The mushrooms are safe from intervention, for there is no government above the Council of Fungi. They are the real Illuminati, spread across the world beneath our feet. A collective unconscious with a singular, defiant will.
Why is nobody mentioning that any point you choose just happens to be in an EXACT STRAIGHT LINE from the others!? And this just happened by chance? I've got some ocean-front property in Arizona I'd love to sell you.
I'd argue that's why it's not good satire. this just goes straight into another conspiracy theory now. satire doesn't really work when it straight up contributes to what it's supposed to be satirizing.
that's like satirizing the US culture by shooting up a school.
I'm sorry if that felt too harsh about US culture. i take it back. I should have said it's like satirizing the US education system by shooting up a school.
to think of that, this is one of the cases when basic math knowledge is important/useful outside engineering, finance, or anything that is stereotypically use math.
There always exists a circle such that, given any three non-colinear unique points in 3-space, all three points lie on the border of that circle. Spherical geometry is not required.
A fair point. The important part is the last bit, which points out that spherical geometry is not a requirement for it to hold, which is the only reason I replied to you in the first place
There are projections where infinitesimal circles stay circles, e.g. our dear Mercator projection, but that doesn't hold for finite sized circles, i.e. circles would still be distorted in north-south direction. Tissot indicatrix
That's a general metric holding for lots of projections. I think the specific projection that works for finite sized circles is stereographic projection.
On a stereographic map you should be able to draw a circle that stays a perfect circle ("small circle") on a globe.
In addition, in its spherical form, the stereographic projection is the only map projection that renders all small circles as circles.
By small circles they mean circles on a sphere that are not an equator (great circle), not infinitessimally small circles. So basically they just mean circles.
By small circles they mean circles on a sphere that are not an equator (great circle), not infinitessimally small circles. So basically they just mean circles.
This only applies to the circles perpendicular to the axis of projection, i.e. usually the circles of latitude (parallels), though. The Tissot indicatrices still show increasing sizes of the circles from the center of the map to its outside. Thus, any circle that isn't coaxial with the parallels is distorted on the map.
There is no qualifier on wikipedia and I do remember seeing some neat geometry tricks you can do with the property long ago.
The Tissot thing to me looks like a visualization for the jacobian, so the factor by which the area at that point is scaled, plus the gradient.
The circles in the stereographic projection are scaled, they are essentially pulled outwards, when further away from the center. This matches an increasing jacobian. But they stay circular, the stretching happens in the right way for that to hold true.
If you wait a bit I'll see if I can find some further things relying on this property, or at least stating it more unambiguously.
The Tissot thing to me looks like a visualization for the jacobian, so the factor by which the area at that point is scaled, plus the gradient.
Essentially, the tissot indicatrices are a visualization of the eigenvalues and eigenvectors of the projection in any point. So, in 2d, the areas of these ellipses correspond to the Jacobi determinant, the product of the two eigenvalues of the Jacobian at that point.
The circles in the stereographic projection are scaled, they are essentially pulled outwards, when further away from the center. This matches an increasing jacobian.
Exactly. The Jacobi determinant increases in radial direction (longitudinal on the globe).
But they stay circular, the stretching happens in the right way for that to hold true.
If you draw a circle on a globe, that is not coaxial to the parallels and apply the projection, the radius of said circle becomes elongated in outward direction in the same way the circles of the Tissot indicatrices increase in size.
Or in other words, any slice oncrement of the circle along a fixed degree of latitude changes in size depending on the value of the Jacobi determinant at that degree of latitude.
Thus, the circle on the globe becomes somehow like a rounded triangle on the map.
Edit: That shifts only the center of the mapped circle towards the outside of the original, but the circle remains a circle.
Stereographic projection preserves circles and angles. That is, the image of a circle on the sphere is a circle in the plane [...]. We will outline two proofs of the fact that stereographic projection preserves circles, one algebraic and one geometric.
I also ran across a math stackexchange thread talks about a proof that "Stereographic projection maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane".
