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[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index] some ideas
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 setsthese are both actually unknotted 2spheres 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 runnerup "fourgetter". 5) It would be nice to have more control on the colorgradient. 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 wcoordinate. 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 wdirection, 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 wdirection. Once we find these we will look at the wvalues of these points and do slicing as follows: between any two consecutive such wvalues, 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 wvalue 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 wcoordinate of V. Arrange these in circular order as they appear on the surface. (There are two possible directionsit 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 wvalues 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 v1v, v2v,..., vkv, 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 v1v, v2v,..., vkv, changes sign four times, then it is a saddle point. If either some of the numbers viv are zero, or if , in the circular order, the numbers v1v, v2v,..., vkv, 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


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