Welcome to The STEM Sessions Podcast. I am your host, Jarl Cody.

I watch numerous outdoorsy type YouTube channels

- Always looking to increase my knowledge and skills for the trail
- Knots, navigation, gear reviews, new areas to explore, first aid

Back in August, Dave Canterbury posted a video entitled “Secrets to Accurate Land Navigation” (link in shownotes)

- He discusses how the distance you measure on a map is not the same as distance physically traveled because of changes in elevation along the path
- Dave claims this fact is excluded in most navigation classes, but is critical to successful navigation by map and compass
- https://youtu.be/hR4xVDeYD5E

I’ve taken many navigation courses, and Dave is correct in that distance deviations due to elevation change is typically not mentioned by instructors

- Classes are typically taught on flat ground so there’s no need
- But I think he overstates their importance, at least for the use cases in the meaty part of the bellcurve

Of course, being an engineer, I needed to prove this to myself

- The following episode is the results of that deepish dive into the topic, and it’s a good example of basic math being relevant in the real world

This is The STEM Sessions Podcast Episode 21 – What a Distance Slope Makes

A few months ago, watched a video discussing how change in elevation causes your distance traveled to deviate from the distance on a map

- Basically, if you’re walking uphill or downhill between two points, you walk a greater distance than if those points were at the same elevation

Main point of video is you need to track this deviation during land navigation with map and compass, because it will add up over the course of your travel and throw off your route

- Engineer in me says while this is mathematically true, it doesn’t matter because the deviation is minimal in circumstances most people will find themselves in

Remember having similar discussions on the trail 20 years ago when handheld GPS receivers first hit the consumer market

- I carried one before most people did, so I’d always get questions about how they worked
- One guy asked if it took elevation change into account
- Told him it tracks elevation as its own parameter, but wasn’t sure if it accounts for change in elevation’s impact on distance traveled
- For most hikes, it didn’t matter

Math behind this discussion is the Pythagorean Theorem

- In a right triangle, the square of the length of hypotenuse is equal to the sum of the square of the lengths of the other two sides

Here’s how we use this theorem of basic geometry in our navigation example

- Think of a piece of graph paper.
- The (0,0) point is where we currently are standing
- X-axis represents distance shown on the map between our origin and destination
- Let’s say the map shows distance of M (for map) to our destination
- So we draw a dot at coordinate (M, 0) representing this distance
- Y-axis represents elevation change
- Let’s say the maps shows our destination to be at an elevation of E (for elevation) compared to our origin destination
- So we draw a dot representing the elevation of our destination at coordinate (M, E)

Draw a triangle connecting the three points

- Should be a right triangle, so Pythagorean Theorem applies
- Distance we will physically travel between our origin point (0,0) and destination point (M,E) is the hypotenuse of the triangle
- Let’s call this distance D
- It is longer than the distance measured on the map (0,0) to (M,0) because of the elevation change (M,0) to (M,E)
- And of course, elevation and distance need to be in same units

So distance we physically travel D is equal to the square root of the quantity M-squared plus E-squared

- Therefore, the bigger the elevation change the more physical distance deviates from map distance

Dave approaches his argument from perspective of cross-country, off-trail navigation via map and compass

- You find your current position and destinations on a map
- Determine the distance between them and direction of travel
- Then divide the journey into segments you can count or pace, say 100 ft or 100 m or 0.1 miles
- As you travel, you manually keep track of distance traveled (there are several ways to do this and I play to make a separate episode about the math behind it)
- At the end of the first segment, you stop and update your course, and travel the next segment

To account for difference between map distance and physical distance, Dave uses the Pythagorean Theorem, laid out similarly as I did

- To calculate the squares and square roots, he says calculate them in your head or use the calculator app on your phone or use a pre-printed look up table in your navigation notebook
- For me, only the last option is acceptable
- What if your phone battery is dead or you lose it?
- How many can do squares and square roots in our heads while tried and stressed?
- Do I even trust the results?
- Heck, let’s be honest, how many of us know how to do square roots in our head in the first place?

Is there a threshold below which we don’t need to worry about this deviation in distance values?

For rare instances when we shouldn’t ignore it, is there a shortcut or rule of thumb we can use to avoid calculating squares and square roots by hand?

