The easiest and most common mat layout is one with the widths of all four borders equal. If you are forcing a picture into a standard-sized frame, however, that’s not always possible. And then there’s the matter of bottom-weighted mats.
Bottom-weighted mats, or mats with the bottom edge wider than the others, were introduced long, long ago. Some say that pictures centuries ago were hung very high on the wall and the bottom width of the mat was increased to compensate for the ‘distortion’ of that perspective. Unfortunately, that story makes no sense; top-weighting would be required to correct for the top being further from the viewer than the bottom. Another explanation involves the notion of a difference between the visual or optical center and the geometric center. Yet others claim it is to compensate for the drop of the mat in the frame due to tolerances necessary to account for expansion, etc. For whatever reason, bottom weighting could be seen as an attempt to fool your audience or overcome optical perceptions, whichever you prefer. As commonly practiced in “finer frame shops everywhere”, the bottom width is generally increased ¼” to 1″, depending on the size of the pictureref.
Using Standard Mats
But how would one incorporate bottom weighting while fitting an image into a standard-sized mat? For example, if the vertical difference between the hole size and mat size is greater than the horizontal difference, and assuming the left and right borders will be the same width, is it better to:
Make the top and bottom borders equal,
Make the top the same size as the left and the right and put all of the extra width on the bottom,
Make the bottom larger than the top by some fixed amount,
Make the differences even more subtle by making the difference between the top border and the side borders the same as the difference between the top and bottom borders?
Let’s clarify your choices with an example. Suppose you want a 4″-high hole that’s 7″ wide in a standard 8″-high by 10″ mat. The horizontal difference between the mat size and the hole size is 10″ – 7″ = 3″, so if you want the left and right borders to be the same, each will be 3″ ÷ 2 = 1½”. The vertical difference between mat and hole size is 8″ – 4″ = 4″.
Would make the top and bottom borders the same, making them each 4″ ÷ 2 = 2″.
Would make the top 1½” like the left and right borders, leaving 4″ – 1½” = 2½” for the bottom border.
Uses the customary bottom weighting, which the one reference I give above lists as ¼” for an 8″x10″ mat (personally, a ¼” bottom weight isn’t worth the trouble). That means the top border would be (4″ – ¼”) ÷ 2 = 1⅞” and the bottom would be ¼” more, or 2⅛” (notice as you check your work that 1⅞” + 2⅛” = 4″). Finally,
Is a tad more complicated. Let’s call the difference between the left or right border width and the top border width “d”, such that
1½” + d = T (for top border width).
Then the bottom border (B) would be
T + d or (substituting the last expression for T) (1½” + d) + d = 1½” + 2⋅d.
Since T + B = 4″, then (substituting for T and B)
(1½” + d) + (1½” + 2⋅d) = 4″, meaning 3″ + 3⋅d = 4″ or 3⋅d = 1″, meaning d = ⅓”,
so (substituting back into our equations for T and B)
T = 1½” + ⅓” = 15/6” and B =15/6” + ⅓” = 21/6”
(again noting that 15/6” + 21/6” = 4″) .
The choice you make would be an artistic decision, but I think A is the most common answer. Choice C could be used for traditional bottom-weighting, as in our example, or could be used for some other more artistic value. Technically, both Choices B and D are possible results of that equation. B would be exactly what you get when you want bottom-weighting and are not restricted to standard mats; it would work best if the resulting difference between the top and bottom borders is not too much greater than the customary bottom-weighting distances mentioned above. In our example, it yields 2½” for the bottom border, which is an inch larger than the other three borders and may just be too much. In our example, C and D are very close, and remain close when we change the amount of weight in C from ¼” to ½” (as shown by the lighter blue opening). D is more subtle than C, but may only be worth the effort when the difference between the left and top borders is small enough to fool somebody. In other cases with different numbers, results may vary.
