tag:blogger.com,1999:blog-1840985738235902482.post1080169770069365712..comments2024-03-27T07:14:48.488-04:00Comments on John the Math Guy: Statistics of multi-dimensional data, theoryJohn Seymourhttp://www.blogger.com/profile/11350487038873935295noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-1840985738235902482.post-52370001271826605192018-01-03T11:09:53.917-05:002018-01-03T11:09:53.917-05:00Thanks for the comment, cate. I have gone back and...Thanks for the comment, cate. I have gone back and forth on this question several times. Right now, I think I disagree with you. :)<br /><br />Let's say we start with a unit sphere, with centroid at the origin. How do I transform this into an arbitrary ellipsoid?<br /><br />First, we can scale the x, the y, and the z axes to account for the length of the three axes of the ellipsoid. Three parameters used up. (Note: I was confused by this for a while. I got hung up thinking -- erroneously -- that ellipsoids were either prolate or oblate. I still have trouble picturing an ellipsoid that is neither.)<br /><br />Second, we orient the major axis. We can do this with two angles, as you have said. I like to think of the two angles as latitude and longitude. Two more parameters, bringing this up to five.<br /><br />Third, the medial and minor axes lie in a plane which is perpendicular to the direction that the major axis points, but we need to account for rotation of the ellipsoid around that major axis. This defines the direction that the medial axis points within the plane. This is one more parameter, bringing us up to six. (Note: Another spot of confusion for me. initially, I missed this one.)<br /><br />Fourth, we need an offset in x, in y, and in z to center the ellipsoid arbitrarily. Three more parameters, bringing us up to nine. <br /><br />Is there a flaw in what I am thinking?John Seymourhttps://www.blogger.com/profile/11350487038873935295noreply@blogger.comtag:blogger.com,1999:blog-1840985738235902482.post-33783123795933428072018-01-02T17:28:10.992-05:002018-01-02T17:28:10.992-05:00To characterize an ellipsoid, only 8 numbers are n...To characterize an ellipsoid, only 8 numbers are needed. Orientation requires just 2 numbers (e.g. two angles, or a vector with module one). Or just the coordinates of center and of one pole (3+3), but then one semi-axis can be calculated, so we need just the two additional distances.catehttps://www.blogger.com/profile/15700217782948121553noreply@blogger.com