Previously we discussed various 2-dimensional geometric shapes or conic sections like the parabola, hyperbola, and ellipse. Now let us consider the 3-dimensional cylinder or cylindrical surface. In mathematics, a general cylindrical surface is defined as follows:
Let C be a curve in a plane, and let L be a line that is not in a parallel plane. The set of points on all lines that are parallel to L and intersect C is a cylinder.
Notice that the above cylindrical surfaces can be irregular and open or closed. When we extend the parabola, hyperbola, and ellipse directly into three dimensions, we simply get parabolic, hyperbolic, and elliptic cylinders. These cylindrical surfaces have the same algebraic expression as the corresponding 2-dimensional conic sections.
Now if we extend the equations into 3-dimensional space by algebraically adding 3rd dimension terms, we get the analogous surfaces of the paraboloid, hyperboloid, and the ellipsoid. The elliptic paraboloid below is given by the equation:
If we simply change the sign of one of the terms above we get the hyperbolic paraboloid below given by:
The hyperboloid has two general forms and one special degenerate form. The first form seen below is called the hyperboloid of one sheet. It is given by:
The special degenerate form of the hyperboloid of one sheet given by:
Is the double-napped cone.
The second form of the hyperboloid is called the hyperboloid of two sheets and is given by:
Finally we have the ellipsoid which is given by:
A special case of the ellipsoid where a=b=c is the sphere.
Click here to see a plethora of different and interesting algebraic surfaces.
This is the second post of this series on conic sections. This time we will look at the same double-napped right circular conical surface but the intersecting plane will travel completely through one half of the surface and not through the base as seen below.
From a practical standpoint, various celestial bodies are known to travel in elliptical orbits. This includes our own planet with the sun at one of the foci. Also the reflective property of an ellipse where any light or sound emitted from one focus is reflected to the other focus is utilized in optics design. Also this phenomenon can be experienced in what are called “whispering galleries”. These are buildings with elliptical geometry such that a person at one focus can hear very easily someone speaking at the other focus due to the reflective property. One example of this that I have experienced myself is in the rotunda of the US Capital building.
In order to derive an algebraic expression for the ellipse we begin with the definition that an ellipse is the set of points in the plane such that the sum of the distances from the point to each of the two foci is constant for every point. A good way to visualize this is to imagine you have a piece of string, two pushpins, a piece of paper or cardboard, and a pencil. If you use the two pushpins to pin down each end of the string such that the distance between the two pushpins is less than the length of the string. Then use the pencil to pull the string taut in every direction and make a mark with the pencil along this perimeter as seen below you will have constructed an ellipse.
This way of looking at the ellipse probably makes it more intuitively clear what is meant by the above definition. You can see that every point on the ellipse has the same distance from one pin to the pencil and back to the other pin. This is the constant length of the string.
We begin with an ellipse centered on the origin and the two foci on the x-axis. One focus at (c,0) and the other at (-c,0). Then the distance from a point (x,y) in the plane to each focus is given by:
Now by the above definition we know that the sum of these distances should equal a constant for every point of the ellipse. Although it might seem strange we will use the constant 2a. The reason for the 2 will become more evident once we reach the end of our derivation which goes as follows.
The reason for the change in the last step is due to the fact that we know that the length of the string, 2a, must be greater than the distance between the foci, 2c. If 2a is greater than 2c, then a is greater than c and a squared must be greater than c squared. Typically a substitution is made at this point of,
Also notice from the substituting expression that a is greater than b. From this equation we can find the location of the major axis (longer axis) vertices and the endpoints of the minor axis (shorter axis). Simply set y=0 in the above expression and we get:
Since as we said before that a is greater than b, then the minor axis is along the y axis and has a length of 2b and the major axis is along the x axis and has a length of 2a. Conversely the equation for an ellipse with the major axis along the y axis is:
Just as with the parabola we can use the same substitutions for translation of axes and rotation of axes to develop expressions for other more general ellipses.
One final point of interest concerning the relationship between an ellipse, a circle, and a parabola. If the distance between the two foci is reduced to zero (i.e. foci are at the same point) the ellipse becomes a circle. If this same interfocal distance goes to infinity, the ellipse becomes a parabola.
The next post in this series soon to follow on hyperbolas.