With Larger Side Borders
If the horizontal difference between the hole size and the mat size is greater than the vertical difference, you could face up to the same number of choices as above, but you are working with less material for the top and bottom borders and I think it is usually better to keep things simple and make those borders equal.
Differing Left And Right Borders?
Do the vertical borders always need be the same size? Although I can’t say I’ve ever seen or read about different-sized side borders, I’m not convinced that uniformity is strictly required. For example, in photography, as in older art forms, there a “rule” of spaceref that says, among other things, that there should be plenty of space on the side of the subject into which it is looking. If you have a “perfectly” centered and close-cropped picture of your mother looking to your left, could a mat with a wider border on the left side create the space that’s lacking in the image? Maybe you could even choose a mat color that is a pastel version of the background to her right (your left)? Maybe a contrasting outer mat could be added with traditional (identical) vertical borders.
I present the above thoughts to give some background and (more importantly) stimulate your own creativity. If you think of other possibilities, I’d be thrilled to have you add them to the comments. Thank you!
A few years ago, Nancy took a photograph of her junior-high-school best friend JoAnne’s father on a tractor at his northern-Florida homestead and gave it to JoAnne. After he died, JoAnne brought the picture back, along with some of the old fence pickets from the property, and asked if we could use them to frame the picture. After a lot of research, planning, and experimentation, this is what we came up with:
The pickets were thin, dilapidated, warped, and dirty. The few articles I did find were about “barn wood” which, although it had a slightly distressed surface, was still thick and sound with straight, flat, parallel and perpendicular sides – none of which applied here. The articles were not all that helpful and not all that well written. I thought this project could be an opportunity to learn something new, and to share it with you. I hope I took enough notes and pictures to show you exactly how this frame was made. At least that’s the plan.
But First . . .
From math class, you may remember that one problem-solving strategy is to solve a simpler problem first and then use that answer to help solve the harder problem. With that in mind, I have an idea to write a series of short articles on working with weird wood to make frames, so that I can draw on that information in the final article about this project. The first article will be about working with pieces of moulding of different widths in the same frame. Then I see a discussion of moulding where the inside and outside edges are not parallel. Maybe then we’ll work with wood with a wavy inside edge. Following that may be a discussion about what to do when your moulding is curved (but with uniform width). But even before the first article, I may have to give a short post about matting techniques. My hope is that by doing all of this it will expand your view of what’s possible and it will stimulate those creative juices of yours. These articles will probably not be consecutive blog posts; another art festival season has just begun and other things will invariably come up as I am writing these pieces. So please be patient and stay tuned. Thank you!
I’ve had a few opportunities lately to help people edit their photographs where they wanted to combine two photos into a composite and were worried about the relative sizes being proper, especially when the camera settings and/or the scene were not identical. Based on these experiences, I’ve created a couple of scenarios to introduce certain concepts.
A Safe Selfie
Suppose you need to add part of a wild animal behind you – to make a safe selfie, if you will. Most of your composite shots, where two objects are moved around in an image with plenty of other size reference points, fall in this category. Generally, combining subjects is a two-part process:
Resize One Picture To Match Pixels Per Inch For The Two Subjects
First, you must know the physical size of the two objects. In the selfie case, you probably already know your own size, but suppose you want to place the head of an animal (whose picture you took from a safe distance) right behind you. In one case, I Googled an animal to get size information but could not find the size of the head of an adult male of that species. They did list shoulder height, however. So then I found a picture of this type of animal online that showed the head and enough of the animal to measure its shoulder height (since my client’s picture of the animal did not have all of these features), and by comparing the two measurements on the picture, found the size of the head. Fortunately, it’s not always that hard. Now, measure the subjects in your two pictures in pixels. Divide the number of pixels by their length in inches. Resize one of the images so that the pixels per inch that you just calculated are the same in both pictures. For example, let’s say your 6-foot height (72″) measures 792 pixels in the first picture. That’s 11 pixels/inch. The alligator or bear’s head, which you found to be 24″ long, measures 192 pixels in the second picture, for 8 pixels per inch. You can either enlarge the ferocious animal or downsize your likeness. If you want to reduce your size, open the first picture in Photoshop. Click on “Image” in the menu, and then “Image Size…”. Make sure the “Resample” box is checked. Multiply the pictures existing resolution (say 300 Pixels/Inch) by the target pixels per inch calculated above (in this case 8 to match your animal) and divide by your starting pixels per inch (11). That gives you 218.182, which is what replaces the existing 300 in “Resolution”. Hit “OK”. Now, you and your animal head are the appropriate sizes, if you plan to put them side-by-side in your picture. If you want to move one in front of the other, its size will change.
