Derive Formula for Sides $a$, $b$, $ c$ of Right Triangle Using Select Differences (b-a, c-a,..), Value of Reductive Difference Path in Applications?

Question

Here is a derivation of a suggested approximation formula for a right triangle $ABC$ with assumed unknown legs of length $a$, $b$ and also an unknown hypotenuse $c$ (perhaps all due to excessive length or an obstruction as a mountain, river,..., denying feasible direct measurements) but an approximate estimate of select differences namely, [$b$ - $a$] and [$c$ - $a$], for example, may be feasible.

Claim: this is sufficient to produce an estimate of the absolute length of $a$. A related formula substituting $b$ with $a$ follows, and therefrom courtesy of the Pythagorean Theorem, an estimate of $c$ is derived, all of which could be used to provide statistics of interest, such as perimeter and area assessments.

Proof: Construct at Point A, an arc of length $a$ dividing the hypotenuse into two segments, $a$ and $c$ - $a$. At Point B, similarly construct an arc of length $a$ dividing leg $b$ into two segments, $a$ and $b$ - $a$. Assume [$c$ - $a$] and [$b$ - $a$] as known quantities (that is, our proposed knowledge of differences). Now, by the Pythagorean Theorem, expand with cited segments inserted:

${ c^2 = a^2 + b^2 }$

Or, upon inserting segments

${ (a + [c-a] )^2 = a^2 + (a+ [b-a])^2 }$ ${ a^2 + [c-a]^2 + 2a[c-a] = a^2 + 2a[b-a]+ [b-a]^2 }$

Or, rearranging in terms of a quadratic equations in terms of the parameter ‘$a$’:

${a^2 + a*2*([b-a] - [c-a])+ [b-a]^2 - [c-a]^2 = 0}$

Example of formula application: Consider a right triangle which has unknown sides which are actually 3, 4 and 5. Thus, $b$ - $a$ has a value of 1, $c$ - $a$ has a value of 2, and the quadratic equation has the form:

${a^2 + a*2*(1 - 2) + (1 - 2^2) = 0 }$

Or:

${a^2 -2*a - 3 = 0}$

which per the quadratic formula has the positive root: (2 + SQRT(4+12) )/2 = (2+4)/2 = 3 as required.

Now, the question of this thread is the potential value of this difference method in field applications. Note, not exactly competitive is an angle (as in trigonometric) based method as, with the example of a right triangle, at least one absolute side length is still required. If all sides are inconveniently large and not precisely known, this would appear to present a relative disadvantage to my basic proposed reductive (as in only differences) methodology. Also, a quadrilateral constructed to mic a land mass, is reducible upon drawing a diagonal and lines perpendicular (say on a photo of the object to assess its area) into the case of right triangles to permit an area assessment.

The last example relates to area determination of large surface objects, but other possible example are not immediately apparent, accept perhaps in astronomical measurements, of which I have little background.

I welcome suggestions on other possible applications, including have I over stated the associated value of this difference method.

Answer 1

Upon further web searching, I can across commentary of some actual workers in the field, to quote a source:

The majority of cartographic work is, as you would expect, performed on computers. There are two ways that I'm aware of that are used to estimate area. One method is to manipulate satellite imagery to accentuate what you want to measure - in this case, land area - and, essentially, count the pixels. It's more accurate than taking some Google Earth imagery and adjusting the contrast, but it will always be an estimate...

And further:

Alternatively, features (coastline, rivers, political borders, etc) are collected as vectors and their associated polygons. Since the vectors of the collected data are mathematically defined, it's easy to calculate the precise area of the resulting polygon. Geometry really is the (practical) answer. As you're aware, area can be calculated using other methods, but they don't really apply because the areas are defined using polygons, and not, for example, parabolas. The issue here really depends on how accurately the polygon represents the actual area. Since you mentioned cartography, modern maps are created with a known level of accuracy - again, depending on its intended application.

And importantly, a qualifying assessment to quote:

If you're seeking "perfect" accuracy, issues also arise because, of course, landmasses are not really geometric, and they're certainly not pixels. On top of that, the shape of a landmass isn't constant. Erosion, land reclamation, changes in sea level all constantly affect the "true" area. What's more, because maps are intended to have a practical use, they're subject to economic factors as well...

Another opinion states:

One can use remote sensing/image processing technologies. For instance, you can create an image ratio that separates one land cover from all others. It's kind of hard to explain what an image ratio if you don't have some knowledge of remote sensing though. In any case, you could create the ratio to show only the water in an area. Then you can run the image through some more processing and determine exactly how much of the image's pixels are showing water. Subtract that from the total number of pixels and you have the pixel count for "non-water...or land/veg".

And an extremely important comment:

If the area of the image in question is known, then one can calculate what percentage consists of land. If it is not know, it can be determined using reference maps.

So, with respect to my question of the significance of the differences in the sides of a right triangle methodology, especially for large area problems, some key points:

1. Geometry is a definite tool in the field for area assessment with a polygonal approach.

2. Acquired pixel information must be interpreted relative to a known area or prior maps.

As such, measuring the differences in pixels compared to a known measurement appears to be a valid reductive procedure for an estimate insertion into a geometric area formula.