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The ‘Spherical’ Earth

In this blog post we examine some of the common misconceptions and facts about our round planet which we all live and thrive on.

In this blog post we will talk about the ‘spherical’ earth that we are all living in. Some people have a common misconception that the earth is flat, and for millennia this has been a carefully discussed topic especially in the middle ages where religious beliefs ruled the entire society.

We will try and debunk utilizing state-of-the-art photos, mathematics for describing our ’round’ earth and finally come with some facts about the earth and common misconceptions which people should be careful not to fall against.

Is the earth flat?

Before starting off on our talk about our beloved earth lets try and answer one of the simplest questions most people have asked themselves: Is the earth flat? To start off with, it certainly seems flat to the naked eye, as the horizon on which you can steer out into creates a seemingly straight line as expected with a flat earth. One interesting video about the flat earth is the ‘Vsauce’ video titled: “Is Earth actually flat?” A contribute to debunking the flat earth society.

Video 1: Vsauce video about the flat earth society and whether or not the earth is a flat disc.

For centuries this fact has puzzled human kind who believed to be the center of the universe and the earth to be a flat plate swimming alongside the back of a turtle, or in some cases, a flat disc where the sun has been placed on a curved space with a stationary earth.

Fighting the ‘Globe’ Theory

One of the latest’s and most famous examples of such a phenomena is the 1893 drawing by Richard Ferguson, a firm believer in the flat earth theory constructed the ‘plausible’ drawing of how the earth might be a curved square space with a stationary centered earth against the entire solar system and how this revolved around the earth system. He constructed this in order to ‘combat’ the dominant ‘Globe’ theory.

The 'square and stationary earth' theory proposed by Orlando Ferguson meant as a fight against the dominant 'Globe Theory'
Figure 1: The flat earth fighter Orlando Ferguson in 1893 fighting against the ‘Globe Theory’ the square space contains numerous biblical passages which is used to justify the wrong assumptions.

Even to this day with our photographic evidence from satellites and space missions, there exists people who remain members of the so-called flat-earth society, a society of firm believers in the flat earth theories that exist.

Photographic evidence of ‘Globe’ earth

After the invention of satellites in the 20’th century we now have photographic evidence of the so-called ’round’ earth as can be seen in Gallery 1. Here a collection of pictures of our blue ’round’ earth have been taken showcasing clearly that the earth is round.

This earth looks a lot like a sphere and surely this is a way better approximation of the shape of the earth rather than the flat earth society experienced. The following shows how to calculate the spherical earths volume, circumference and surface area based on a supplied radius of the earth.

Related read: Top ten misconceptions about ChatGPT, Top ten misconceptions about Earthquakes, Top ten misconceptions about Tsunamis

The Spherical earth

Now lets dive down into the mathematical description of our spherical earth. A sphere is most easily described by its radius, r, from which you can calculate the volume, V, and surface area, A, along with the two-dimensional circumference S.

(1)   \begin{align*} V &= 4/3 \cdot \pi \cdot r^3 \\ A &= 4 \cdot \pi \cdot r^2 \\ S &= 2 \cdot \pi \cdot r \end{align*}

For the determination of the radius of the earth there exist multiple values based on a variety of methods a great overview over the different methods is given in a Wikipedia article:

The most common values ranges between 6.384 km – 6.528 km for a radius of the earth. This quite different range details the differences in roundness that the earth resembles, as the mountains and sea-floor valleys aren’t accurately mapped in a perfect spherical world.

The non-spherical earth

The spherical earth which we live on is therefore not exactly round as first expected as the mountains valleys and later realized poles of the earth aren’t accurately mapped in a spherical system. The

In fact when mathematicians worked on theories of spherical earths and tidal forces, wind speed generations, Coriolis forces to mention a few. These are facts about the earth which a full perfect sphere cannot perfectly answer.

The slight variations in shape and size of the earth have significant impacts on the way our planet behaves. For example, the mountains and valleys affect wind patterns and ocean currents, and the non-uniform distribution of mass leads to variations in gravity, causing fluctuations in the Earth’s rotation.

Despite these complexities, the spherical model of the earth has proven to be incredibly useful for navigation, satellite communication, and understanding the world around us. But it’s important to recognize that the earth is not a perfect sphere and that its imperfections have real-world effects.

Mathematical description of imperfections

The description of the deviations from the spherical shape is mathematically described through the flattened ellipsoid equations where an ellipsoidal disc can be described simply through two different equations concerning the radii typically described in the equations of state.

The equations used to describe the surface area, volume and in a similar format as that of a spheroid except for the fact that the parameters; a, b and c are introduced instead of the value of r.

Ellipsoids for describing the Globe type earth which is more accurate than the spheroid version.
Figure 2: Illustration of the three typical ellipsoids with different values of the radii parameters a, b and c. For the special case where a=b=c we have a spheroid. See source.

(2)   \begin{equation*} (\fraq{x}{a})^2 +(\fraq{x}{b})^2 + (\fraq{x}{c})^2 = 1 \end{equation*}

Now for the cases where surface areas of ellipsoids need to be calculated the expressions are quite a bit more complicated as the surface integrals depend on the chosen parameterization of the different values of a, b and c among others.

The parameterization typically found used in spherical projections correspond to spherical coordinate systems utilized in many different aspects of life and engineering in general. The parameterization follows to be the following:

Volume of an ellipsoid:

(3)   \begin{equation*} V = \frac{4}{3} \cdot \pi \cdot a \cdot b \cdot c \end{equation*}

While the general formula for spherical coordinates and therefore parameterized surface area of the ellipsoid is the following:

(4)   \begin{equation*} \begin{bmatrix}   x = a \sin(\delta) \cos(\theta) \\   y = b \sin(\delta) \sin(\theta)\\   z = c \cos(\theta) \end{bmatrix}  =  R \cdot  \begin{bmatrix}   \cos(y)\cos(\lambda) \\   \cos(y)\sin(\lambda) \\    \sin(y) \end{bmatrix} \end{equation*}

Where -1/2 \leq y \leq 1/2 and 0 \leq \lambda \leq 2 \pi. The definition of R is described as

(5)   \begin{equation*} R = \frac{abc}{\sqrt{c^2 (b^2 \cos^2 \lambda + a^2 \sin^2 \lambda) \cos^2 y + a^2 b ^2 \sin^2 y}}, \quad -\pi/2 \leq y \leq \pi/2, \quad 0 \leq \lambda \leq 2 \pi \end{equation*}

These equations are used in various applications from satellite tracking systems to navigation systems for vessels and ship, to map creation and transformations utilized for creating specific map types.

Now that we have established that the earth is round one may consider approving of this theory and buying a globe to support the ‘globe’ theory. This helps the website and content creator if you decide to click and buy one of the recommended globe spheres below.

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