Use A Vanishing Point To Resize One Subject For Changing Distance
Now you can use vanishing points to maintain the correct sizes as you move your objects into place. We’ve already explained that process in Using The Vanishing Point To Keep
The Size Right When Moving Wildlife Around. I would like to point out that as long as your object stays the same distance from the camera, or in the same focal plane, you can move it up, down, and all around without changing size. If you move it closer to the camera, it should get larger. When you move it away, make it smaller. Once you resize it for its new distance, you can again move it up, down, and all around within that new focal plane at no extra cost. Also, once you find the horizon in your picture, it doesn’t matter which point along that horizon line you use as the vanishing point; all of them will resize your object correctly. Pick a point that is conveniently off to one side far enough to make long enough construction lines to give you some precision when changing size.
A Beach Scene
I also helped somebody with a beach scene that invoked two simpler special cases of the resize problem. The base or background image was a wide-angle beach scene and the photographer wanted to add objects that they took with a zoom lens at the same scene that same day.
The photographer’s intent, in this case, was to shoot objects floating on the water near the horizon with a strong zoom lens and add them to the picture so that they looked closer. An object floating in the water is restricted to a specific plane in such a way that its distance from the horizon is directly related to its distance from the camera (within a camera’s normal field of view), which is the determining factor in that object’s relative size. As long as the horizon is in the picture, this is no problem. Whether you add that object at its original pixel size (as magnified by a telephoto lens) or even if you scale it further in Photoshop (by holding down the shift key to preserve the aspect ratio as you move a corner of the selected border while using the Move tool, for example), as long as you keep the horizon of the added object directly on the same line as the horizon of the background, the invisible construction lines from an invisible vanishing point will ensure the size and placement of your object are in agreement. If the horizon is not in the picture, then you need to look for other size references and handle as in the first general case discussed above.
Birds (or other airborne objects) are even easier. Their position is unrestricted and, more importantly, there are no other size references in view so there is really no way of knowing how large the object really is or how far away, meaning that if it were a ball, there would be no way for you to tell if it is a large ball far away or a smaller ball up close. If you are familiar with the object and know how large cooper hawks are, for example, then your brain will automatically assign the hawk an appropriate distance based on its size when trying to make sense of the picture. You can put that hawk just about anywhere and the viewer won’t know the difference. Obviously, if you put a pigeon in a hawk’s talons then each would act as a size reference for the other and at least their relative sizes would have to match. If they were not touching (or near enough to imply an interaction), there would be no such restriction.
There are other positional clues besides size to think about. On a sunny day, an object’s shadow provides positional information, namely the object’s relationship to the sun, which must be consistent throughout the image (for best results).
