Introduction

Introduction to Conic Sections

Conic Sections        A conic section or more simply, conic, is the intersection of a plane and a right circular conical surface. This intersection is a curve (or curves) that can be specified as a quadratic equation in terms of x and y.        This surface is generated by rotating a line in the Y-Z plane (see next figure) about the Z axis with the vertex at the origin. The Z axis the axis of rotation and is the axis of the cones. Each cone is referred to as a nappe (pronounced: năp).   There are 7 possible ways to pass a plane through this conical surface: The plane passes through the shared vertex of each nappe, that is the vertex of the cone.      The plane lies along the sides of the nappes and passes through the vertex of the cone.   The plane contains the axis of the conical surface.     The plane passes through the conical surface normal to it axis. Normal means that the axis is at right angles no matter how the angle from the axis to the plane is measured. The green and blue angle measurements illustrate this requirement. (These three lines represent 3D axes for the 3 dimensional coordinate system we use where all three lines must be at right angles with each other.)   Imagine tilting the plane a bit. When we do this we stretch the circle a bit from one end to the next, that is we elongate it. The resulting intersection is an ellipse.   Let's tilt the plane further until it becomes parallel to the edge of a nappe. The intersection created is the parabola.   Let's tilt the plane further. When we do this the plane will pass through both nappes. We get two intersections here, one in each nappe, that defines the hyperbola.     Before proceeding to studying each conic, we should appreciate the history behind them.   Euclid cataloged the conics, his work became lost over time. Apollonius of Perga, Greece, 3rd century wrote eight books on the conics. Not having algebra he had to argue his points using figures, i.e., diagrams. Kepler used Pollonius' ideas to model planetary motion with the ellipse, and Newton took this model further with his theory of gravitation.   Now, let's talk about the names. circle: the plane is perpendicular to the cone's axis. ellipse (ĭ - lĭps '): (Greek: to leave out) the plane tilts further, leaving out one nappe. parabola (pə - răb' - ə - lə): (Greek: beside, compare) the plane is parallel to an edge of the cone. Only one nappe can be cut in this case. hyperbola (hī - pur ' bə - lə): (Greek: hyperbole: exceeds ) the plane exceeds parallel. Both nappes must be cut.     The right circular cone for review: This is another FREE ALGEBRA PRINTABLE presented to you from the Algebra section of K12math.com

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