 # A Couple of Lingering Geometric Questions...

‘Lingering’, because they never answered them in grade school or high school.

The closer you slowly travel to the North Pole (or South Pole, for that matter), the slower the earth revolves. When you finally reach dead center, i.e., when you reach the point (deliberate mathematical term, here), does the motion finally just STOP? I know I should know the answer, but again, no one ever taught me:o.

Also, I have a few questions about three-dimensional Cartesian coödinates (right term?).
For example, x[SUP]2[/SUP]+y[SUP]2[/SUP]+z[SUP]2[/SUP] is a three-dimensional circle, in other words a sphere (or should they all be cubed–whatever:smack:). How do you represent just a line in three-dimensional space? I think z=1, e.g., is a plane, isn’t it? And please, feel free to offer a crash course in three-dimensional geometry as well. I for one would welcome it. (Also, I was about to ask, how you represent a point in three-dimensional space. But that’s easy, isn’t it? (1, 1, 1) would be a point, wouldn’t it?)

BTW, another thing that would be a point, would be where three planes intersect, no?
I know, where two planes intersect, is a line (they did tell us that much in HS geometry:rolleyes:). At first, I though therefore, four lines intersecting, would be a point. But actually, I think it would be just three. Am I right?

Thank you in advance for your nice replies:). And this one may require a lot of math and illustration. So let me say in advance, feel free to do so:). Thank you again.

:):) The earth turns on its own axis at a certain rate, about once per 23.93 hours. At all points along the axis itself, including the poles, there is no linear motion with respect to the axis, like the centre of a spinning record.

you mean x[SUP]2[/SUP] + y[SUP]2[/SUP] + z[SUP]2[/SUP] = 1

Define it as the intersection of two planes, for example, or via two distinct points on the line.

Yes.

First of all, in Euclidean space two planes may be parallel, and so forth, so the following only holds for generic configurations: two planes intersect in a line; so, if you add another plane (i.e., you have a total of three linear equations), the intersection will be a single point. For example, x - 1 = y - 1 = z - 1 = 0.

If you want to define a point by intersecting lines, you only need 2. You can, of course, most simply specify a point by giving three coordinates (a, b, c), as you point out.

Imagine a record playing on a record player, and a small person is standing on the record near the edge. That person is rotating and also moving in a circle. Now he walks toward the center of the record. When he gets there, he stays in one place, but continues to rotate. Same deal with the North Pole.

Close. x[SUP]2[/SUP]+y[SUP]2[/SUP]+z[SUP]2[/SUP]=1 (or any other constant) is a sphere.

With dertain restrictions, yes. Two planes define a line as long as they’re not parallel (parallel planes never intersect). Three planes define a point as long as no two of them are parallel, and all three of them don’t intersect along the same line. That last part is kinda hard to visualize. Picture an old-style Risk token like this, except with six points instead of five. Those would be three planes intersecting along a common line, instead of at a single point.

There are two degrees of freedom in choosing a line in a plane, so it is tempting to express a line with two parameters, e.g.
y = mx + b
where m is the slope and b the “y-intercept.” However this fails for lines parallel to the y-axis (m “wants to” become infinite). To avoid this, either use three parameters (ny + mx + b = 0) or replace (x,y) with polar coordinates (r,θ) yielding a line equation r cos(θ-β) = d. Many of us might feel that the latter equation is a “bridge (of donkeys?) too far.”

Similarly, a plane in three-space has three degrees of freedom but, again to avoid an infinite slope, you need four parameters … or a switch to polar coordinates.

To place a line in 3-space there are only four degrees of freedom, but to describe it as the intersection of two planes (each denoted with 4 parameters) uses eight parameters! There are various ways to reduce these eight parameters down to 7, 6 or even 5, but to get all the way down to four you may need Euler angles.

For the reasons you mention, these spaces of all lines, all planes, etc., are not Euclidean and are more easily described using homogeneous coordinates. For instance, the (4-dimensional) space of all lines in 3-dimensional space may be realized inside the Klein quadric in 5-dimensional projective space. The 3-dimensional space of all planes sits naturally inside a 3-dimensional projective space.

septimus, why would you use four parameters to define a plane? Three will suffice: For instance, the coordinates of the point of the plane closest to the origin. And if you’re worried about the case where the origin is on the plane, specify the point in spherical coordinates.

Another common (though less efficient, in terms of parameters) way to describe a line in three dimensions is through a set of parametric equations. You give three equations, giving each of x, y, and z as a function of a fourth parameter t (which depending on the context, may or may not stand for “time”). For instance, the set of equations could be something like

x = 3t + 2
y = -4t - 3
z = 17.2t +12.5

The drawback to this method is that there are many different sets of equations which will describe the same line, and it can be difficult to tell whether two sets of equations are equivalent. The benefit, though, is that it can also describe all manner of other three-dimensional curves, and if t does represent something meaningful like time, it gives you extra information there as well.

I cannot speak for septimus, but, while the space of planes is three-dimensional, it is not a Euclidean space, so, no matter what you do, there will be some issues with the coordinates. The entire thing looks like real 3-dimensional projective space with a point removed, I believe, so YMMV what the nicest way is to parameterize that; four homogeneous coordinates is not unreasonable, though (for instance, write ax + by + cz = d; then your coordinates are [a, b, c, d].)

Your parametric description of a line is basically specifying a single point on the line, and a direction vector. The space of all lines is 4-dimensional, but you are using 6 numbers where 4 will suffice The easiest way to visualize 3-D space is to think of it as stacked planes one over other. Like a new ream of paper (the latitude / longitude follow the same logic).

