Flat Earth – The Horizon, Curvature and Angles

Over the past few months I’ve been working on the curvature equation for a circle.  It doesn’t seem like very exciting stuff but it has enormous implications.  During this process I thought I had found this equation but it turned out to be incorrect.  I had fixed what I thought was the error but that turned out to be incorrect as well.

The nagging problem stems from the way one would experience curvature if they truly lived on a ball.  All the current methods of finding curvature don’t really have a good explanation and upon further study are shown to be calculating something other than curvature.  So what do we mean by curvature?  I will answer this question and provide a new and rational equation for curvature and show why the other equations do not work.

Ball Earth Math

This famous document that has made the rounds in the flat earth community is not so much incorrect as it calculates an irrelevant number.  Though it does properly calculate the value for X (or the “drop” along the axis), X is not the value we are looking for.

As you can see, the line X is tilted so it is parallel to the axis.  This is where the calculation becomes misleading.  What we really need to do is extend the line in order to intersect the line of site projected from point 0 (in the Ball Earth Map document).  Please see my curvature image to see an example of this.



Distance² x 8 / 12

The equation “distance² x 8 inches / 12” does not solve for curvature but only for the hypotenuse of the distance: “Height = R-R*(cos(θ))” and “Length = sin(θ)*R”.  Since these two values only represent the length and height of the distance travelled along the hypotenuse; and, since you cannot have a circle without height and length of equal value, you cannot have an arc without height and length of unequal value; therefore, curvature cannot simply be height (or the amount of “drop” along the axis) nor the value of the hypotenuse.

As an example, 1° of circumference is equal to 24901/360°=69.1 miles.  If we plug in that value to the equation we get “√((3959-3959*(cos(1°))² + (sin(1°)*3959))² = 69.09 miles.  This is not the value for curvature.  However, this does resolve the hypotenuse.  The equation was being used to measure curvature by plugging in the arc distance (the distance travelled along the ball) not the hypotenuse; therefore, the resultant value will only solve the value of the hypotenuse.  All we are calculating with this equation is the point at which two different lines of site intersect on a ball not the height required for an object to be visible from point A.

Two points would intersect if we “forced the line” and built along the ground from each point (A and G).  The two lines would have to be 3959 miles long.  This is equivalent to a building at point D being 1639 miles in height.  Since we build either along the surface or perpendicular to the Earth’s surface (i.e. buildings), we need to calculate something else.

Line of Site, the Hypotenuse and θ

So what are we actually trying to calculate?  If the hypotenuse is not the distance nor the length along the axis, then what is?  The confounding problem is related to the line of site of the observer.  Once we establish the position of the observer, all the other pieces fall into place.  If you examine the image below, you will notice that a line of site moves away from point A towards infinity.  So we need to take the observer as being at point A and does not move.

Next, several dashed lines at various degrees have been drawn until they intersect with the line of site originating at point A.  The equation ((1/(COS(θ)/R))-R) calculates the hypotenuse from θ to the point at which it intersects with the line of site from point A minus the radius.  What we are calculating is the height necessary for an object to be above the surface, at a particular arc distance away or at angle θ, in order for it to be visible to the observer at point A.  This is the key.

As an example (if the ball theory is to work), as a ship goes over the horizon, the mast (and the rest of the boat) will begin to tilt as per angle θ.  Though this is a tiny angle at first, it nonetheless must follow that angle.  The mast does not start tilting back to stay parallel with the axis of the ball, it stays fixed to the boat.   As the boat continues along the circumference, the mast would need to extend in height to remain visible to the observer at point A.  This is an effect of curvature.

In the diagram below, I put the original curvature calculation beside the equation I have proposed.  You can see that at smaller distances the two values are very similar.  However, as you move past 2° the values begin to diverge more rapidly.

A simple way to calculate the “distance to horizon” is to divide the distance travelled by 69.17 miles which equals θ and plug that into ((1/(COS(θ)/R))-R). 

Click me


So what do we mean by curvature?  For example, if I travelled from point A to G, I would have experienced 6225 miles of the total circumference but there is not 6225 miles of curvature between point A and G.  If you look at line D1 – D0, you will see that there is only 1159 miles of arc height (or the maximum height of the arc between two points).  It also happens to be the value of X at 45° or 90°/2.  This works for any arc length.

If you know the distance travelled, you can calculate the equivalent θ travelled.  Using a variation of the Ball Earth Math equation, R-R(COS(θ/2)), we can solve for X which gives us the maximum height of the arc.

At a distance of 6225 miles (or ¼ of the globe) the maximum height of the arc has been shown to be 1159 miles.  In a similar fashion, the ship and the observer would have to lift off the surface at 45° and travel for 1159 miles to see each other.  However, we can see that all this is doing is altering the line of site for both the boat and the observer.


The definition of curvature is the degree to which a curve deviates from a straight line, or a curved surface deviates from a plane.  The curvature of a circle is defined mathematically as the reciprocal of the radius (ie. κ = 1/r).  All this tells us is that as the circle becomes larger κ becomes smaller.  This does not help us figure out what the effect of curvature is to a person living on a ball.

