Assignment 11
Problems
Optional: Orthographic Map
You do not have to turn this problem in if you do not want to, but I recommend it because it gives a good concrete example of the kind of computations you have to do to relate the angle you see when looking at a map to the map-angle-the true angle between the vectors on the earth that are being depicted.
Exercise 1 Can you find the coordinates \((x,y)\) of a point on the map where the vectors \(\langle 1,0\rangle\) and \(\langle 0,1\rangle\) only make a \(45\)-degree angle with one another?
Hint: can you make the problem easier for yourself by restricting \(x\) and \(y\) to lie on some line, so the problem ends up having one variable instead of two?
You can see how this would make such a map difficult to use for navigation: it would look like the map is telling you to turn \(90\) degrees but in reality you should only turn half that!
Because having to do computations like these constantly when working with a map is a huge technical headache, mathematicians much prefer conformal maps, where the angle you see in the Euclidean plane accurately represents the map-angle, and all of this is unnecessary. (This is why the main map employed by mathematicians, Stereographic Projection, is conformal).
Optional: Archimedes Map
You do not have to turn this problem in if you do not want to, but I recommend it because it is both an interesting example of a map, and it provides the final step in Archimedes’ argument relating the area of a sphere to the area of its bounding cylinder.
Exercise 2 Read the section of the textbook on Archimedes Map (in the Examples chapter). Give a proof that this mao is infinitesimal area preserving following the outline below:
- Show that at each point \(p\in M\) the vectors \(\langle 1,0\rangle\) and \(\langle 0,1\rangle\) are sent by \(\psi\) to orthogonal vectors on the sphere.
- Find their lengths on the sphere (ie the map-lengths), and use this data to find the area of the infinitesimal rectanlge they form.
- Observe that everything beautifully cancels and the area is still one, even though the square was stretched into a rectangle!
Since the infinitesimal area is unchanged by the map at each point, we can finish Archimedes proof via integration (which I do below, using your exercise)
Theorem 1 The surface area of the unit sphere is the same as that of its Archimedes map: that is, the same as the area of its bounding cylinder.
Proof. Archimedes map captures every point of the sphere except the north and south poles. Since points have zero area, this omission has no effect on our actual question so we can proceed to calculate with the map \(M\).
\[\area(\SS^2)=\iint_{\SS^2} dA_{\SS^2}=\iint_{\psi(M)}dA_{\SS^2}= \iint_{M}dA_{\map}\]
But now we know that \(dA_{\map}=dA_{\EE^2}\), that’s what you’ve proved in the exercise above! So we can sub this out, and then realize the resulting integral is just the definition of the Euclidean area of \(M\) in the plane:
\[=\iint_M dA_{\EE^2}=\area(M)\]
Stereographic Projection:
Read the stereographic projection section first (we will cover the necessary bits in class as well)
The stereographic projection map has many uses in mathematics beyond just representing points of the sphere on the plane. Because it is conformal (angle preserving), its often used as a tool to help build more interesting conformal maps between regions of the plane, following this general recipe:
- Start with a region on the plane.
- Use \(\psi\) to map it to the sphere.
- Do something to the sphere, moving the region around
- Use \(\phi\) to put it back on the plane.
The overall composition is a map between two regions on the plane, that was created by going to the sphere and back! In these exercises, we will deal with a fundamental example of this, and construct a map from the unit disk onto half of the entire plane!
The strategy above is summarized for this case in the following three figures:
Exercise 3 (Disk and Half Plane: Construction) Let \(\DD\) be the unit disk \(\DD=\{(x,y)\mid x^2+y^2<1\}\) and let \(\mathbb{U}\) be the upper half plane \(\mathbb{U}=\{(x,y)\mid y>0\}\). Let \(T\colon \DD\to \mathbb{U}\) be the map described above. Prove that $T can be expressed as
\[T(x,y)=\left(\frac{2x}{1+x^2+y^2-2y},\frac{1-x^2-y^2}{1+x^2+y^2-2y}\right)\]
By building it step by step:
- Start with \((x,y)\) in the unit disk.
- Apply \(\psi\) to get the disk onto the sphere.
- Rotate the sphere about the \(x\) axis in the appropriate way so that the south pole goes to \((0,1,0)\).
- Apply \(\phi\) to return to the plane.
This map is conformal - meaning that it preserves all angles! And even more than that, it takes generalized circles to generalized circles.
Exercise 4 (Disk and Half Plane: Understanding) Prove that these claims are in fact true: that our new function is conformal, and sends generalized circles to generalized circles. Hint: what kinds of maps is it built out of? What do each of these maps to do angles, or to generalized circles (on the plane) / circles (on the sphere)?
Use this to “transfer” this picture of polar coordinates in the unit disk onto the plane, via our new map.
Spheres of Radius \(R\):
The chapter on stereographic projection deals with the unit sphere. It is not too hard to generalize what we have done to spheres of other radii, and while this may not sound super exciting at first, it actually turns out to be absoltuely fundamental to how we are going to discover hyperbolic space! So, it is a rather important exercise to work this all out for yourself.
The good news is you have this entire chapter as a guide, where I’ve worked out many of the details for the case of the unit sphere. The formulas will be quite similar, but there’ll be \(R\)’s inserted in various places: so the second piece of good news is that I’ll give you the formulas that you need to derive! That way, you can check your work.
Definition 1 Let \(\SS^2_R\) be the sphere of radius \(R\) in \(\EE^3\). Then the chart \(\phi\) for stereographic projection of this sphere is defined geometrically exactly as in the original version: given a point \(p\in\SS^2_R\), \(\phi(p)\) is where the line connecting \(p\) to the north pole \(N=(0,0,R)\) intersects the \(xy\) plane.
Exercise 5 Show that the formulas for both the chart and the parameterization of stereographic projection here are as follows:
\[\phi(x,y,z)=(X,Y)=\left(\frac{Rx}{R-z},\frac{Ry}{R-z}\right)\]
\[\psi(X,Y)=(x,y,z)=\left(\frac{2R^2 X}{X^2+Y^2+R^2},\frac{2R^2 Y}{X^2+Y^2+R^2},R\frac{X^2+Y^2-R^2}{X^2+Y^2+R^2}\right)\]
(It might help to look back at Proposition Stereographic Projection Formula, and attempt Exercise exr-stereo-parameterization).
Running through the same arguments as in the chapter above (which you don’t have to write down), its straightforward to check that this new map is a conformal map between \(\SS^2_R\) minus \(N\), and the plane. This means its parameterization \(\psi\) both preserves angles and stretches all vectors by a uniform length: we can use this fact to compute the dot product for this map.
Exercise 6 At a point \(p=(X,Y)\) on the plane, what is the factor by which a vector \(v\in T_p\EE^2\) is stretched when mapped onto \(\SS^2_R\) by the parameterization of stereographic projection? Hint: we know the factor is the same for all vectors: so pick an easy vector to calculate with and find its length!
Once you know this, follow the argument style of Theorem Stereographic Dot Product to compute the map-dot product on the plane, and show that it is equal to
\[(v\cdot w)_\map = \frac{4R^4}{(R^2+X^2+Y^2)^2}(v\cdot w)\]