Quadric Surfaces
Suppose f(x, y, z) is a polynomial of degree two(quadratic polynomial). The graph of the equation f(x, y, z)=0 is called a Quadric surface. Analysis of the most general quadric surface requires difficult tools from Linear Algebra analogous to the rotation of coordinates in the plane. We shall examine only a few of the most common quadratic polynomials.
We start with equations of the form :
The graph is symetric in each of the three coordinate planes, because if (x, y, z) satisfies the equation, then so do all eight points ( x, y, z). Therefore an accurate graph in just the first octant suffices to determine the whole graph.
Spheres
Consider the graph of the equation:
The equation simply says ||x|| = a, hence the graph consists of all points at distance a from 0.This is a sphere of radius a with center 0.
Ellipsoids
Consider the graph of:
Since squares are non-negative, each point of the graph satisfies:
This means the graph is confined to the box:
Suppose –c<zo<c. The intersection of the graph and the horizontal plane z=zoconsists of all points (x, y, zo) that satisfy:
This curve is an ellipse.It is as large as possible when zo=0, and it becomes smaller and smaller as zo c or zo -c. Thus each such cross-section by a horizontal plane is an ellipse, except at the extremes zo= c, where it is a single point.
The same argument applies to plane sections parallel to the other coordinate planes. This gives us enought information for a sketch. The surface is called an ellipsoid (Fig. 1.).In the special case a=b=c, it is a sphere.
Fig 1. Ellipsoids.
Hyperboloids of one sheet
Consider the graph of:
Each horizontal cross-sectoin is an ellipse:
no matter what zois.The ellipse is smallest for zo=0 .However as zo or zo- ,the ellipse gets larger and larger.
The surface meets the y, z-plane in the hyperbola:
and it meets the z,x-plane in the hyperbola:
This information is enough to sketch the surface, called a Hyperboloid of one sheet.(Fig. 2a.)
Fig. 2. hyperboloids.
Hyperboloids of two sheets
Consider the equation:
If (x, y, z) is a point of the surface, then:
Hence z2 ≥ c2. This means either z ≥ c or z ≤ -c, that is, there are no points of the surface between the horizontal plane z = c and z = -c.
If zo 2 >c2, the horizontal plane z = zo meets the surface in the curve
an ellipse. Also the surface meets the y, z-plane and the z,x-plane in the hyperbolas
respectively.The surface breaks into two parts, and it is called a Hyperboloid of two sheets.(Fig 2b.)
Paraboloids
The surface
is called an elliptic paraboloid. Since z ≥ 0,the surface lies above the x,y-plane, and it is symmetric in the y,z- ir z,x-planes. Each horisontal cross-section
is an ellipse and these ellipses grow larger as zo increases. The graph meets y,z- andz,x-planes in parabplas (Fig. 3 a.)
Finally we consider the hyperbolic paraboloid, the locus of:
It is symmetric y,z- and z,x-planes. The horizontal planes z=zo>0 meet it in hyperbolas whose branches open out in the y-direction. The horizontal planes z=zo<0 meet it in hyperbolas that ope out in the x-direction. The y,z-plane meets the locus in the parabola z=y2/b2,which opens upwards, and the z,x-plane meets it in the parabola z=-x2/a2, which opens downwards. The best description is "saddle-shaped".(Fig. 3b.)
Fig. 3. Paraboloids.