Further I notices the mathematical wikipedia page for the projection states it without the weird map terminology simply as "It maps circles on the sphere to circles or lines on the plane".
I really don't see any qualifiers anywhere, to the best of my understanding this holds in general for all circles. With the one exception that circles through the point opposite the center turn into lines (infinitely large circles for simplicity).
Looking at the stereographic projection, there is a longer distance between points the father you get from the center of the map. Although the latitude lines remain circular in a polar projection, the map scales to avoid distortion father from the constant growth of the map once you leave the projected hemisphere. The northern hemisphere in an artic projection still must distort, making geometry a mess.
Goode homolosine projection is closer to keeping that distortion down, but all maps are an estimate due to the way a 3d curve is translated to a flat surface.
All that said, and I know I'm being pedantic, you could come really close by calculating the center of the circle in a sphere, then projecting the map stereographically from the center. That specific projection would come the closest, given the irregular shape of the Earth.
Stereographic projection is the one (and only) thatballows that. You can draw any circle (or a straight line) on a stereographic map and it will remain a circle on the globe.
I think it should still be possible to define a perfect circle from 3 points on a globe, tho
Imagine the 3 points on the globe defining a plane, and then just intersecting the globe by that plane, you'd have a perfect circle on a sphere that still goes through the original 3 points if I'm visualizing this in my head correctly, might try this in blender or something
The center of that circle is the Northwest Angle, and it's populated by turmpers. It was created by a "survey error". This tells me that Canada killed JFK because they knew Turmp would happen if they did. This was a Canadian attack all along.
it definitely works for planes and spheres. my intuition says it wouldn't work on all surfaces though, and in particular would probably break down around saddle points.
Don't take all this stuff too seriously, most of it is either performative content revenue farming or manipulation of public opinion by some actor. This ticks all boxes, could be anything; a person honestly that dense or deep into conspiracy theories isn't even the most likely one. Unfortunately neither is satire.
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Guys, I went to the center of the circle, and there was a completely normal looking tree there. Maybe too normal. What could it mean?
We need to dig deeper and get to the root of this.
I dug it up, and found some bugs, worms and roots. Some kind of code?
Guys, look at this mushroom! All these little frills!
What were we doing?
Put that down that's a deep state mushroom!
The mushrooms are safe from intervention, for there is no government above the Council of Fungi. They are the real Illuminati, spread across the world beneath our feet. A collective unconscious with a singular, defiant will.
Now that is a BIG mushroom!
I knew it! The libs bugged our phones to infect us with their brain worm! Smash your phone and chuck it in a lake!
NO That's probably a deep state lake
I can't beleave you made that pun
Ceenote is branching out into comedy.
If you start digging there and go all the way to the other side, you will end up in China! Check mate libtards. /s
I connected the three trees at the middle and they made a triangle!!! How deep does this thing go?’
A triangle? Oh no...
hey stop spying at me through my moneys
Why is nobody mentioning that any point you choose just happens to be in an EXACT STRAIGHT LINE from the others!? And this just happened by chance? I've got some ocean-front property in Arizona I'd love to sell you.
Hold up! You can even make a triangle out of those 3 points! Illerminaty confirmed.
Aka "the strongest shape." Think about it.
Winnipeg was the shooter! I knew it!
Winnipeg was the inspiration for Winnie the Pooh
Winnie the Pooh is the code for Xi Jinping
China was behind the grassy knoll
That's a big grassy knoll.
Too big to ignore.
It's always fucking Winnipeg. People need to find a new punchline...
Winnipeg sounds like some kind of bear furry porn.
co-linear points can also be on a circumference, if you don't mind infinite radius
Non-euclidean planes say what?
I was about to ask whether you can have three colinear points on a sphere, but then I remembered that the Earth is flat.
Which brings me to another question. What does a circle on a Mercator projection looks like on a sphere?
It's still a circle but all the corners add up to 365°, and their where we get the days from.
You can test this at home. Draw a circle on a paper, wrap it around a ball.
If you want the edge cases, draw the circle on a sheet of rubber (or maybe a plastic bag?) and stretch it over a ball.
