Part 1: Coordinate Surfaces

Surfaces are the analog in space of curves in the plane. When you think of "surface," several familiar objects may come to mind. Spheres, cylinders, and planes are all surfaces, but is a cone -- with its sharp point -- a surface? We will consider several classes of surfaces in this module and try to give at least an intuitive answer to the question "What is a surface?"

In the xy-plane, a single equation in either or both variables usually represents a curve of some sort -- a one-dimensional object. In Cartesian coordinates, if we set one of the coordinates equal to a constant, the resulting "curves" are horizontal and vertical lines. In polar coordinates, r = constant represents a circle, and  = constant represents a ray emanating from the origin.

In space, the simple equations obtained by setting a coordinate equal to a constant represent surfaces. In Cartesian coordinates, setting any of the three coordinates equal to a constant c defines a plane in space.

1. In your worksheet, plot each of the equations x = 5, y = 5, and z = 5.
2. Describe in words the planes represented by each of the equations x = c, y = c, and z = c, where c is any constant.

In cylindrical coordinates, the surface r = c is a cylinder (left-hand figure below), and  = c defines a plane through the vertical axis (right-hand figure -- negative values of r give the part of the plane on the other side of the z-axis).

3. In your worksheet, plot each of the equations r = 3 and  = 2.
4. What is the coordinate surface z = 1? Plot it in your worksheet to confirm.
5. Write a Cartesian equation of the cylindrical surface of radius c in the left-hand figure above. [Hint: Think about the distance of any point (x,y,z) on the cylinder from the z-axis.]
6. Show that your equation in step 5 is equivalent to r = c in cylindrical coordinates.
   

In spherical coordinates, we get spheres by setting  = c (left figure below). Setting  = c is the same as for cylindrical coordinates, except never takes negative values. Finally, setting  = c defines a cone at the origin as in the right figure below.

7. In your worksheet, plot the coordinate surfaces  = 4,  = 1, and  = 1 in spherical coordinates.
8. The equation in Cartesian coordinates of the sphere of radius c is x² + y² + z² = c². Explain how this equation becomes  = c in spherical coordinates.

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Copyright CCP and the author(s), 2001-2002