When you shoot someone’s face with a wide-angle lens from a close distance, it will not look the same as when you shoot the same face from far away with a telephoto lens. An example of this is shown in the fourth image from the top at Choose the Right Lens to Make Flattering Portraits (the only image that’s in color). I’ve seen some experts blame this on lens distortion (as the guy in this otherwise great video at Focal Length for Storytelling – How Lens Choice Affects Your Images, but I don’t consider that lens distortion. There is such a thing as lens distortion, but in this case, the subject’s nose really does look bigger and the ears really do disappear behind the cheeks if you were to close one eye and look at that person from 3″ in front of their face. I call that a perspective shift and it is strictly a matter of angles and geometry, not lens issues. The “distortion” occurs when you take that image out of context by changing the perspective, which happens quite noticeably when you move an object from very far away to very close (or vice versa) in your image or if you take a 180° panorama, for example, and print/display it small enough to cover only 15°. This perspective shift is virtually impossible to correct in Photoshop, so don’t go too wild while moving things around in your picture. (Interestingly, it is by a lack of any perspective shift that you can catch somebody who created a reflection in their picture by just adding a flipped subject in post-processing. The explanation of this tangent to today’s topic would require a separate article, however.)
To see the Note click here.To hide the Note click here.
Well, that’s about everything I know on this subject. Please feel free to contribute your own hard-earned understanding of this issue for the betterment of photography in the comment section below. Thanks.
As we discussed on the Services page of our website, we digitally “stretch” our image before wrapping it around the edge of our gallery-wrapped canvas images. Here’s how we do that:
Our gallery wraps are either 3/4” thick or 11/2“. On the thin ones, I usually take the 1/4” strip along the edges and stretch it to 1″, thus having an extra 1/4” to wrap around to the back side to cover for variations in the printing and stretching processes. On the larger ones, I take 1/2” and stretch it to 2″ (thus leaving 1/2” on the back). I wouldn’t stretch the image more than four times its original size, but you could go less. To do that, you would effectively be taking a wider margin to wrap around the side.
As an example, if I want a 12” x 18” image stretched around a 11/2” frame, I would crop the image to 13” x 19”. Then, after putting guides 1/2” in from each edge and another guide right on each edge, I would increase the canvas size 3” in both dimensions to get 16” x 22” with the image centered.
To see the Note click here.To hide the Note click here.
Click Image ⇨ Canvas Size…
Put a check in the Relative Box
Make Width and Height 3 Inches
Make sure Anchor dot is in center of the grid
I would then use a scale transform to digitally stretch the outermost 1/2” to 2” wide, filling the canvas.
To see the Note click here.To hide the Note click here.
Make sure Snap is checked in the View Menu
Use Rectangular Marquee tool to select the 1/2” strip between the guides along one of the edges
Click Edit ⇨ Transform ⇨ Scale
Place the mouse cursor over the little square in the middle of the outer edge of the selected area and drag to the edge of the canvas
Hit the check mark to finish the transform
Repeat Steps 2 through 5 with the 1/2” strips along the other three edges
(Actually, I first do the four corner squares separately, but since only a small bit along the edge of those squares has any chance of being seen, you could include them in either the horizontal or vertical strips (or even both)).
Then I add a blank (transparent) edge around the image representing the canvas I need for stretching the canvas around the frame by increasing the canvas size by double the required margins in both dimensions, the same way we did above. That margin would be at least the width of the moulding along the bottom (1″ for the 11/2” moulding we are using now) and enough extra to get a grip with the canvas pliers (for me that’s at least 3/4“). That would make the image’s final dimensions at least 191/2” x 251/2“. When I am finished, I add layers with cut lines, fold lines, staple lines, positioning marks for the hanging hardware, etcetera, but that is a personal matter beyond the scope of this article.
That’s about it. Feel free to leave comments or questions.
Our latest image, Eclipse Over Long Pine Key, of the solar eclipse in August in the Everglades was by far our most complicated yet. While spending hours and hours overcoming challenges in post-processing, I wondered if I was wasting my time – would anybody even be interested in the results and was each of these steps really necessary, or was I just over-thinking a problem again. You be the judge.
Another Brilliant Idea
Although I wrote a blog article with suggestions for taking eclipse photographs with either your smartphone or camera, we had no plans to take any ourselves. Then, about twenty hours before the start of the event, I got another ‘brilliant’ idea – the concept for a multi-Gigapan panorama and time-lapse image of the solar eclipse.
