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Unformatted text preview: 3-4 Convex Sets and Separation
A) Convex a. A set M is called convex if for every two points P and Q of the set, the entire segment PQ lies in M i. b. Is a plane convex? c. What if we cut the plane with a line? B) Postulate 9 The Plane Separation Postulate a. Given a line and a plane containing it. The points of the plane that do not lie on the line form two sets such that i. Each of the sets is convex, and ii. If P is in one of the sets and Q is in the other, then the segment PQ intersects the line. C) 3 definitions (Half-Planes/edge/opposite sides of L) a. Given a line L and a plane E containing it, the two sets described in the Plane Separation Postulate are called half-planes or sides of L, and L is called the edge of each of them. If P lies in one of the halfplanes and Q lies in the other, then we say that P and Q lie on opposite sides of L. D) Postulate 10 The Space Separation Postulate a. The points of space that do not lie in a given plane form two sets, such that i. Each of the sets is convex, and ii. If P in one of the sets and Q is in the other, then the segment PQ intersects the plane E) 2 definitions half-spaces/face a. The two sets described in the Space Separation Postulate are called half-spaces, and the given plane is called the face of each of them b. Note: while every line in space is the edge of infinitely many halfplanes, every plane in space is a face of only two half-spaces Homework number 9 ...
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- Spring '09
- Algebra, Euclidean geometry, Plane Separation Postulate