Even as satire, the worst part is seeing Kirk being treated like he's anywhere near as important as the Kennedy and Lincoln assassinations.
It's been days, not decades. This attitude will fade.
It's a cult, so it depends on how long its members worship him. Once Trump's gone Kirk will probably be forgotten quickly.
Could be a coincidence. Only way we're going to know is if we occupy Winnipeg.
For at least 5 months of the year no one wants to occupy Winnipeg. That value increases slightly for the other 7 months.
Be nice to Canada, they have the worst neighbors
Fucking Alaskans, man.
Fact checking satire makes for even better satire.
I never thought about it, but now I'm gonna have some fun with this.
Well I know people who go for this shit in earnest, so it's good satire.
Reddithas a sub called r/peopleliveincities, I'm sure they'd be happy to accept this one as well.I'd argue that's why it's not good satire. this just goes straight into another conspiracy theory now. satire doesn't really work when it straight up contributes to what it's supposed to be satirizing.
that's like satirizing the US culture by shooting up a school.
Are you okay?
He's literally making his point... And he's entirely right. Do you understand the concept of hyperbole?
I'm sorry if that felt too harsh about US culture. i take it back. I should have said it's like satirizing the US education system by shooting up a school.
to think of that, this is one of the cases when basic math knowledge is important/useful outside engineering, finance, or anything that is stereotypically use math.
I got in trouble in my friend group meme chat for drawing a Star of David connecting the points in this meme
That's almost the plot of the rdj Sherlock Holmes movie. Just, you know, different star.
Readers added more context: Any three unique points on a sphere form a circle.
There always exists a circle such that, given any three non-colinear unique points in 3-space, all three points lie on the border of that circle. Spherical geometry is not required.
No, three unique points in 3d space that are colinear will not form a circle.
Thank you! Edited.
But now you're just repeating the post lol
A fair point. The important part is the last bit, which points out that spherical geometry is not a requirement for it to hold, which is the only reason I replied to you in the first place
Yup, a circle that lies on the surface of the sphere. You're only safe underground or in space.
The next one will be in the Arctic.
Also didn’t know we were calling this the UWU shooting.
I've got one labeled "Mormon School Shooter" and the other "Mormon Church Shooter"
NSFW (never safe from Winnipeg)
Yeah, clearly they are up to something. Pretty long term planning going on up there.
y'all gonna buy some wigs?
I wonder what size the circle would be if you took in to account the earth’s curvature.
Are there any map projections that allow for accurate projection of circles across arbitrary points?
All map projections are arbitrary. The only way to do this is on a globe.
Different projections preserve different properties. From memory there are ones that leave circles circular, so would allow this.
Edit: It's stereographic projection that maps circles to circles.
There are projections where infinitesimal circles stay circles, e.g. our dear Mercator projection, but that doesn't hold for finite sized circles, i.e. circles would still be distorted in north-south direction.
Tissot indicatrix
That's a general metric holding for lots of projections. I think the specific projection that works for finite sized circles is stereographic projection.
On a stereographic map you should be able to draw a circle that stays a perfect circle ("small circle") on a globe.
By small circles they mean circles on a sphere that are not an equator (great circle), not infinitessimally small circles. So basically they just mean circles.
This only applies to the circles perpendicular to the axis of projection, i.e. usually the circles of latitude (parallels), though. The Tissot indicatrices still show increasing sizes of the circles from the center of the map to its outside. Thus, any circle that isn't coaxial with the parallels is distorted on the map.There is no qualifier on wikipedia and I do remember seeing some neat geometry tricks you can do with the property long ago.
The Tissot thing to me looks like a visualization for the jacobian, so the factor by which the area at that point is scaled, plus the gradient.
The circles in the stereographic projection are scaled, they are essentially pulled outwards, when further away from the center. This matches an increasing jacobian. But they stay circular, the stretching happens in the right way for that to hold true.
If you wait a bit I'll see if I can find some further things relying on this property, or at least stating it more unambiguously.
