Monday, December 17, 2018

"You can see the curvature!" High-altitude balloon footage

Hello everyone. Last time I showed you some simple techniques you can use to make the curvature more visible. I have proven that these techniques do not add the curvature to a photo. They only help to elicit one if it was already there. Today I am going to be showing you the results of my analysis of this footage from a high altitude balloon. The footage was taken using two GoPro cameras equipped with 4.35mm lenses. One of them had a infrared filter. In every image you will actually see three frames. Two versions in infrared, and one taken without filter. I recommend you to visit Dwayne Kellum's channel. He has more videos like this. I took some frames from various altitudes and applied both a reference line and a vertical stretching to all of them. The reference lines are not entirely horizontal, because the frames I chose were slightly tilted. I tried to catch and connect both ends of the curvature. You should see that somewhere in the middle, Earth's curvature is always visibly above the reference line. It is not a straight line flat-Earthers want it to be. This is even more visible after stretching. Let's see the results.

33.9 km (111233 feet)

Let's start with the image from 33.9 km altitude. First a raw image

Earth curvature from a high-altitude balloon 33.9 km alt

Let's add some straight lines

Earth curvature from a high-altitude balloon 33.9 km alt

And finally let's stretch the image

Earth curvature from a high-altitude balloon 33.9 km alt

And add some lines for reference

Earth curvature from a high-altitude balloon 33.9 km alt

Earth curvature from a high-altitude balloon 33.9 km alt

29.5 km (96802 feet)

Let's move to the image from 29.5 km altitude. First a raw image

Earth curvature from a high-altitude balloon 29.5 km alt

Let's add some straight lines

Earth curvature from a high-altitude balloon 29.5 km alt

And finally let's stretch the image

Earth curvature from a high-altitude balloon 29.5 km alt

And add some lines for reference

Earth curvature from a high-altitude balloon 29.5 km alt

Earth curvature from a high-altitude balloon 29.5 km alt

16.5 km (54195 feet)

Let's do the very same procedure to the image taken from 16.5 km altitude

Earth curvature from a high-altitude balloon 16.5 km alt

Some straight lines added

Earth curvature from a high-altitude balloon 16.5 km alt

Stretched version

Earth curvature from a high-altitude balloon 16.5 km alt

Some more reference lines

Earth curvature from a high-altitude balloon 16.5 km alt

Earth curvature from a high-altitude balloon 16.5 km alt

Conclusions

As you can see, the curvature is easily visible in footage taken from 33.9 km, 29.5 km and 16.5 km. Of course I could keep going and apply the same procedure to the frames from lower and lower altitudes and the curvature would still be visible. But of course I would eventually end up with completely unreadable pictures. Stretching affects a photo quality, and I would have to stretch more and more, because as you should now know, the closer you are to Earth's surface, the more flat it appears. I think you get the point. The curvature is visible, and you don't have to go to space to see it.

In fact, the very first image from this post should be enough to convince you about the existence of the curvature. Like I said before, the images from space are the ultimate proof, and the topic for a separate post, but as you can see, pictures taken from a balloon are good enough to prove that the Flat Earth theory is wrong.

So, that's it for today. If you enjoyed this post, I’d be very grateful if you’d help it spread by emailing it to a friend, or sharing it on Facebook. Definitely you should share it with a flat-Earther, if you know one. And feel free to leave a comment below.

Sunday, December 2, 2018

"You can see the curvature!" foreword

Hello everyone. In my previous posts I mainly focused on the fact that it is hard to see the curvature on daily basis, and what is the reason of it. I've shown you how big our planet is, how tiny we are and also how observing a curve or a sphere from a very close distance make them appear straight and flat respectively. In the next couple of posts I want to show you that it is possible to see the curvature. Even from the ground. And I am going to show you how to do this. I'm going to start with sharing some techniques helpful during photos analysis. We will use them later. They help to "elicit" the curvature if there is one in the photo. They are especially useful, when working with low altitude footage, where the curvature is not immediately apparent, because of the reasons I described in my previous posts. You can clearly see the curvature from higher altitudes, without any image manipulations though. This will be discussed soon. All this involves some picture manipulation in software like photoshop or gimp, so I bet flat-Earthers are now laughing and screaming their usual arguments regarding faking the evidence. But, like I will show you in a minute, these techniques do not add the curvature. They only help to elicit one if it was already there. Let me show you how this works.

Adding a straight line

First technique is just adding a straight line near the horizon. How simple is that! You can do it in 2 minutes flat-Earthers and you will see, that even in the pictures you keep sending me as a proof of a flat horizon, the curvature is visible. All you need to do is to add a reference line and you will see that the horizon is not flat anymore.