However, by showing rational examples I have demonstrated the effect of curvature is the height necessary for an object to be above the surface, at a particular arc distance away or at angle θ, in order for it to be visible to the observer at point X.  Without altering the line of site of the observer and keeping the object (in this case a boat) on the surface of the earth, we can measure the effect of curvature.

I would propose then, that what we are looking for is effective curvature and it is defined as “The height necessary for an object to be above the surface, at a particular arc distance away or at angle θ, in order for it to be visible to the observer at point X and X being a stationary position.”



Perspective, Geometry and Flat Planes


Recently I’ve was in a rather colourful chat with a proponent of the globe model.  It was actually quite useful as it forced me to fix a few errors on one of my previous posts.  I’ve been working on a model for the flat plane using trigonometry and from that I developed a couple of axioms.

Dealing with people who are both closed off to new ideas and have anti-social tendencies can make a rather potent brew of a personality.  However, I see it as a way to strengthen the flat plane model rather than harming it.  Just like you need the resistance of the ground beneath your feet to walk, we also need resistance from the globe proponents – without it, we couldn’t develop the models we have.

With that in mind, I’ve been wanting to examine perspective in relation to Euclidian geometry.  This particular globe proponent relies heavily on geometry to bash the heads of flat plane folks.  The thing that dawned on me was the important difference between perspective (which works in 3 dimensions – x,y,z) and Euclidian geometry which operates in only 2 dimensions (x,y).  Of course the Euclidian axioms have been extended and expanded to incorporate solid geometry in 3 dimensions but not his original “Elements”.

Here are his original Axioms:

  1. “To draw a straight line from any point to any point.”
  2. “To produce [extend] a finite straight line continuously in a straight line.”
  3. “To describe a circle with any centre and distance [radius].”
  4. “That all right angles are equal to one another.”
  5. The parallel postulate: “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”

Each of the axioms operates in only 2 dimensions involving only lines and angles (and a point).  Axiom 5 is where the globe proponent kept hammering – if the interior angles on the same side are equal to two right angles, then they will never meet and therefore, “bunching” of objects at the horizon are impossible.  To get an idea what is meant here, please watch Brain Mullin’s great video:

Assuming you’ve watched this video, you can see what is meant by “bunching”.  The opponents of this video argue that he is applying perspective incorrectly because he can’t show where “…train tracks cross”.  Now I’m not exactly sure if they are just using hyperbole or not but the idea is that you can’t “bunch” parallel lines.  This is obviously the case if you apply Euclidian geometry in 2 dimensions.  Just apply Axiom 5 to any set of parallel lines and they are absolutely correct.

However, what these opponents have done is applied Axiom 5 to a non-Euclidian example – namely an object in 3 dimensional space.  I’ve come up with a simple example to explain this:

  • On a 8 x 11.5 piece of paper and draw an isosceles triangle and hold it directly in front of you face.  This image is presenting itself in 2 dimensions.


  •  Now, while sitting at a desk, place the piece of paper flat on the desk with the top of the triangle pointing away from you.

The triangle is now presented in 3 dimensions and takes on the laws of perspective.  If you push the paper away, you will notice that the top and bottom of the triangle begin to “bunch” together.  Eventually, the top and bottom will merge together.  The two sides of the triangle never have to cross or change angles or alter in anyway – the object stays the same but it appears to change.

You can do the same thing with parallel lines.  Draw two parallel lines on another sheet of paper and hold it up to your face.  Just as predicted in Axiom 5, the lines will never cross.


But now place the paper on the desk in front of you – what happens?  The lines begin to merge together like this:


Again, if you push the paper away from you, the bottom and top of the lines will also begin to “bunch” together.  This is the point of Brian’s video but is ignored by the opponents of it.  Again, they are applying 2 dimensional logic (which is correct if it stays in 2 dimensions) to a 3 dimensional observation.

As I was trying to explain in my previous post, viewing angles are not linear with distance and proceed by doubling (1,2,4,8,16,32,64) while the distance needed to travel to achieve that viewing angle is only half.  For example, a mountain that is 2.5 miles high can be seen at a distance of 144 miles with a 1 degree viewing angle.  You would need to travel half that distance (72 miles) to see it at 2 degrees.  And again, half that distance again  (36 miles) to see it at 4 degrees, etc.

Because of this affect, you get “bunching” of the object as it approaches the horizon.  This phenomenon is recognized in the image rendering world:

Objects Bunching on the Horizon

The artists who work in this field do everything in their power to mimic reality so they take into account real affects not the opinions of either global or flat earth communities. But you get my point (I hope) that bunching on the horizon is a real phenomenon that has been modeled and expressed mathematically.

So what does this mean?

If this “bunching” phenomenon is valid (which is seems to be) then the flat earth community has a scientifically and mathematically based model that can explain things like:

  • Viewing angles of Polaris at different locations in world
  • How the sun/moon “rises and falls”
  • How light from the sun/moon increases/decreases with distance
  • How objects disappear at the horizon

I’m looking at modeling this phenomenon in a real 3D rendering system.  Hopefully it will provide some necessary information for those looking into our world and our place it it.