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Post by mondo on Oct 21, 2022 7:02:52 GMT
I don't understand why in the second paragraph on page 208 it is said that the effect of a uniform linear transformation on a circle $C$ will create an image $E$ which is a "Conic section", which must be an ellipse. Along the lines they also say about the change of coordinates, all that sounds a bit cryptic to me. What kind of transformation is it here that a circle was mapped to ellipse?
PS: I have one more question to the second to last paragraph on the previous page (p.207) "Suppose we fold it like a letter being placed in the envelope [two creases]. Three different points will end up above a single point on a table.." - why three? If I fold once I have two points, I fold one more time I have four etc, so why 3?
Thank you.
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Post by Admin on Oct 22, 2022 12:26:02 GMT
Mondo
The transformation is a uniform linear transformation as stated on page 208 in the second paragraph of subsection 2 of section VIII. The transformation will induce a change of coordinates and the image quadratic will be another quadratic which will be an ellipse. This is all explained in the book on page 208.
You're not thinking about it correctly. Think of A4 paper and horizontally long envelopes about a third of the size of A4. Fold the bottom third of the paper towards the top and then the top third of the paper towards the bottom. Now do you see it? Think!
Vasco
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Post by mondo on Oct 22, 2022 19:23:52 GMT
Mondo The transformation is a uniform linear transformation as stated on page 208 in the second paragraph of subsection 2 of section VIII. The transformation will induce a change of coordinates and the image quadratic will be another quadratic which will be an ellipse. This is all explained in the book on page 208. But they never give any formula for this transformation. How do we know that circle will be mapped to ellipse? How are the coordinates changed, according to what? Maybe it is not important here and this "uniform linear transformation" is just some transformation $X$, we don't care how it is performed we just want to analyze its effect? You're not thinking about it correctly. Think of A4 paper and horizontally long envelopes about a third of the size of A4. Fold the bottom third of the paper towards the top and then the top third of the paper towards the bottom. Now do you see it? Think! Vasco Ok, I see it now - I think this image depicts that -> images.squarespace-cdn.com/content/v1/55ac3d89e4b07d0961f93c14/1517079421941-MO54PAUJYQBYDB99LAU0/Crease_mountain_valley_assignment_DSC_0542.JPG?format=1000wThanks. |
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