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CHAPTER FOUR: QUADRIC SURFACES
- Hong Kong University Press, HKU
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CHAPTER FOUR QUADRIC SURFACES On the plane a linear equation in two variables defines a line. In space a linear equation in three variables defines a plane. A line in space is the intersection of two planes; it is therefore defined by two linear equations in three variables. A quadratic equation in two variables defines a quadratic curve on the plane. Quadratic curves on the plane are called conics because they are plane sections of a circular cone in space. In Chapter Three we have seen that there are only three types of conics. A quadratic equation in three variables defines a surface in spaces. These surfaces are called quadric surfaces. While there are only three types of conics, we shall see in this chapter that there are numerous, in fact no less than nine, types of quadric surfaces. Plane sections of a quadric surface are defined by a quadratic equation in three variables and a linear equation in three variables. Eliminating one variable, we see that they are defined by a quadratic equation in two variables and a linear equation in three variables. Therefore plane sections of quadric surfaces are conics. This consideration provides us with a convenient way to study the shape of a quadric surface. The various types of quadric surfaces together with their equations and plane sections are individually studied in the following sections. 4.1 Surfaces of revolution We have seen that a circular cylinder is generated by rotating a straight line about an axis parallel to it. If the axis of revolution is chosen to be the z-axis then is an equation of the circular cylinder. A circular cone is generated 167 Vectors, Matrices and Geometry by rotating a straight line about an axis that intersects it. If the axis is chosen to be the z-axis and the vertex to be the origin, then is an equation of the cone. In general the surface obtained by rotating a curve about an axis is called a surface of revolution. The circular cone and the circular cylinder are the simplest examples of such surfaces. In this section we only consider surfaces of revolution generated by rotating conics about their axes of symmetry. Together with the circular cones and the circular cylinder, they are known as quadric surfaces of revolution. Since the axis of rotation is an axis of symmetry of the generating curve, every such surface is symmetric with respect to a plane that passes through the axis of revolution. Take a circle and rotate it about its diameter to obtain a sphere. If the centre of the circle is at the origin, then an equation of the sphere is x 2+ y2 + z2 =r2 . Every plane section of a sphere is either a point or a circle. Next we consider an ellipse being rotated about one of its axes. The resulting quadric surface of revolution is called an ellipsoid of revolution. As there are two axes of symmetry to an ellipse there are two different kinds of ellipsoids of revolution. We consider first the ellipsoid obtained by rotating an ellipse r about its major axis. To obtain an equation of the ellipsoid we place the z-axis on the major axis and the y-axis on the minor axis of the given ellipse r. Then the ellipse r on the yz-plane is defined by y2 z2 b'2+ a2 =1 (a>b). Consider an arbitrary rotation p about the z-axis. Under p the yzplane is taken to a plane E which contains the z-axis, and the ellipse r to an ellipse per) on E. Points on per) are therefore points of the ellipsoid of revolution under consideration, and conversely every point of the ellipsoid is on one such rotating ellipse per) for some p. On the rotating plane E we choose a pair of auxilary coordinate axes; one of them is taken to be the original z-axis and the other one 168 [3.14.142.115] Project MUSE (2024-04-17 21:52 GMT) Quadric Surfaces labelled as the r-axis to be the image of the y-axis under p (see Figure 4-1). Then on the rz-plane E, the rotating ellipse per) is defined by (1) Now a point X = (x, y, z) on E will have coordinates (r, z) on the rz-plane E where (2) If furthermore X lies on per), then its coordinates (r, z) also satisfy equation (1). Hence the coordinates of...