\( \newcommand\der{\operatorname{der}} \newcommand\Der{\mathrm{D}} \newcommand\dd{\operatorname{d}} \newcommand\ang[1]{#1^\circ} \newcommand\parenthesize[1]{\left(#1\right)} \newcommand\dif{\mathrm{d}} \newcommand\Dif{\Delta} \)

1Visualization

  • Visualize the surface of the Earth as a sphere.
  • Visualize a person at a point on the surface of the Earth.
  • Visualize yourself looking at that person and the Earth from outer space.
  • This document has been designed to be understandable by replacing "manifold" with "sphere".

2Spherical coordinate systems

2.1Mathematician's heading

To simplify the explanation of the coming spherical coordinate system, we introduce "heading".

Our definition of heading begins with pointing east (positive x-axis), and then goes counterclockwise, so is north (positive y-axis), is west (negative x-axis), and is south (negative y-axis).

Note that our heading differs from the navigator's heading, which begins with pointing north and then goes clockwise.1

2.2? The spherical coordinate system

The spherical coordinates \((r,a,b)\) mean "set heading to \(a\), set elevation to \(b\), and then go the distance \(r\)".

The spherical coordinate system adds another angle component to the polar coordinate system.

We now describe how to map the spherical coordinates \(S(r,a,b)\) to Cartesian coordinates \(C(x,y,z)\). The slogan to remember is that \(S(r,a,b)\) means “set heading to \(a\), set elevation to \(b\), and then go the distance \(r\)”. Another slogan is “face east with your entire body, then turn your entire body left by angle \(a\), then turn your head up by angle \(b\), and then look at distance \(r\)”.

Let's read slower as we imagine the drawing.

Let \(O\) be the center of both the spherical coordinate system and the Cartesian coordinate system. Let their centers coincide.

A negative angle \(-a\) means the angle \(a\) but in the reverse direction.

Draw the point \(D\) at \(C(r,0,0)\), which means that \(D\) lies on the positive x-axis, at distance \(r\) from \(O\).

With the positive z-axis as the axis of rotation, rotate the vector \(OD\), by angle \(a\) toward the positive y-axis (or toward the negative y-axis if \(a\) is negative). Call the resulting vector \(OE\). Thus, the point \(E\) is at \(C(r \cos a, r \sin a, 0)\), which is still on the xy-plane.

Then, rotate the vector \(OE\), by angle \(b\), out of the xy-plane, toward the positive z-axis (or toward the negative z-axis if \(b\) is negative). Call the resulting vector \(OF\).

Then \(F\) is the point described by \(S(r,a,b)\).

2.3Relationship with Cartesian coordinate systems

Let's say that the point \(F\) is at \(S(r,a,b)\), which is equal to \(C(x,y,z)\). By the definition of rotation and the congruence of triangles, we can convert spherical coordinates to Cartesian coordinates as follows:

\[\begin{align} x &= r \cos a \cos b \\ y &= r \sin a \cos b \\ z &= r \sin b \end{align} \]

We can convert Cartesian coordinates to spherical coordinates as follows:

\[\begin{align} r &= \sqrt{x^2 + y^2 + z^2} \\ \tan a &= y/x \\ \sin b &= z/r \end{align} \]

but we have to pick the angles that make the signs correct.

Thus, we have just explained what is meant by the slogan “set heading to \(a\), set elevation to \(b\), and then go the distance \(r\)”.

2.4Which spherical coordinate system?

Note that our \(b\) is elevation, not azimuth. The relationship between elevation and azimuth is \[ \text{azimuth} = \ang{90} - \text{elevation}. \]

If the elevation is zero, then the spherical coordinate system reduces to the polar coordinate system on the xy-plane. That method embeds the polar coordinate system into the xy-plane. Thus the angle \(a\) is called heading angle, polar angle, or longitude. The angle \(b\) is called elevation angle or latitude.

Azimuth is angle from zenith. In this case, zenith is the z-axis.

ISO standard? Azimuth? Elevation?

What2

3Solid angles

4Distance on sphere, great circles, geodesics

  • Generalize straight lines to geodesics: shortest one-dimensional object connecting two points.
    • Let there be a sphere of radius \(r\).
    • Let X be a point on the sphere.
    • Let Y be a point on the sphere.
    • ? the sphere's shortest one-dimensional subspace that connects X and Y.
    • The metric (which determines "shortest") is implied from context.

5Manifold

  • A manifold is a locally Euclidean space.
    • "Manifold" generalizes "smooth surface".
    • For now, think of "manifold" as "smooth surface".
    • Example manifold: a sphere.
    • Another example: a smooth two-dimensional surface in three-dimensional ambient Euclidean space, such as a curved sheet of paper.
    • A manifold is a "space that locally resembles Euclidean space near each point".
    • The manifold's ambient space is implied from context.
    • Formally, every manifold is a space (a set of points).

6An important shift in thinking

  • To begin studying differential geometry, we shift our thinking from global to local:
    • Previously: We associate some things with the entire space.
    • Now: We associate some things with each point in a manifold.
    • Previously, we have a space:
      • A metric is associated with the entire space.
      • A coordinate system is associated with the entire space.
    • Now, we have a manifold (which is also a space) but:
      • A local metric is associated with each point of a manifold.
      • A local coordinate system is associated with each point of a manifold.
      • A manifold can be globally non-Euclidean but is locally3 Euclidean.

