some ideas

[roseman at dimension4.math.uiowa.edu (Dennis M. Roseman)]

Daeron:

Here are some various things I have been thinking about in the last  
week.

On Friday on the drive back to Iowa, I began to think about some  
things.  


One of the things I thought about was several things that I had meant  
to look at and never did.

One thing was the spun trefoil this is (I think) trefoil.off

How does this look with matrices:

0	0	0	1
0	1	0	0
0	0	1	0
1	0	0	0

and

1	0	0	0
0	0	0	1
0	0	1	0
0	1	0	0

1) Do these look different from each other ?.  I know we used one of  
these but I don't think we did both.  I think the one we didn't do is  
the one which in the controller, we see the red line segment as a  
point and the green and blue segments touch the "visual edge" of the  
sphere.

2) Also, I never did see the result of slicing in a hyperplane other  
than  w = some_constant.  

 


3)I made some new data sets--these are both actually unknotted  
2-spheres which have the same default projections as two of the  
examples we looked at.  I ftp'ed these to my GCJune92 directory,   
They are spunUnknot and oneTwistSpunUnknot.  I've worked on some  
other examples but they are not done yet.

4)  Remark: your 4D module with some of the modifications we have  
talked about and perhaps some other future changes is going to be  
something of considerable use, I think.  So I think it could use a  
snappy name.  I have a couple of suggestions.  The first is  
"foursight" since it is a four dimensional viewer.  Far sillier is my  
runner-up "fourgetter".

5)  It would be nice to have more control on the color-gradient.   
Part would be choices of color or coloring scheme.  But also as it is  
now, the color gradient shows "height" along the current projection  
direction.  It would be handy to be able to chose one direction for  
the color gradient and another direction for projection.  For example  
rotate a knot, project it, but the color gradient would be from the  
original default position.

6)  Here is something that I think would not be hard to write but  
would be very useful.  The idea is an intelligent slicer.  This would  
find ALL interesting slices.

	The slicing we did in early June just did slicing evenly  
spaced in the w-coordinate.  

	The idea is to slice at values that are the important ones.   
(To simplify the discussion, I will talk about slicing in the  
direction orthogonal to the w-direction, but more generally we have  
interest in slicing in other directions).
	
	Step 1 is to find the critical points on the surface with  
respect to the w-direction.  Once we find these we will look at the  
w-values of these points and do slicing as follows:  between any two  
consecutive such w-values, say w' and w" we will find values w1  and  
w2  so that w' < w1 < w2 < w" and so that the intervals [w, w1], [w1,  
w2], and [w2,w"] are of equal length.  We do all our slicing at the  
hyperplanes with w-value w1 and w2.  If we do this for all possible  
w'  and w" we find, we will have a VERY interesting and useful  
slicing.
	
	Now I need to tell you what a critical point is.  There are  
five possible kinds of critical points: local maxima, local minima,  
saddle points and degenerate critical points.
	
	So let v be a vertex of our surface.  Consider the edges  
which have V as a vertex, suppose that v is the w-coordinate of V.   
Arrange these in circular order as they appear on the surface.   
(There are two possible directions--it doesn't matter which you  
chose).  Consider the vertices on those edges in our list which are  
opposite to V on that edge.  We get a circular list of numbers: v1,  
v2, ...vk.  These are the w-values of those vertices.  

	
	If all of vi's are less than v, then we are at a local  
maximum.
	
	If all of vi's are greater  than v, then we are at a local  
minimum.
	
	If, in the circular order, the numbers v1-v, v2-v,..., vk-v,  
changes sign twice, then it is NOT a critical point, and we ignore  
it.  (Most points should be like this.)
	
	If, in the circular order, the numbers v1-v, v2-v,..., vk-v,  
changes sign four times, then it is a saddle point.
	
	If either some of the numbers vi-v are zero, or if , in the  
circular order, the numbers v1-v, v2-v,..., vk-v, changes sign six  
times or more, then we have a degenerate critical point.
	
7)  Another idea, I think we talked about this is to have write a  
function that will slice a knot evenly and throw away every other  
slice.  By rotating this about and peeking in the gaps, it might give  
us a quick way to get a good look of a lot of things at once.  If the  
knot is big & fat, we might have to separate the visible slices by  
some factor so that we can peek inside better.

Dennis