So take the ream of paper. The paper sheet in the middle is the X-Y axis when Z = 0. Now you just have to describe the points on the paper above and below the middle sheet like:

A. A cylinder : this will be a circle with the same diameter placed on each sheet of paper at the same location. So for example, x2 + y2 = 1, will be a cylinder (with the center axis as the Z axis). By default, if you do not specify a relationship of the Z variable, then it means that the same shape is drawn on all the sheets.

B. Line - if your line lies on one of the sheets of paper, then you specify which sheet (distance from the middle sheet) and how it is drawn on that sheet. If the line cuts across sheets, you specify the points on each sheet

C. Sphere : As you see from above, the sphere is a bunch of circles that is the biggest in the middle sheet and gets smaller as you go up or down (like the latitudes). So you specify the circle diameter on each sheet.

I have found this visualization to help 3D beginners. YMMV.

x[SUP]2[/SUP]+y[SUP]2[/SUP]+z[SUP]2[/SUP] = r[sup]2[/sup] is a sphere of radius r (consisting of the set of all points in three dimensions whose distance from the origin is r).

One way is with parametric equations, which Chronos gave an example of. For each value you plug in for the parameter t, you get the x, y, and z coordinates of a point on the line.

Another way would be with an “equation” with two = signs, asserting that three things all have to be equal. For example, x=z=0 would be the y-axis, consisting of all points whose x and z coordinates are 0. x=y=z would be the line that passes through the points (0,0,0) and (1,1,1) and (2,2,2) and etc.

Yep, it’s parallel to, and one unit above, the xy-plane.

It may be that most people don’t really study three-dimensional analytic geometry (with coordinates and equations and stuff) until third-semester Calculus. Here’s one link to a discussion of 3D coordinates in the context of Calculus III, though there’s no actual calculus on this particular page.

Often but not always. Image the front wall, side wall, and ceiling of a room. Those three planes intersect in a point up in the corner of the room. Now imagine the front wall, ceiling, and floor. Those three planes don’t intersect at all. It’s also possible for three planes to intersect in a line.

Thanks on the Euler angles, and the great Wiki cite and its refs.

Makes eminent sense (to my high-class brain) for airplane and ships. They also cite other applications; IANAMechanical Engineer, and IAMNAGraphics Programmer, but I’ve only seen used (and once busted my brains to learn the basics) are rotation matrixes. Perhaps they are more extensible (if that’s even a word in mathematics descriptions of formal methods)?

ETA: Just sawWiki general article on rotation formalisms with specific section on conversions. Answers undoubtedly there, undoubtedly too difficult for me. Into the breach…

Putting coordinates on an abstract geometrical figure such as a line or a plane involves making a arbitrary choice of frame of reference, which needs to be taken into account. The Wikipedia article on Grassmannian spaces explains (albeit briefly) how the Grassmannian Gr(k, R[sup]n[/sup]) of k-dimensional linear subspaces of n-dimensional Euclidean space can be regarded as the quotient O(n) / (O(k) ×O(n - k)), where O(n) is the real orthogonal group.

Now, an element of this orthogonal group O(n) can be written down as an n×n matrix whose rows, and columns, form an orthonormal basis. If n=3, then you have the familiar case of rotations in three dimensions and Euler angles. If you needed to, certainly you could work with matrices in O(4) (or SO(4)) and a correspondingly larger number of Euler angles, and so on in higher dimensions.

NB a 1-dimensional linear subspace of 3-space, for example, always passes through the origin. The OP asked about arbitrary lines, so you have to do something like putting them in (mostly) one-to-one correspondence with points in Gr(2, R[sup]4[/sup]), that is, one dimension higher. But, anyway, it’s pretty straightforward to represent them using matrices, put coordinates on the space, etc. 3 and 4 dimensions are probably most familiar to engineers and scientists.

I became familiar with unique decompositions of linear transforms, often using angles, when working as a comp. sci. researcher designing signal processing methods. With this in hand I proved a simple fundamental theorem about a certain class of linear transforms. Later I found that a Prof. of Comp. Sci. had written a 357-page book on this exact topic, almost every result wherein was a trivial corollary of that fundamental theorem (which he never derived)!

That book-writing professor is now a top researcher at Microsoft, while I’m the forgotten hermit in an Asian jungle! Sometimes I think if I had even a smidgen of ambition and social skills I might have made something of myself. :o

DPRK, I think there’s a slight chance that you might have lost the OP somewhere in there.

Fair enough. All one should get out of all that is that yes, one can generalize Euler angles to higher dimensions, and, yes, one can represent planes of any dimension in any-dimensional space using matrices; that last one should anyway be pretty clear as each row of the matrix gives a linear constraint (like Ax + By + Cz + D = 0) and you simply have a list of simultaneous linear equations.

The number of parameters required to specify k-dimensional planes in n-dimensional space then works out to (k+1)(n - k).

Our angular velocity is the same, one rotation per day, everywhere on our planet … it is our linear velocity that is reduced as we move up in latitude (= 1,040 mph x ( 1 - sin(lat) )) …

Also … the equation for a sphere is r = a in spherical co-ordinates where a is constant … cute eh? …

You have it backwards.

The Earth revolves faster when you move to the poles and slower when you move to the equator. Remember the spinning ballerina example?

Took me a moment to figure out what you meant by this. Pretty clever.

I always heard it for figure skaters.

And yeah, moving mass toward the axis of rotation will cause an object to spin faster, but I really don’t think that’s what the OP was going far, and may just be confusing.

Not sure a single person’s mass will increase the rate of rotation all that much … a little sure but nothing that we could measure … if the OP is moving along a line of longitude, then he has a force being applied to him, which changes his momentum … the figure skater analogy only applies when momentum is constant, she can’t be holding rocket motors pointed in inappropriate directions …

The forces acting on a person moving along a line of longitude are all internal to the Earth-person system, and are no different than the forces within the figure skater’s muscles.