Review : Generating Solids   3 Nickels awarded! Tagged with:  academic    posted on 15 Jun 2007 12:37:18 IST
Cylinders Given a line L and a curve C in a plane P, the cylinder with generator L and directrix C is the surface obtained by moving L parallel to itself, so that a point of L is always on C. If L is parallel to the z-axis, the surface's implicit equation does not involve the variable z. Conversely, any implicit equation that does not involve one of the variables (or that can be brought to that form by a change of coordinates) represents a cylinder. If C is a simple closed curve, we also apply the word cylinder to the solid enclosed by the surface generated in this way (Figure 1, left).    Figure 1: Left: an oblique cylinder with generator L and directrix C. Right: a right circular cylinder. The volume contained between P and a plane P' parallel to P is     where A is the area in the plane P enclosed by C, h is the distance between P and P' (measured perpendicularly), l is the length of the segment of L contained between P and P', and is the angle that L makes with P. When =90° we have a right cylinder, and h=l. For a right cylinder, the lateral area between P and P' is hs, where s is the length (circumference) of C. The most important particular case is the right circular cylinder (often simply called a cylinder). If r is the radius of the base and h is the altitude (Figure 1, right), the lateral area is 2rh, the total area is 2r(r+h), and the volume is rh. The implicit equation of this surface can be written x+y=r;     Cones Given a curve C in a plane P and a point O not in P, the cone with vertex O and directrix C is the surface obtained as the union of all lines that join O with points of C. If O is the origin and the surface is given implicity by an algebraic equation, that equation is homogeneous (all terms have the same total degree in the variables). Conversely, any homogeneous implicit equation (or one that can be made homogeneous by a change of coordinates) represents a cone. If C is a simple closed curve, we also apply the word cone to the solid enclosed by the surface generated in this way (Figure 1, top).    Figure 1: Top: a cone with vertex O and directrix C. Bottom left: a right circular cone. Bottom right: A frustum of the latter. The volume contained between P and the vertex O is     where A is the area in the plane P enclosed by C and h is the from O and P (measured perpendicularly). The solid contained between P and a plane P' parallel to P (on the same side of the vertex) is called a frustum. Its volume is     where A and A' are the areas enclosed by the sections of the cone by P and P' (often called the bases of the frustum).       Surfaces of Revolution. The Torus   A surface of revolution is formed by the rotation of a planar curve C about an axis in the plane of the curve and not cutting the curve. The Pappus--Guldinus theorem says that: The area of the surface of revolution on a curve C is equal to the product of the length of C and the length of the path traced by the centroid of C (which is 2 the distance from this centroid to the axis of revolution). The volume bounded by the surface of revolution on a simple closed curve C is equal to the product of the area bounded by C and the length of the path traced by the centroid of the area bounded by C. When C is a circle, the surface obtained is a circular torus or torus of revolution (Figure 1). Let r be the radius of the revolving circle and let R be the distance from its center to the axis of rotation. The area of the torus is 4Rr, and its volume is 2Rr.                                              Quadrics     The five nondegenerate real quadrics    Figure 1: The ellipsoid   Figure 2: Left: hyperboloid of one sheet (3). Right: hyperboloid of two sheets (4).    Figure 3: Left: elliptic paraboloid Right: hyperbolic paraboloid Surfaces with equations are cylinders over the planes curves of the same equation .A surface with equation can be regarded as a cone over a conic C (any ellipse, parabola or hyperbola can be taken as the directrix; there is a two-parameter family of essentially distinct cones over it, determined by the position of the vertex with respect to C). The real nondegenerate quadrics (1), (3), (4), (7), and (8) are shown in Figures 1--3. The surfaces with equations (1) --(6) are central quadrics; in the form given, the center is at the origin. The quantities a, b, c are the semiaxes. The volume of the ellipsoid with semiaxes a, b, c is . When two of the semiaxes are the same, we can also write the area of the ellipsoid in closed form. Suppose b=c, so the ellipsoid x/a+(y+z)/b=1 is the surface of revolution obtained by rotating the ellipse x/a+y/b=1 around the x-axis. Its area is The two quantities are equal, but only one avoids complex numbers, depending on whether a>b or a<b. When a>b, we have a prolate spheroid, that is, an ellipse rotated around its major axis; when a<b we have an oblate spheroid, which is an ellipse rotated around its minor axis. Given a general quadratic equation in three variables, ax+by+cz+2fyz+2gzx+2hxy+2px+2qy+2rz+d=0,	About the Author: pirate1_from_jee (596) Scorching goIITian  108  [136 rates] total posts: 211     Offline
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