To start the last week of March Mathness, because it’s a Monday, I’m going to go easy on y’all with some light, easy topics. (Maybe I can lull you into paying attention for the major topic of the month: matrix rotation.)
It has occurred to me that, if I’m talking about math in March, I absolutely must mention one of my all-time favorite mathematical objects, the Mandelbrot. I’ll try to get to that today, but the main topic is a simple something that I ran into while working on my 3D model of the big island of Hawaii.
The question was: How many miles are there per degree of latitude?
Just to be clear, we’re talking about the east-west distance along a line of latitude. The north-south distance, along lines of longitude, does not vary. (By “distance” I mean the distance from one degree to another.)
What makes the question a bit interesting is that it is slightly more complicated than asking: How many miles are there per degree of longitude?
The reason they’re different questions is that the circles of latitude get smaller as we move from the equator to the poles.
How many miles (or kilometers) per degree along a line of latitude depends on your latitude. At the equator, the circle of latitude is “full size,” but at the poles, it shrinks to zero size.
By “full size” I mean that the circle of latitude is the same size as any circle of longitude.
More importantly, “full size” is the maximum size any such circle can be. The center of these circles coincides with the center of the Earth.
Therefore, at the equator, the east-west distance per degree is (a) at its maximum and (b) the same as north-south distance per degree.
As we move north, the east-west distance gets shorter and shorter until, at the poles, it becomes zero.
How can we calculate the distance at any given latitude?
The first thing we can do is calculate the distance per degree along the equator. (That will also be the distance per degree for any north-south meridian.)
This is very simple if we remember the formula for circumference (2πr):
Where R is the radius of the Earth (3958.8 miles or 6371.0 kilometers). Plugging in R, we get:
Which, by the way, is 60 nautical miles (nmi). A nautical mile is defined as one minute of (oddly, under the circumstances) latitude. That is: 1/60 of a degree.
In kilometers, one degree is:
So now we know the maximum, the distance per degree along the equator, which is the same as the distance per degree along any meridian: 69.094 miles or 111.19 kilometers.
The next step in solving this starts with characterizing the size of a circle of latitude at a given latitude.
Essentially, we want its radius. Then we can calculate its circumference and divide that by 360.
The insight we need comes from recognizing what controls the circle’s size.
The latitude, obviously, but we need to see the latitude as the angle of a line drawn from the center of the Earth to a point on its surface.
At the equator, the angle is 0°, and at the poles, it is 90° (technically ±90° for north (+) and south (-)).
Otherwise, the angle is somewhere between 0° and ±90°.
Now consider a line drawn from a given latitude horizontally to the Earth’s axis (blue line, aka y-axis, in the diagram to the right).
The length of this line is the same as the distance as along the x-axis (green line) if we draw a vertical line straight down from our latitude point.
The point where the vertical line meets the x-axis is the cosine (of the angle) times the radius.
(The point where the horizontal lines meets the y-axis is the sine times the radius. That number isn’t useful here, and we’re ignoring it. I only mention it in case you wondered where it was.)
The bottom line is that this distance is the radius of the circle of latitude. We now have a direct connection between that radius and the angle (latitude).
This gives us the formula for a degree along a line of latitude:
It’s as simple as that.
If we remember that cos(0°) is 1, and cos(90°) is 0, we notice two things:
Firstly, at 0° (i.e. at the equator), this equation is the same as the one above.
Secondly, at 90° (at the poles), this equation gives us zero, which is what we’d expect at the poles.
Lastly, we can plug some figures in to see how it all works out:
Something I find a bit amusing about all this is that, although the answer is simple enough to figure out, it requires the dreaded trigonometry, which so many thought they’d never use in life.
The truth is, at least in my case, I’ve used “trig” quite a bit over the years!
It crops up all the time in image generation, especially 3D image generation. (Even in 2D work you often want to generate circles or circular things.)
There’s a reason why any decent calculator has sin and cos keys! Trig is super useful and pops up in all sorts of places.
Now, part of what happens is that, having a tool in the toolkit means being able to whip it out in situations where it’s useful. Not having it, one may never realize how it could be used.
So it’s hard to tell if I’ve found trig so useful just because I have it available or because it really does crop up that often. It’s also likely that my interests bring me both to trig, itself, and the things it’s useful for.
It’s worth not being afraid of; I want to return to that later.
For whatever it may be worth, my intention is to wrap up March Mathness this coming Sunday (the last day of March) with a discussion of math phobia.
For now I’ll just mention what an odd thing it seems to me that people seem so pleased with themselves when they say they don’t do math. It’s almost as if it makes one more human to be bad at math.
I keep imagining people being the same way about reading and writing.
(It also amuses me in the context of AI fans who’ve long exclaimed how much the brain is “like a computer.” Ha! As if.)
The thing is, it has mostly to do with the dreadful way math is taught. It’s so bad, that I can’t really blame anyone who fears, hates, can’t do, or just plain has no use for, math.
I do feel bad for people missing out on all the fun, beauty, and utility.
The other consideration is that there is a big difference between math that is actually useful and math that is viewed as necessary or even foundational. Sometimes that view is out of date.
For instance, does anyone really need to know long division?
Maybe how companies set prices, or understanding lotto odds, or how populations grow, would be more valuable in understanding our modern world.
That said, there is something I find valuable about the rigor mathematical training gives one. There are other practices that do the same (computer programming can be one of them if approached correctly).
At this point I think I’ll leave talking about the Mandelbrot for another day.
It seems too much of a topic shift, and I have written about it before, so I don’t really have a good sense of what more to say about it. (I figured I’d think of something if I got started.)
I have been enjoying a YouTube channel, Maths Town, that specializes in some really (really, really) nice Mandelbrot zooms. I’m utterly enthralled by them.
Here’s one of my (latest) favorites (look at these on the best monitor you have at the best internet speed you have and the best resolution you can):
Here’s another (very different look, but same Mandelbrot underneath):
What makes them both so strikingly beautiful and so striking different is the palette and technique used to render the image. It’s the same old Mandelbrot in all cases.
Here’s one more that’s a little more Mandelbrot-y in how it’s rendered:
You’ll find many more, along with variations on the Mandelbrot itself (inversions, larger exponents, distortions) on the Maths Town channel.
I’ll try to re-visit the Mandelbrot topic later this week.
Have a good latitude, my friends!