7Tangent spaces

  • Definitions for this section:
    • We can derive a vector space \(V(A)\) from a space \(A\) by collecting every possible subtraction of every pair of elements of \(A\).
      • An \(A\)-vector is a member of \( V(A) = \SetBuilder{x-y}{x,y\in A} \).
      • A vector can be created by subtracting two points.
    • Let \(S\) be a space whose ambient space is \(A\).
    • Let \(p\) be a point in \(S\).
    • A plane in ambient space \(A\) is \( \SetBuilder{x}{n \cdot (x-b) = 0, ~ x \in A} \), where \(n \in V(A)\) is a vector that is normal to the plane, and \(b \in A\) is any point on the plane.
  • Understand that \(T_p(S)\), the tangent space of \(S\) at \(p\), is a vector space of every \(A\)-vector that is tangent to \(S\) at \(p\).
    • The ambient space is implied from context.

8Curvature

  • Understand osculating circles (kissing circles).
  • Understand that the curvature (scalar curvature or Gaussian curvature) of a curve at a point is the reciprocal of the radius of the circle that osculates the curve at that point. Formally, \( K = 1/R \).
    • A straight line has zero curvature everywhere.
  • Osculating circles readily generalize to osculating hyperspheres:
    • Replace normal lines with normal hyperplanes.
  • Understand Riemann curvature tensor [1]?
  • Ricci curvature function

9Local coordinate systems

  • Understand local coordinate systems.
  • A local basis at \(p\) is a basis for the tangent space at \(p\).
    • Recall:
      • Such tangent space is a vector space.
      • A basis corresponds to a linear coordinate system.
  • geodesic normal coordinate system?

9.1Local coordinate systems on a sphere

  • Consider a sphere with center \(O = (0,0,0)\) and radius \(r\).
  • At point \((r,0,0)\), we can have these local coordinate axes (local basis vectors): \((0,1,0)\) and \((0,0,1)\). The sphere's tangent space at \((r,0,0)\) is a linear combination of those two local basis vectors. Thus a \((r,0,0)\)-local Cartesian coordinate tuple \((u,v)\) corresponds to the ambient Cartesian coordinate tuple \((r,u,v)\).

10What does "parallel" mean?

  • What does it mean for two lines to be parallel ("beside one another")?
    • In Euclidean geometry, two lines are parallel iff either they are the same line or they never meet. (Recall that a line extends infinitely in both directions.)
    • Generalize: two vectors are parallel if they have the same direction.
  • What does it mean for two vectors to have the same direction?
  • ? https://en.wikipedia.org/wiki/Vector_field#Vector_fields_on_manifolds
  • Deforming a manifold deforms its vector field in the same way?
    • Draw two different parallel horizontal lines on a sheet of paper.
    • Roll the paper into a cylinder.
    • The lines becomes circles!
  • Two great circles of a sphere are "parallel" in what sense?
  • Generalize "parallel" to "parallel transport".
  • affine connection4
    • Roll a plane along a manifold such that the plane is always touching the manifold. (Recall: tangent = touching)

11Local metrics

  • The local metric at point \(p\) is \( g(p) \).
    • Note that \( g \) is a function that takes a point and produces a function that takes two tangent vectors. We follow currying convention: \( [g(p)](u,v) = g(p,u,v) \).
  • Difference from ordinary metrics:
    • A metric takes two points, but a local metric takes two tangent vectors.
    • Interpret \(g(p,u,v)\) as the local distance between point \(p+u\) and point \(p+v\).
      • Remember point–vector addition.
      • The points \(p+u\) and \(p+v\) often lie outside the manifold, although this error is small for short tangent vectors.
    • \( d(p+u,p+v) \) and \( g(p,u,v) \) may differ. The ambient space's global metric and the manifold's local metric may not coincide. For example, the distance between two opposite points on a sphere is:
      • \(2r\) if we drill through the sphere (this is the ambient space's metric);
      • \(\pi r\) if we stay on the sphere (this is the sphere's metric).
  • If \(g(p)\) is linear (with respect to what?), we can state it as a metric tensor.
    • \( g(p,\dif u,\dif v) = \sum_i \sum_k g_{ik} (\dif u_i) (\dif v_k) \)
  • The differential form of the three-dimensional Euclidean metric? \[ (\dif s)^2 = (\dif x)^2 + (\dif y)^2 + (\dif z)^2 \]

12Cartography (mapmaking)

13Local trajectory?

13.1The length of a curve in a manifold; integral of local metric?

14Tensor notation and terminology

  • Understand that \(g_{ij}\) notates a tensor, not is a tensor.
    • \(g\) is the tensor.
    • \(g_{ij}\) is a real number.
  • ? covariance and contravariance with respect to what?

15Synthetic differential geometry?

16Chart, atlas?

  • Are we interested in this detail?
  • A chart (a coordinate chart) for \(S\) is a homeomorphism between a subspace of \(S\) and a subspace of an Euclidean space.
  • An atlas is a collection of charts.
  • "There are ways of describing curves without coordinates […]"5

17Bibliography

[1] Arnold, V.I. 1989. Mathematical methods of classical mechanics. Springer-Verlag.

  1. https://en.wikipedia.org/wiki/Cardinal_direction#Additional_points↩

  2. https://en.wikipedia.org/wiki/Spherical_coordinate_system↩

  3. https://en.wikipedia.org/wiki/Local_property↩

  4. <2020-01-27>↩

  5. <2020-01-26> https://en.wikipedia.org/wiki/Coordinate_system↩