What I pictured were images of the eclipsing sun as it soared barely over the tops of the skyscrapers in downtown Miami. This image would be taken from a balcony somewhere between the tenth and twentieth floor so you could capture an interesting street view in the foreground. Of course, it would be a Gigapan to give plenty of detail (we’ve been influenced and inspired by fellow Miami photographer Robert Holmes), but one panorama would not be enough. Ideally, as you took each of your carefully placed sun shots, you would need to shoot an area of the city directly beneath it so that the constantly changing light intensity and the buildings’ shadows would change in synchrony with the sun. The sun photos would be taken with a camera with about 16-stops of neutral density filter. For ease of execution, it might be better to have a second camera for the Gigapan unit. Each associated Gigapan panorama would need to have enough overlap with its neighbor to be able to stitch them together and go high enough to capture the sun’s position so you could correctly add in the better sun shot later, and give enough headroom for the sun’s path.
Reality Sets In
Some of the challenges of this project were unanticipated and, as you will see, some were self-inflicted. Although this was our most ambitious project yet and would turn out to be tremendously challenging for the technical support crew (me), Nancy, with her artistic eye, still made the aesthetic decisions. First of all, we don’t do cityscapes. We had less than a day to get ready for this shoot, and already had a doctor’s appointment for Nancy’s mom scheduled for the morning of the eclipse. We brainstormed and searched Google Earth to make a list of possible sites. Then, after dropping off Nancy’s mom from the doctor the next day, we headed to the Everglades. The pines on the island in the lake at the Long Pine Key Campground were our first choice. We got there around one o’clock, but I was disappointed in the height of the trees, so although the equipment takes over a half an hour to set up (and the eclipse started at around 1:30), we headed to our second choice, Pine Glades Lake, about six miles away. It proved to be completely inadequate for our needs, but the side trip ate up another hour of valuable time. By the time we returned to Long Pine Key, found the ideal location, and set up the Gigapan, the eclipse was already near its peak (around 2:45). This may still be doable, I told myself; for the sun I can just flip a copy of the pictures we get in the second half of the event. I set up the Gigapan to combine all the missed areas in the first panorama, and then took six more Gigapans after that. Since I only packed one tripod, as I operated the Gigapan Nancy had to lie on her back, pull the 100-400mm lens back to 100mm so she could find the sun on the LCD monitor in Live View, zoom out to 400mm, focus, and push the button every five minutes. Shortly after 4 o’clock, after shooting our seven Gigapans and twenty sun shots, we packed up and went home.
The Real Work Begins
Stitching The Panoramas
Photoshop does pretty well putting together smaller panoramas, but bogs down as the number of images grows (for small panoramas, Canon’s PhotoStitch, which came with the camera, does an even better job). Gigapan Stitch, which comes with their motor drives, does pretty well on the larger panos, but I usually use Kolor’s Autopano Giga because it has more choices in projections, better control options, and does better at eliminating ghosting (which happens when things (including even trees and branches) move between shots). I’m still climbing the learning curve, which added to the time needed to stitch together all seven panoramas. I believe there must be a way to combine the stitched panoramas into one large image with Autopano, but I couldn’t find it in time for this project, so had to warp and stitch the individual panoramas together by hand. Unlike my imagined cityscape, the ground location of the camera and the intervening lake make the concerns about the shadows much less significant. Because of that, and the fact that each of the panoramas was larger than strictly required, I was able to cover the field with only two of the seven panoramas, but then added the last panorama – the one with clouds.
And Now For The Hard Parts
First, The Bad News
When I saw the first of the sun photos on my computer screen, I was amazed that one could capture such details as sunspots with a regular camera. And then it occurred to me that this meant I wouldn’t be able to just flip all the sun shots to re-create the shots we missed, and that since we didn’t have the sun camera on a tripod, I would have to find a way to make sure the suns had the right orientation with the horizon if I wanted this image to be anatomically, or should I say astronomically correct. Also, although I expected the sun in the panoramas to be blown out, I thought I’d still be able to use its position to place the new sun. Wrong! The blown out area was much too large to be useful.
