In Figure 9.1 the infinitesimal changes for each polar coordinate are plotted for a random point and a dashed circle is shown for all points with the same radius.įigure 9.1: Two dimensional Cartesian and polar coordinates on a flatĪssuming an object is placed at a certain position, which can be parameterized as \((x,y)\), or \((r,\phi)\), a general infinitesimal change in its position will place it in the coordinates \((x+dx,y+dy)\), or \((r+dr,\phi+d\phi)\). In Figure 9.1, the basic axes of these two coordinates are plotted.Īn infinitesimal change in the direction of each axis is written as \(dx\) and \(dy\).įor each point, the infinitesimal changes are parallel with the respective axes and are not shown for clarity.Īnother very useful way of parameterizing this space is through polar coordinates.įor each point, we define a radius (\(r\)) and angle (\(\phi\)) from a fixed (but arbitrary) reference axis. One of the most basic ways to parameterize this space is through the Cartesian coordinates (\(x\), \(y\)). To start, let’s assume a static (not expanding or shrinking), flat 2D surface similar to Figure 9.1 and that the 2D creature is observing its universe from point \(A\). The start of the analysis might seem too mundane, but since it is difficult to imagine a 3D curved space, it is important to review all the very basic concepts thoroughly for an easy transition to a universe that is more difficult to visualize (a curved 3D space embedded in 4D). Here, we will be working with this creature in its efforts to analyze distances in its 2D universe. To be more understandable (actively discuss from an observer’s point of view) let’s assume there’s an imaginary 2D creature living on the 2D space (which might be curved in 3D). Once the concepts have been created/visualized here, we will extend them, in Extending distance concepts to 3D, to a real 3D spatial slice of the Universe we live in and hope to study. Hence, here we will assume a 2D surface/space and discuss distances on that 2D surface when it is flat and when it is curved. The emphasis here is on visualization of the concepts for a beginner.Īs 3D beings, it is difficult for us to mentally create (visualize) a picture of the curvature of a 3D volume. There are many references available for such reviews that go deep into the abstract mathematical proofs. However, this section is not intended to be a fully thorough and mathematically complete derivation of these concepts. However to be generic (and allow its measurement if it does in fact exist), it is very important to create a framework that allows non-zero uniform curvature. The observations to date (for example the Planck 2015 results), have not measured 200 the presence of significant curvature in the universe. Next: Extending distance concepts to 3D, Previous: CosmicCalculator, Up: CosmicCalculator
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