Surfaces, Graphs, Level Curves, and Level Surfaces
Some simple surfaces in 3D space
Besides planes, there are some simple surfaces in that can be described by simple equations.
Cylinders
Suppose is a closed curve on a plane. A cylinder can be generated by projecting along the axis perpendicular to the plane. For example, if the closed curve is a unit circle on the xy-plane. Then the equation
describe the cylinder of (a circular cylinder) parallel to z-axis because for any cross-section at , is on the cylinder as long as i.e. and are independent of . Hence does not appear in the equation.
In the applet below, draw the cylinder of the circle on the yz-plane centered at with radius parallel to x-axis.
Spheres
The equation for the sphere centered at with radius is clearly as follows:
The above equation describes the fact that the distance between any point and equals .
If we compress/stretch a sphere along the three axes by some factors, we will get a more general kind of surfaces called ellipsoid. For example, you can draw the ellipsoid in the applet below.
Cones
Consider the right circular cone with apex at the origin such that it is open upwards and its semi-vertical angle is . The equation of the cone is as follows:
If we want to include the cone below the xy-plane, we can use the following equation:
More generally, if the semi-vertical angle is , the equation of the cone becomes
where .
You can draw a cone in the applet below.
Multivariable functions
In this chapter, we mainly study multivariable functions i.e. real-valued functions of more than one variable. Suppose is a real-valued function of two variables. We can express this function as follows:
where is any point in the domain of , a subset of containing all the possible inputs of the function. Then a unique real number is assigned by the function for each . The range of is the set of all possible outputs of the function i.e. the set of all possible values of .
Example: Suppose . It domain is and its range is .
Similarly, we can also consider a function of three variables , or in general, a function of variables .
As we know, we can visualize a function of one variable as a graph in i.e. it is a set of all points of the form . Similarly, we can visualize a function of two variable as a graph in i.e. it is a set of all points of the form .
In the applet below, you can draw the graph of and then find the point on the graph corresponding to the point .
You can try the following functions:
Level curves
Consider the graph of . The contour curve is the intersection of the graph and the horizontal plane for some real number . The level curve is the projection of the contour curve onto xy-plane. It can be represented by the equation .
In the applet below, the level curve is shown in the left panel and the corresponding contour curve is shown in the right panel. You can view the level curves for the following functions:
Level surfaces
For functions of three variables , it is not easy to visualize its graph since it is a three-dimensional object in four-dimensional space. However, we can visualize the intersection between the graph and as it is a two-dimensional object i.e. a surface. It is called the level surface of . Its equation is .
Example: Let . For , its level surface has the equation i.e. a sphere centered at the origin with radius .
In the applet below, you can visualize the level surface for various functions.
(Note: GeoGebra can only handle the plotting of level surface for quadratic polynomials in three variables.)