Is There An Astronomer In The House?
But how would I determine the proper position and orientation? For position, I used “The Photographer’s Ephemeris” (TPE) app on my phone (cost: $3) to find the azimuth (compass bearing) and altitude (angle of elevation) of the sun from the location of the camera at any time during the eclipse and I had taken the compass bearing and/or the angle of elevation of a few of the features in the image during the shoot. With this information, I mapped out a grid on a separate Photoshop layer, placed a small circle at the location of each sun, and then used the Pen Tool to mark the sun’s whole trajectory.
For the orientation, I checked the web and even Facebook for pictures or information showing the orientation of the sun and moon’s path across it, but could not find the information I needed. By then it was several weeks after the eclipse, but I thought I could just go out and take new photographs of the sun with the camera on a tripod to get its orientation. My first image, at around 9 am, showed a different sunspot pattern than shown during the eclipse. A photograph taken around 1:30 pm showed that same pattern, but the sun had rotated clockwise about 66° in relation to the horizon from the first shot. This mission wasn’t going to be easy.
I opened each sun photo in Photoshop and erased the black background. On new layers, I found the center (which became slightly more challenging as the missing piece became larger), placed a circle on the sun’s edge, and drew horizontal and vertical crosshairs over it. I duplicated all of those construction lines and moved them to represent the moon. I placed all the layers into a group labeled with the time of the shot so it would be easier to combine these images into one Photoshop file. In this master sun file, I started with the shot taken at the peak of the eclipse and rotated that group so that the moon’s center was directly over the sun’s. As I rotated the other suns to align their spots with the first, the moons’ centers formed a horizontal line about half a radius above the sun’s center. Even better, the distance of each moon’s center from that of the peak moon along that line was basically proportional to the time difference between the two. Because of that, I was able to find the moon’s position at any time and replace the sun photographs that I missed.
The Sun’s Trajectory
I noticed that the sun’s 66° difference in orientation in relation to the horizon on my two later test shots seemed to match the change in angle of the sun’s trajectory in relation to the horizon during that interval. I’m still not confident that I’ve got my mind wrapped around all of the intricacies of these three moving celestial bodies, so this could be a coincidence, but I decided to run with this notion. After plotting the sun’s trajectory in its separate layer, we decided for aesthetic reasons (call this artistic license) to compensate for the disappointing tree height by compressing that layer downward (which could possibly have been an accurate representation if the declinations of (or the latitudes directly below) the sun and moon had been somewhat less than the 121/2° they were at the time). To do this, I simply made a selection with the Rectangular Marquee Tool, the bottom edge of which was on the horizon and the other three sides were large enough to include the whole trajectory line. Then, using a scale transform, I just lowered the top edge of the selected area to taste.
The Moon’s Path
I was pretty sure that the moon’s path was not parallel to the sun’s but didn’t know how much to tilt it. I heard that the plane of the moon’s orbit differed by 5 degrees from the earth’s, but didn’t know how that related to the problem. In the master sun file, I saw that the each sun had to be tilted from 0 to over 27 degrees with an average of 10°. I chose to rotate everything in the master sun file 8 degrees.
After deriving a trajectory for the sun, at first I selected and placed the appropriate photographs at five-minute intervals. When the eclipse started just after local apparent noon (when the sun crossed our meridian of longitude and was at its closest to being directly overhead), it was moving the fastest through its trajectory and the five-minute suns were much further apart than they were at the end of the eclipse three hours and 77 degrees of azimuth later. Nancy wasn’t happy. Instead of equal time, I considered next an equal-azimuth-change approach, but that would have the suns getting further apart at the end of the trajectory instead of the beginning, so Nancy decided on having the suns the same distance apart on the image, the third easiest of the three approaches because of the curved trajectory and requiring use of the Pythagorean Theorem.
