Surfaces

Let C be a curve, then we define a cylinder to be the set of all lines through C and perpendicular to the plane that C lies in.

We can tell that an equation is a cylinder is it is missing one of the variables.

Recall that the quadrics or conics are lines , hyperbolas, parabolas, circles, and ellipses.  In three dimensions, we can combine any two of these and make a quadric surface.  For example

is a paraboloid since for constant z we get a circle and for constant x or y we get a parabola.  We use the suffix -oid to mean ellipse or circle.  We have:

•        x2          y2          z2
+           +            =  1 is an ellipsoid
a2          b2          c2

•           x2         y2         z2
-          -           +            =  1 is a hyperboloid of 2 sheets while
a2         b2          c2

•        x2          y2          z2
+           -            =  1 is a hyperboloid of 1 sheet
a2          b2          c2

Surface of Revolution

Let y = f(x) be a curve, then the equation of the surface of revolution abut the x-axis is

y2 + z2 = f(x)2

Example

Find the equation of the surface that is formed when the curve

y   =   sin x          0  <  x  < p/2

is revolved around the y-axis.

Solution

This uses a different formula since this time the curve is revolved around the y-axis.  The circular cross section has radius sin-1 y and the circle is perpendicular to the y-axis.  Hence the equation is

x2 + z2 = (sin-1 y)2

Cylindrical Coordinates

We can extend polar coordinates to three dimensions by

 x = rcosq      y = rsinq      z = z

Example

We can write (1,1,3) in cylindrical coordinates.  We find

and

so that the cylindrical coordinates are

(, p/4, 3)

Spherical Coordinates

An alternate coordinate system works on a distance and two angle method called spherical coordinates.  We let r denote the distance from the point to the origin, q represent the same q as in cylindrical coordinates, and f denote the angle from the positive z-axis to the point.  The picture tells us that

r  =  r sin f

and that

z  =  r cos f

From this we can find

 x = rcosq = r sin f cosq      y = r sinq = r sin f sinq      z = r cos f

Immediately we see that

x2 + y2 + z2 = r2

We use spherical coordinates whenever the problem involves a distance from a source.

Example

convert the surface

z  =  x2 + y2

to an equation in spherical coordinates.

Solution

We add z2 to both sides

z + z2  =  x2 + y2 + z2

Now it is easier to convert

r cos f + r2 cos2 f  =  r2

Divide by r to get

cos f + r cos2 f  =  r

Now solve for r.

cos f                    cos f
r =                            =                     =  csc f  cot f
1 - cos2 f                sin2 f