any points in determine distances), as well as to the hyperbolic plane. One by-product of this analysis, incidentally, is the observation that the Guth-Katz bound for the Erdos distance problem in the plane, immediately extends with almost no modification to the sphere as well (i.e. This gives a proof of Proposition 1.ĭetails of the correspondence are provided below the fold. The analogue of the lines in this setting become great circles on this sphere applying a projective transformation, one can map to (or more precisely to the projective space ), at whichi point the great circles become lines. This group has a natural interpretation as the unit quaternions, which is isometric to the unit sphere. The latter can then be lifted to a double cover, namely the spin group. The starting point is to view the Euclidean plane as the scaling limit of the sphere (a fact which is familiar to all of us through the geometry of the Earth), which makes the Euclidean group a scaling limit of the rotation group. In this post I would like to record some observations arising from discussions with Jordan Ellenberg, Jozsef Solymosi, and Josh Zahl which give a more conceptual (but less elementary) derivation of the above proposition that avoids the use of ad hoc coordinate transformations such as. Certainly it was clear from general algebraic geometry considerations that some bounded-degree algebraic description was available, but why would the be expressible as lines and not as, say, quadratic or cubic curves? The claim follows.Īs seen from the proof, this proposition is an easy (though ad hoc) application of elementary trigonometry, but it was still puzzling to me why such a simple parameterisation of the incidence structure of was possible. ) Elementary trigonometry then reveals that if maps to, then lies on the perpendicular bisector of, and depends in a linear fashion on (for fixed ). (Note that such rotations also form a Zariski-dense subset of. Identify a rotation in by an angle with around a point with the element in. Proof: A rigid motion is either a translation or a rotation, with the latter forming a Zariski-dense subset of. Grade 12 Geometry Lesson 1 Euclidean Geometry: Introduction (Triangles and parallel lines) Mlungisi Nkosi 113K subscribers Subscribe 109K views 11 months ago Maths Grade 12 Euclidean. Proposition 1 One can identify a (Zariski-)dense portion of with, in such a way that for any two points in the plane, the set of rigid motions mapping to forms a line in. One of the tools used in the proof (building upon the earlier work of Elekes and Sharir) was the observation that the incidence geometry of the Euclidean group of rigid motions of the plane was almost identical to that of lines in the Euclidean space : Parallel Postulate: f a straight line intersects two other straight lines, and so makes the two interior angles on one side of it together less than two right angles, then the other straight lines will meet at a point if extended far enough on the side on which the angles are less than two right angles.In a previous blog post, I discussed the recent result of Guth and Katz obtaining a near-optimal bound on the Erdos distance problem.A circle may be described with any given point as its center and any distance as its radius.A straight line may be extended to any finite length.A straight line segment may be drawn from any given point to any other.Things which are halves of the same things are equal to one anothe.Things which are double of the same things are equal to one another.Things which coincide with one another are equal to one another.If equals are subtracted from equals, the remainders are equal.If equals are added to equals, the wholes are equal.Things which are equal to the same thing are equal to one another.The shortest distance between two points is a straight line.The interior angles of a triangle sum up to 180°.A solid has ashape, size, andposition, and can be moved from one place to another.The study of plane geometry and solid geometry.
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