An Idea of a Moment
When I was in high school, I thought it would be fun to derive an equation from scratch for a cone.
A Misconception
Now a cone is not what most people think – a two-dimensional object the shape of an upside down ice cream cone, though it is similar. A cone is really like an upside-down ice cream cone with another ice cream cone that is right side up, with the two tips touching and with a center line running right through the middle of the fat ends. In addition, the cone’s wide ends stretch out both ways to infinity.
We Begin
Now using the basic x-y-z diagram typically used in 3D analytical geometry, I drew a cone (Figure 1) with its cross-over point at the origin, x=y=z=0, or according to accepted notation, (0, 0, 0). Not being able to draw the cone to infinity, I chose to stop the image at y=h at the top and y=-h at the bottom. I define the radius of the cone at y=h as r. Then,
Equation 1. x2 + z2 = r2
Equation 2. r/y = tan (theta) or in other words, r=y tan (theta)
So,
Equation 3. x2 + z2 = y2 tan2 (theta)
Now the angle is a constant in the case of a cone. So this boils down to
Formula: x2 + z2 = ay2
Some Tests
1. Making the constant a equal to zero.
Let’s test this out. If a = 0, then x2 + z2 = 0 Since squared numbers, if real, have to be less than or equal to zero, this is nonsense, unless both x and z are equal to zero. This thus reduces to a line along the y-axis. This makes sense. An infinitesimally thick cone is a line.
2. Making the constant infinite.
If a =∞ , then y could be anything except possibly zero and x2 + z2 = ∞, which would be a cylinder infinitely wide centered along the y-plane. This makes sense.
3. Making z or x or y equal to zero.
If, on the other hand, a was an ordinary number greater than zero and less than infinity, and if z = 0, then x2 = ay2 which can be changed to the following two equations:
x = by where b =√ a
x= -by where b = √a
These two equations constitute two lines crisscrossing in the xy-plane at the origin, (0, 0, 0). This is what a cone becomes when the third dimension, z, is eliminated.
Something similar occurs if x, instead of z, becomes zero, but using the zy-plane without reference to x.
The equations in this instance became:
z = by where b = √a
z = -by where b = √a
If y becomes zero the result is the single point x=y=z=0, or (0,0,0). This is like making the cone have a zero height. This is what we would expect.
In Summary
So our conclusions are correct, and we have the basic equation of a cone with midpoint at the origin and line of symmetry as the y-axis being:
x2 + z2 = ay2
It would have been far more complicated to determine the equation of a cone floating about somewhere in space, with midpoint (x,y,z), and with symmetry not about the x-axis, y-axis, or z-axis. While it would have been difficult, it would have been just as rewarding. No doubt much could be learned from such an undertaking. How about it? Will you try it?
