side of the sky and sometime during Civil twilight. In Figure 3, above, notice how the time we arrived at (5:13 AM) has the moon 2 minutes before Civil twilight.

Wondering what Civil twilight is? It is the legal equivalent to either dusk or dawn. Dusk when the sun has set, dawn when the sun has not yet risen. Signs that say park hours are “Dawn to Dusk” mean something quite precise. But those times change daily. For more click on the word “Civil” in the Ephemeris and it will tell you! Or take a look here.

The ideal exposure for detail in a full moon is about 1/100 of a second at ISO 100 and f/9. But atmospheric conditions, and the moon’s altitude may significantly affect the settings to use.  The best choice of aperture is to stop down enough for a sharp shot that keeps the foreground through to infinity (the moon’s focal distance) in focus.  If your foreground is at or beyond your hyperfocal distance (as it most probably will be), you’re good to go.

The problem, of course, is that your foreground is probably not going to fare well unless it is also well lit – so bracketing your exposures is always a great idea. The darker the twilight, the wider the bracketing needs to be.

Verifying The Sight Lines

After all the calculations and planning, a group of Bay Area Night Photographers ran out at the crack of before dawn to capture the “Full Moon Set behind Coit Tower“. One of the bleary-eyed ambitious photographers was Phil McGrew. Phil get’s extra kudos for going the morning before the planned event (that’s two thermoses worth of coffee) and here is what he got:

Photo 5: Coit Tower? And the Moon by Phil McGrew

The moon is in the right spot, but, whoops, there is something else in the shot, too! A big square building blocking the view behind the tower.  A more thorough scouring of the map in Figure 4 might have revealed the problem (see Figure 5).  Behind the Coit, and set up on a hill are a series of apartment buildings.  From almost anywhere else on Treasure Island, or Fort Baker in Marin, the Coit tower sits all by itself on the skyline.

Figure 5: Oops! (Click to see it larger)

What are the takeaways here:

  • There is no substitute for direct observation from the planned location. Any number of things can be a problem from light posts, billboards, trees and shrubs to, well hulking square buildings in the line of sight.
  • Extra scrutiny of the sight lines in TPE *might* save one from a needless trip to get a direct observation.
  • Knowing the local topography helps as does picking a structure or formation that clearly stands above the surrounding area.

Phil also discovered that the lack of brightness on his foreground meant he had to choose between exposing for moon detail, or exposing for the foreground. In Photo 5 he nailed a great foreground exposure and might be able to tease some moon detail out of the RAW file.  Or he could resort to…

One Last Trick – HDR

First I am a hater of images that have been composed by dropping a well exposed (oversized) moon into a separately taken landscape. There are technical challenges to embrace here so why not embrace them! Besides my desire as a scientist and engineer is to maintain reality through honest acquisition.

I am not, opposed, however, to using technology to overcome the limits of technology. Namely a camera can not readily capture the range of exposure – brightest to darkest – that the human eye can so a trick called “High Dynamic Range” photography (also called tone compression, tone mapping or image fusion) is sometime a necessity.

In the morning of June 15th, moonset behind Coit Tower was the target as describe earlier. That evening, moon rise behind the Transamerica Building was the goal.

You can click the diagram to the left to see where we were. As kismet would have it, the very parking space that I had calculated at the correct spot was open and I pulled in!

The haze was heavy, contrast was low. But in the end, the moon peeked (and peaked) right on schedule and right where it was supposed to go. It is always satisfying when things work out like that. More satisfying if the weather is great.

Rising Moon Collage

Photo 6: Moon rising over San Francisco (through the haze)

The fifth shot in the panel above is like all of the others in that it is a three-shot bracketed exposure combined using Photomatix Pro. The three shots were:

Figure 7: Bracketed Exposures

A wider bracketing range may have helped, the haze was quite thick. Using Photomatix Pro, playing with the knobs a bit I got this result:

Yellow Moon [5_032356-8tm]

Photo 7: High Dynamic Range Composite of 3 Images

I can only imagine what having a clear day to shoot in might have accomplished.

Best of luck on your alignments!

Comments, questions, praise, quibbles over the math – we’ll listen. Find us on Facebook.  Or attend one of our workshops. Want to keep it cheap, hook up with me, Steven in the Bay Area Night Photography group.