For the latter question, weed to relate elevation E to map distance M, because those are the two known quantities we can rely on

- Let’s go back to our triangle plotted on graph paper
- Elevation E would be the rise of the triangle, and map distance M would be the run of the triangle
- Slope of a line is defined as rise divided by run, so in our case, slope of the hyptonenuse is the ratio between elevation and map distance

Slope a good starting comparison, because ratios are often easy to estimate without a calculator or complex division

- When comparing two numbers you can quickly determine if one number is about half of the other
- Or 10% of the other
- Once you have those two estimates in mind, combine them to make other estimates
- 25% is half of 50%
- 60% is 50% plus 10%
- And so on

Physically, what does slope represents the steepness or incline of the path

- Let’s say our elevation E is 2 feet and our map distance M is 10 feet
- Slope is E/M or 0.2
- For every ten units of map distance, elevation increases by two units
- If M extrapolates to 20 feet, E would be 4 feet
- This would be considered a 20% incline or grade
- Note neither the ratio or incline equals the angle of the slope.

At the opposite extreme, a slope of 1 means for every ten units of map dist, elevation increases by ten units

- Unit for unit parity
- Slope of 1 is a 100% incline
- When pictured on a graph this would be a 45-degree angle

Here are a few real world reference examples

- Wheel chair ramps can have slope of 1:12 or 0.08 or 8%, which is a 5 degree incline
- US interstate highways have a max grade or incline of 6%, or about one to 16.5, which is a slope of 0.06
- Train tracks have a grade of 1.5% to 4%, which is one to 25 on the high end

Next step is to relate slope (let’s call it S) to the difference between map distance M and real distance D

- D is the square root of quantity M squared plus E squared
- E is equal to slope S times M
- Combine equations and D is equal to M times square root of quantity one plus S squared

Square root of quantity one plus S squared determines the difference between M and D

- For example, a slope of .1 (or a 10% incline) results in a multiplier of .005 or half a percent
- A slope of .25 or 25% incline results in a multiplier of .031 or a 3.1 percent increase
- A slope of .5 or 50% incline results in s multiplier of .118 or 11.8%

Used Excel to determined and plot these multipliers for slope values from 0 to 1

- Plot is not linear
- Between slopes of 0 and 0.2, the multiplier increases from 1 to 1.02
- From slopes of 0.2 to 0.4, the multiplier increases from 1.02 to 1.077
- So the greater the slope, the more physical distance deviates from map distance

How does this help?

- At the very least, you can put this table or plot in your navigation notebook so you can estimate physical distance without performing complex division in your head or on a calculator
- If you estimate a slope of .25 the table or plot tells you to increase map distance by about 3%
- Makes the math much easier to do in your head
- You can find the plot in the shownotes

But it also helps you decide when you can ignore this deviation entirely

- No universally accepted difficult scale for terrain
- But typically, easy trails have less than a 5% grade and highly difficult trails will have sustained grades of 20% or more
- For example, if the slope I’m walking is 0.25, the multiplier is about 1.03 or 3%
- Over a map distance of 500 feet, do I care about an extra 15 feet? Probably not, especially if I stop at 500 feet to reevaluate my position
- What about over a mile of map distance? Does the extra 160 feet matter?
- Maybe in bush where I can’t see the horizon or any landmarks to orient me
- In those cases a handful of feet might matter
- But how true did I stick to my compass bearing? How accurately did I count my distance?
- I think effort would be better spent reducing error in those two factors first

Obviously whether or not you chose to ignore the extra distance caused by elevation gain or loss should be based on your personal experience, the terrain and weather conditions, and your goals for the outing

- Are you looking for an unmarked food or water cache, making precise navigation key?
- Are you looking for cabin in thick bush that you can’t see until your standing on top of it?
- Most people will never be in those situations
- For most, land navigation is about using a map and compass to find a landmark or self-rescue and terrain is the bigger factor
- In those extreme cases, by the time the deviation becomes relevant, the slope is so severe, I’d rather choose an easier, albeit longer, path to begin with

Thank you for listening to The STEM Sessions Podcast.

This episode was researched, written, and produced by Jarl Cody.

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Until the next one, stay curious.