Lecture 2
... Proof. Suppose that f = 0 defines a circle. Then f (x, y) has the form (x − a)2 + (y − b)2 = r2 . Thus F is equal to (X − aZ)2 + (Y − bZ)2 = r2 Z 2 . Set Z = 0. Then X 2 + Y 2 = 0, which has the solution [1 : ±i : 0]. Conversely suppose that F = 0 contains the points [1 : ±i : 0]. Then F (X, Y, 0) = ...
... Proof. Suppose that f = 0 defines a circle. Then f (x, y) has the form (x − a)2 + (y − b)2 = r2 . Thus F is equal to (X − aZ)2 + (Y − bZ)2 = r2 Z 2 . Set Z = 0. Then X 2 + Y 2 = 0, which has the solution [1 : ±i : 0]. Conversely suppose that F = 0 contains the points [1 : ±i : 0]. Then F (X, Y, 0) = ...
CROSSING NUMBERS AND DISTINCT DISTANCES The
... given by N points is ≥ cN(log N)−1 . However, this approach does not bound the number of distances from a single point. It looks completely plausible that for any N points in the plane, one of the points determines ≥ cN(log N)−1 (or even ≥ cN(log N)−1/2 )) distances with the other points. This would ...
... given by N points is ≥ cN(log N)−1 . However, this approach does not bound the number of distances from a single point. It looks completely plausible that for any N points in the plane, one of the points determines ≥ cN(log N)−1 (or even ≥ cN(log N)−1/2 )) distances with the other points. This would ...
Internal geometry of surfaces
... S = the area of the entire sphere SABC = the area of the triangle ABC ...
... S = the area of the entire sphere SABC = the area of the triangle ABC ...
4.11 Curriculum Framework
... The study of geometric figures must be active, using visual images and concrete materials (tools such as graph paper, pattern blocks, geoboards, geometric solids, and computer software tools). ...
... The study of geometric figures must be active, using visual images and concrete materials (tools such as graph paper, pattern blocks, geoboards, geometric solids, and computer software tools). ...