Another artistic decision was the sun’s size. After re-creating the missing sun shots for the times dictated by the new spacing strategy, I copied all the necessary sun folders into the finished panorama file. As I started moving them into position, we decided they were not large enough, so I deleted all of them from the panorama file, increased the number of pixels in each direction of the master sun file by 25%, and recopied the appropriate folders into the panorama file. As I moved each sun into position, I rotated it so that the bottom edge of that group was parallel to the tangent (slope) of the trajectory path at that point. One could possibly ‘justify’ these actions by arguing that the same effect could have been achieved by just moving further away from Long Pine Key.
Finally, we decided to end the string of suns as it went behind a cloud rather than continue to the trees. The problem was that the cloud and the sun were in their respective positions at different times, meaning the cloud was not properly backlit as the juxtaposed position of the last sun dictated. It was my younger brother who first pointed out that “flaw”.
As I was setting up the first panorama near the peak of the eclipse, I didn’t notice that the sun was almost three f-stops dimmer than normal (or almost one eighth as bright). I think this shows the eye’s and mind’s ability to compensate for different conditions (although it could just show how oblivious I can be to my surroundings when I’m focussed on a challenge). I determined the exposure level of the camera in the usual way. As I took the later panoramas, however, the camera noticed the change in light intensity. I started getting more and more blinkies, and before the fifth panorama, I added our last (3-stop) neutral density filter.
Since the sky of each panorama was blown out in a large area around the sun, I had to restore its color uniformly across the image and then darken the sky around each sun appropriately. To do that I used Photoshop again to find the relative area of each sunHow, and then use that area to determine how much darker its part of the sky should be.
Well, that’s about everything we considered. You’ve got a little over six years to get ready for the next solar eclipse in this country (April 8, 2024), so don’t wait ’til the last minute to prepare (like some people I know). Hopefully, by learning from the tribulations and mistakes of others, you can make your life easier while still making better pictures. Good luck! And feel free to leave comments (or questions).
There are mathematical or drafting programs that may do a better job of finding areas of all sorts of seemingly random two-dimensional shapes, and I may have used one or two of these as a student, but I haven’t had any of them on my computer for many moons. So when I recently needed to compare the size of the visible sun at different times during a solar eclipse so I could compare exposure levels, I was out of luck. But then there was Photoshop. I just finished this article about how to find an object’s area, and put it on our website at www.beehappygraphics.com/find-area.html, mainly because I mentioned the technique in an earlier blog post, and was about to mention it again in an article I promised about the challenges of our newest eclipse image. This probably isn’t the most common task you will be doing, but when you need it, this can be handy. Enjoy!
Sadly, we had no winners to this contest. Here is a solution to that math problem:
There is more than one way to solve this problem, but we will be exploiting three different relationships. First, in preserving the aspect ratio, the length of the image (we’ll call L) is 11/2 times the width (W). . Then, adding up the components making up the overall width of the mat, the image width (less two overlaps of 1/8“) plus two mat widths (M) would equal 16 inches. By the same token, the image length (less same overlaps) plus two mat widths would be 20 inches.
If you replace the L in the last equation with its W equivalent from the first equation, and then add 1/4” to both sides of both equations to combine constants, you are left with the following two equations to solve with two unknown variables:
From here you can use linear algebra (matrices) or algebraic manipulation to simplify until you are left with just one variable. For example, just subtracting the bottom equation from the top (subtracting the left sides separately from the right sides of each equation), you will wind up with
which means the image width is eight inches, which means its length is twelve inches, and the mat guide would be set to 41/8“.
I’ve come up with one more printing-inspired math problem, which I will share as soon as I master a new plug-in for this blog. After that, I’m not sure. Response has been weak, but the former teacher in me feels a need to keep pointing out opportunities to use some of this stuff you learned in school (or is it just to torment those students who were the most difficult – I’m not telling). This isn’t really costing anything, and I give enough warning for the math-averse to stay clear. Stay tuned.