Post and Thm Notes
... * A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane. * Through any two points there is exactly one line. * Through any three points there is at least one plane, and through any three non-collinear ...
... * A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane. * Through any two points there is exactly one line. * Through any three points there is at least one plane, and through any three non-collinear ...
Points, Lines, and Planes Notes
... Essential Question: How can you identify points, lines, and planes? Warm Up: ...
... Essential Question: How can you identify points, lines, and planes? Warm Up: ...
Algebraic Geometry I
... 4. Assume char k 6= 2, 3. Prove that a smooth plane cubic C has nine distinct inflection points. Equivalently prove that mp (C, Hess(C)) = 1 for every inflection point p of C. (Hint: you may assume that the inflection point p is [0, 0, 1] and the tangent line to C at p is y = 0. Then prove that if f ...
... 4. Assume char k 6= 2, 3. Prove that a smooth plane cubic C has nine distinct inflection points. Equivalently prove that mp (C, Hess(C)) = 1 for every inflection point p of C. (Hint: you may assume that the inflection point p is [0, 0, 1] and the tangent line to C at p is y = 0. Then prove that if f ...
Conic section
In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, and as a quadric of dimension 1. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a non-circular conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity.Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well.The conic sections have been named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties.