As a schoolboy I used pencils for drawing, mainly tanks and cars. Then I learned a little bit of perspective and tried to draw Greek temples and roads converting to a point on the horizon.
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| Fascinated by pencils |
It was just few years ago that I have learned about the concept pencils in geometry. I remember it was a paper by a Czech mathematician Metod Saniga, "Pencils of Conics] a Means Towards a Deeper Understanding of the Arrow of Time":
Abstract: The paper aims at giving a sufficiently complex description of the theory of pencil!generated temporal dimensions in a projective plane over reals[ The exposition starts with a succinct outline of the mathematical formalism and goes on with introducing the definitions of pencil-time and pencil-space, both at the abstract (projective) and concrete (affine) levels. The structural properties of all possible types of temporal arrows are analyzed and based on symmetry principles, the uniqueness of that mimicking best the reality is justified. A profound connection between the character of different "ordinary" arrows and the number of spatial dimensions is revealed.
The subject interested me, but the math was too difficult at that time. Now it is time to return to this subject.
Definition 16.1. Two different oriented nonpoint spheres in S3 are in oriented contact if they have a common point x ∈ S3 , they are tangent to each other at x, and if they also have the same orientation defining unit normal vector n at x.
Definition 16.2. Given a point x ∈ S3 , a pencil of oriented spheres at x is the set of all oriented spheres in pairwise oriented contact at x.
Let us now examine these concepts. For r ∈ [0, 2π) and m a unit vector in R4 , the equation of an oriented sphere Sr(m) is
m · x = cos(r), x · x = 1, (16.1)
while the normal unit vector n at x is (for a non-point sphere)
n(x) = (m − cos(r)x) / sin(r). (16.2)
Suppose we have two spheres Sr(m) and Sr'(m') in contact at some point x0 . Let n0 denote the common unit normal vector at the common point x0 . Thus
n0 = (m − cos(r)x0)/sin(r) = (m' − cos(r')x0)/sin(r'). (16.3)
It follows that
m = sin(r)n0 + cos(r)x0 , (16.4)
m'= sin(r')n0 + cos(r')x0 . (16.5)
For a nonpoint sphere r≠ 0 and r≠ π, so that | cos(r)| < 1. Therefore, given x ∈ Sr(m), the unit vectors m and x in (16.1) are linearly independent – they span a two-dimensional subspace of R4 . Any vector tangent to the sphere at x is orthogonal to these two vectors. So the tangent space to the sphere at x is the orthogonal complement of m and x. But from (16.2) it follows that the subspace spanned by m and x is the same as the subspace spanned by mutually orthogonal unit vectors n and x. Therefore the spheres Sr(m) and Sr'(m'), having the same x0 and n0 , are automatically tangent to each other at x0 . The requirement of them being tangent to each other in Definition 16.1 is therefore redundant.
Substituting (16.4) into (16.1) we see that the spheres of the pencil defined by x0 and n0 are intersections of planes with the sphere S3
(sin(r)n0 + cos(r)x0 ) · x = cos(r), x · x = 1. (16.6)
As an illustration Fig. 16.1 shows the several circles from the pencil of oriented circles for x0 = e2 and n0 = e3 . In Fig. 16.1 we have used only the values of r in (0, π].
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| Figure 16.1: Pencil of oriented circles for x 0 = e 2 and n 0 = e 3 , r = 2kπ/10, k = 1, . . . , 10. |
This is because the spheres Sr(m) and Sr+π mod 2π (−m) are the same. We added the point sphere r = π, which reduces to the point x0 , because it naturally ‘wants’ to be included.
In the next chapter we will study the representation of pencils of
oriented spheres in Q - that is the space to which, as we will see, they
naturally belong, and the acquire a definite geometrical and physical
meaning.
Ark, sometimes I get the impression that you read my mind. A few days ago I was thinking about the fact that stereographic projections of plane sections of a sphere are arcs of a circle representing the projected sphere. This circle is exactly the Poincaré model of Lobachevsky space. If the section plane passes through the origin, then the arc approaches the circle at an angle of pi/2 and is a "straight line" in Lobachevsky space.
ReplyDeleteBut I still don't understand what this construction has to do with a pencil :)
Sr'(m) in contact ->
ReplyDeleteSr'(m') in contact
In (16.4) x should be bold
In (16.5) r should be primed not zeroed.
"Any tangent vector to the sphere is orthogonal to these two vectors."
ReplyDeleteReally ?
if they have a common point ->
ReplyDeleteif they have exactly one common point
Therefore the unit vectors m and x in (16.1) ->
ReplyDeleteIn (16.1) x is a pencil.
Fixed. I think. Thanks:
ReplyDeleteThis "In (16.1) x is a pencil." I do not understand.
You did not fix:
DeleteSr'(m) in contact ->
Sr'(m') in contact
In (16.1) x is not concrete - there are lot of x-es, but you wrote that m and x "span a two-dimensional subspace"
"Sr'(m) in contact" is under (16.2)
DeleteShould be better now. Thank you!
ReplyDeleteSorry, I don't understand in the newly appeared text,
ReplyDelete"| cos(r)| < 1. Therefore, given x ∈ Sr(m), the unit vectors m and x in (16.1) are linearly independent"
For vectors m and x to be independent, their product m · x = cos(r), should be zero, right?
But cos(r) is not zero, it is only not equal to 1 or -1.
What did i miss here?
For two vectors to be orthogonal their scalar product must be zero. Orthogonal vectors are automatically linearly independent, but there are vectors that are linearly independent but not necessarily orthogonal. Then there is an orthogonalization procedure. Two vectors are linearly dependent only if one is proportional to the other. Otherwise they are linearly independent. Since both are unit vectors it means that one must be plus or minus the other. But we have excluded this case by considering only nonpoint spheres.
DeleteIs it better now?
Yes, thanks a lot! I am so inattentive, have missed that vectors m and x are UNIT.
DeleteArk, I am still have a couple of questions about pencils:
ReplyDelete(1) "Definition 16.2. Given a point x ∈ S3 , a pencil of oriented spheres at x is the set of all oriented spheres in oriented contact at x".
Trying to visualize this bundle of spheres somehow, I thought that probably this definition should say that pencil is a set of oriented spheres which are in contact PAIRWISE? In other words, can spheres with different normal vectors belong to one and the same pencil?
(2) "Figure 16.1: Pencil of oriented circles for x 0 = e 2 and n 0 = e 3 , r = 2kπ/10, k = 1, . ."
It seems to me that the circles in this pencil do not touch each other. Is it right that they are not in contact? So, it is not a bouquet of circles. And what about the spheres? They must be in contact by definition!
Yes, I will add "pairwise".
DeleteI have replaced Fig. 1 showing the pencil from a different angle.
Now Fig. 1 is very illustrative, thanks a lot. When I was looking at the previous version, it seemed that the intersecting planes do not pass through the x0 point and I tried to imagine a more intricate construction than it actually is.
Delete