Have a look at the picture below. It is a frame from a high altitude balloon footage, which has been taken at 95718 feet = 29.17485km altitude. You can find the entire footage there. I recommend you to visit Dwayne Kellum's channel. He has more videos like this. I've chosen a little bit tilted image but it does not matter.

Earth curvature from a high-altitude balloon

And the same picture with additional red line. You can clearly see that in the middle the horizon is visibly above the red line and it meets the line at its both ends.

Earth curvature from a high-altitude balloon

Changing height of a picture

Second technique is also quite simple. This time all you have to do is to stretch your image vertically. But doing this is clearly an image alteration, you may say. It is indeed. But like I already said, these techniques do not produce the curvature out of nowhere. They can't create a curved line out of a straight one. Let me show you some examples.

Here is a regular, horizontal, straight line. Nothing special.

straight line

When you stretch it, or in other words, increase its height, it becomes, yes you guessed, a straight line. Have a look at the picture below. The only difference is that the line is now thicker. So like I said before, stretching does not produce a curvature out of nowhere.

stretched straight line

Now let's consider a curved line, like the one below.

curved line

Here's how it looks like when it got stretched. Notice the change. Not only it is thicker, but also far more curved.

stretched curved line

Ok, you have seen some simple examples. Let's see how stretching affects real objects. Below you can see a 6x6x6 Rubik's cube. Every side of a Rubik's cube is made of square stickers arranged in a chessboard-like pattern. Black plastic the cube is made of creates a net of straight lines separating colorful stickers. I've added two arrows for a reference.

Rubik's cube

And that's what happened to the cube when I stretched the image. Please notice that the pieces' edges are still straight lines. So again, stretching does not produce the curvature out of straight lines. Remember that flat-Earthers when you accuse me of fabricating photos next time.

stretched Rubik's cube

And for the completion, let's stretch the image I've shown you at the beginning.

Earth curvature from a high-altitude balloon - stretched

Looks like the curvature is clearly visible now. In the photo below, I've added two lines. The red one indicates the horizon. It is a curve, no doubt. Please also notice the green line too. It is a reference line that proves the stretching I've done hasn't affected straight lines the box with information is made of. See for yourselves flat-Earthers.

Earth curvature from a high-altitude balloon - stretched

Fish-eye lens problem

A large number of photos of Earth taken from space or high-altitude balloons are being dismissed by flat-earthers because they were made using fish-eye lenses. They say that any curvature present in those photos is not real and is just the result of the fisheye distortion and thus it cannot be used as evidence. But the truth is that even if fish-eye lenses were used, footage still can be used to prove the existence of curvature. Any photographic lens has a very useful property: a straight line will also appear straight as long as it goes through the center point of the photo. If the horizon goes through that central point, we can tell if it is a horizontal line or a curve. If it appears curved, then it must be curved in real world. Let's see some examples of how the fish-eye lens work.

Let's apply a fish-eye distortion to a Rubik's cube picture. I don't have the lens, so I had to do it this way. Like I said, every side of a Rubik's cube is made of square stickers arranged in a chessboard-like pattern. There are lots of straight lines. So it should be easy to see the distortion after applying the effects, and also what has been left unaffected. Have a look at the picture.

Rubik's cube fish-eye

Did you notice anything? Two straight lines remained unaffected after I had applied the distortion. I've marked them using red in the picture below.

Rubik's cube fish-eye

If you still have trouble with this, I've added more reference lines to the photo, so you can see how everything except the lines that cross the center point is affected.

Rubik's cube fish-eye

This proves what I've written before. You can use the fish-eye lens to prove that the horizon is not flat. Because if it was flat, it would also appear flat in the picture taken using fish-eye lens. Of course the horizon must be kept in the center of the picture.

Conclusion

In this post I described some simple techniques I'm going to use in my subsequent posts. I've proven that they do not add the curvature to photos. They merely make the already existing curvature more visible, and can be very useful if you want to prove the Earth is round based on low-altitude photos. I've also debunked the flat-Earthers' claim that any curvature of our planet shown in photos is the result of the fish-eye effect. Not only that's not true, but also a footage taken by fish-eye lens can be used to prove the existence of curvature. Next time we will go through pictures taken from different altitudes and see a lot of curvature!

Monday, November 19, 2018

"Why can't I see the curvature?" part 3

How to measure the curvature?

