The equator is a circle that goes around the earth. Does that mean that it is a line?
I'm guessing that it is because it is infinite. At the same time, it's overlapping. The real question is, is a circle technically an infinite line?
$\endgroup$ 32 Answers
$\begingroup$From wikipedia we can extract:
Thus in differential geometry a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.
Hence the Equator can be interpreted as a line in the geodesic sense.
$\endgroup$ $\begingroup$I assert that the equator is not a line.
From Wolfram MathWorld:
A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions.
If you look at Earth in terms of orthogonal unit vectors (i.e., $\mathbf i$, $\mathbf j$, $\mathbf k$ or equivalents) then the equator occupies at least two dimensions.
If you look at Earth in terms of spherical coordinates, where the origin is Earth’s center, then equator is of the form $r=k$ for a constant $k$. Thus only one dimension is used to define it. However, its finiteness/infiniteness is debatable:
- If your coordinate system is defined such that $\theta$ is restricted to vary on $[0,2\pi)$ then the equator has a finite length of $2\pi k$.
- If your coordinate system is defined such that $\theta$ can vary across $(-\infty,\infty)$ then the locus (or set of points defined by a criterion in some way) $r=k$ is infinite.
However, notice that $r=k$ for $\theta\in[0,2\pi)$ and $r=k$ for $\theta\in(-\infty,\infty)$ — though they appear differently in numerical form — overlap entirely on a graph. This is part of the reason why $\theta$ is almost always restricted to $[0,2\pi)$ (which is mentioned in the linked article).
Thus, in the conventional sense, the equator is not a line.
You also ask if a circle is an infinite line. In response to that, I ask you, what is the length of that line, or in other words, its circumference? There are many reasons to not think of a circle as an infinite line. I’ll leave that investigation to you; perhaps you’d like to write a paper on it one day....
$\endgroup$