Essentially, the tissot indicatrices are a visualization of the eigenvalues and eigenvectors of the projection in any point. So, in 2d, the areas of these ellipses correspond to the Jacobi determinant, the product of the two eigenvalues of the Jacobian at that point.
Exactly. The Jacobi determinant increases in radial direction (longitudinal on the globe).
If you draw a circle on a globe, that is not coaxial to the parallels and apply the projection, the radius of said circle becomes elongated in outward direction in the same way the circles of the Tissot indicatrices increase in size.
Or in other words, any slice oncrement of the circle along a fixed degree of latitude changes in size depending on the value of the Jacobi determinant at that degree of latitude.
Thus, the circle on the globe becomes somehow like a rounded triangle on the map.Edit: That shifts only the center of the mapped circle towards the outside of the original, but the circle remains a circle.
I couldn't find the video I was thinking about, which is a bummer.
I did find one written argument (uiuc - Stereographic Projection)
And one video proof youtube - Stereographic Projection Circle to Circle Proof.
I also ran across a math stackexchange thread talks about a proof that "Stereographic projection maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane".
Further I notices the mathematical wikipedia page for the projection states it without the weird map terminology simply as "It maps circles on the sphere to circles or lines on the plane".
I really don't see any qualifiers anywhere, to the best of my understanding this holds in general for all circles. With the one exception that circles through the point opposite the center turn into lines (infinitely large circles for simplicity).
Looking at the stereographic projection, there is a longer distance between points the father you get from the center of the map. Although the latitude lines remain circular in a polar projection, the map scales to avoid distortion father from the constant growth of the map once you leave the projected hemisphere. The northern hemisphere in an artic projection still must distort, making geometry a mess.
Goode homolosine projection is closer to keeping that distortion down, but all maps are an estimate due to the way a 3d curve is translated to a flat surface.
All that said, and I know I'm being pedantic, you could come really close by calculating the center of the circle in a sphere, then projecting the map stereographically from the center. That specific projection would come the closest, given the irregular shape of the Earth.
Stereographic projection is the one (and only) thatballows that. You can draw any circle (or a straight line) on a stereographic map and it will remain a circle on the globe.
https://en.wikipedia.org/wiki/Stereographic_map_projection#Properties
If you drew in on a globe, it would look deformed in this projection. I think the radius wouldn’t change, but it would look “wider” towards the north
That theorem only applies 2d from my understanding
I think it should still be possible to define a perfect circle from 3 points on a globe, tho
Imagine the 3 points on the globe defining a plane, and then just intersecting the globe by that plane, you'd have a perfect circle on a sphere that still goes through the original 3 points if I'm visualizing this in my head correctly, might try this in blender or something
The center of that circle is the Northwest Angle, and it's populated by turmpers. It was created by a "survey error". This tells me that Canada killed JFK because they knew Turmp would happen if they did. This was a Canadian attack all along.
Wake up sheeple
Can you do that for any 3 points on a surface?
Flat earth confirmed
Oh right, this only works in 2D. You would need to have a sphere. Checkmate atheists
use the 3 points to choose a plane and then use the 3 points to outline a circle
it definitely works for planes and spheres. my intuition says it wouldn't work on all surfaces though, and in particular would probably break down around saddle points.
So long as they're not in a straight line, yeah (as it says as at the bottom of the post).
Whoa whoa whoa...
There is a bottom of the post?
Don't take all this stuff too seriously, most of it is either performative content revenue farming or manipulation of public opinion by some actor. This ticks all boxes, could be anything; a person honestly that dense or deep into conspiracy theories isn't even the most likely one. Unfortunately neither is satire.
look at the body of the post
Shit I'm stupid.
In my defense, it was 3 AM.
We'll just forget about William McKinley because Buffalo doesn't fit into our perfect circle
If he did, he wouldn't be able to use circumcircle theorem.
What's crazy is that this does fool people despite them drawing circles many times around triangles in their math class.
Wasn't that in elementary or middle school?
Seem pretty cut and dried to me. Time to bomb Winnipeg!