Let's start with a simple quiz. In the picture below you can see 5 cuboids. One of them is off by 1 degree from the others. Can you tell which one is it, just by looking at them?

leaning buildings

Let's try again, this time with two cuboids. Can you find them?

leaning buildings

As always, you can check the code I used to generate my images.
You can find it there: https://gist.github.com/piotoor/f927143325b8bea21b7e4f8a389daa5d
The answers can be found here

It wasn't that easy, right? Now imagine telling such a small difference between two objects separated by 111 km instead of couple of centimeters. Ok, but why 1 degree and why 111 km you may ask. Well, 111 km is the distance between two objects on Earth's surface that are off by 1 deg of each other. Have a look at this picture:

curvature arc length

If you choose theta to be 1 degree, those green buildings will be separated by 111km. Here's how to calculate the distance for any value of angle. We can work out the length of an arc by calculating what fraction the angle is of the 360 degrees for a full circle. A full angle has an associated arc length equal to the circumference. In case of Earth, it is 40 000km. The formula is quite simple:

After rearranging, we get:

Let's put our values into the equaton and calculate arc_len:

As you have already seen, it is hard to tell the difference, when one object is off by only 1 degree. Now, to give you a better picture of this, let's calculate the angle theta for smaller distances, closer to our everyday experiences.

So lets' rearrange our equation again to get a formula for theta:

And compute theta values for 10km and 1km:

As you can see, theta values for small distances are so tiny, it is certainly not possible to notice such a small differences with naked eye. Now, lets talk about the most crucial thing regarding the effect described above. If you look straight ahead observing what's in front of you, objects are leaning OUT FROM YOU, not from left to right, like in the examples I've shown at the beginning. This fact makes seeing the difference harder.

Don't believe me? Let's see if you can tell the difference when objects are leaning from you. In the picture below, you can see 5 cuboids again. This time your job is to tell which one is leaning back 1 deg, which one 2 deg, which one 3, 4 and 5 degrees. That's how you would see a row of buildings. Can you match those values with the buildings?

Answer can be found here
Code: https://gist.github.com/piotoor/f486d0261a4c4aa17bc9116f4cfcb7e1

All the stuff above should convince you, flat-Earthers that the fact that it is hard to see the curvature (0.009 degree for every 1km - impossible to notice with naked eye) does not imply that there is no curvature at all. It is just a matter of scale, which you seemingly don't understand. Moreover, the more the object is distant from you, the smaller it appears, because of how the perspective works It is also "below you", because of the curvature (it can be inferred from this picture). All that makes noticing the lean even harder.

"How far can I see?"

Let's also try to figure out how far we can see. Assuming our view is not obscured by buildings, trees etc. of course. Have a look at the picture below.

earth horizon

In the picture you can see a simple representation of Earth. I've marked all relevant points and segments. The distance to the horizon, d, is one side of a right-angled triangle whose other two sides are h + r and r. Using this and The Pythagorean theorem, we can derive a formula for calculating d, the distance from observer P to the horizon F.

After rearranging:

It is a general formula, so we can caluculate how far we can see from any height. Let's consider some common cases. I've chosen some arbitrary values of h. Let's also calculate the value of alpha, so we will see how much objects located near the horizon would be leaning out from the observer. But how to do that? Well, we will use FPC right triangle again and one of the trigonometric functions - tangent. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side:

In our case, after calculating d, tangent value is known and alpha is what we need. To obtain our angle, we need inverse tangent function - arctangent:

Ok, we are now ready to do some calculations.

from the ground (h = 1.7m)




from a plane (h = 11km)




from a high-altitude baloon (h = 30km)




from the ISS (h = 370km)



As you can see, even from the ISS astronauts see only a fraction of Earth's surface (they can see the curvature though. It is even possible from a high-altitude balloon, but more about this next time). And while standing on the ground you can literally see pretty much nothing :) Even if your view is not obscured by anything and weather conditions are perfect.

Conclusion

All the stuff I've posted so far was intended to explain how big our planet is, how small piece of it we are allowed to see and how does it impact our ability to see the curvature. As a matter of fact, it is possible to prove that Earth is round even from the ground, but it is quite tricky. However you can already see the curvature from a high-altitude balloon quite easily. The bigger the altitude, the easier it is. I initially intended to discuss this in this very post, but it has already grown quite big, so I will move it to the next one, so stay tuned!

I almost forgot. Here are the answers to my little quizzes:
First picture: yellow cuboid
Second picture: yellow and red cuboids
Third picture yellow, red, blue, orange, white cuboids are leaning out from the observer by 0, 1, 2, 3 and 4 degrees respectively

Did you manage to answer right?

Equations in this post have been generated using mathurl.com.
So, that's it. If you enjoyed this post, I’d be very grateful if you’d help it spread by emailing it to a friend, or sharing it on Facebook. Definitely you should share it with a flat-Earther, if you know one. Feel free to write a comment.