Copyright © Karl Dahlke, 2022

If you want to mail a letter, you need an envelope;
the envelope contains the letter.
In mathematics, an envelope is an intersection of shapes or regions.
The envelope is the containment of all those regions.
For example, let the region R_{t}
be a square in the xy plane, centered at the origin, having a side of length 2, and rotated through an angle of t.
There is such a region R_{t} for each t from 0 to 90 degrees.
The intersection of all these squares is a circle, centered at the origin, with radius 1,
also known as the unit disk.
This is the envelope of all these regions.

An envelope can exist in 3 or more dimensions, or even an abstract topological space.
An interesting shape in 3 space is the intersection of the 3 cylinders x^{2} + y^{2} ≤ 1,
x^{2} + z^{2} ≤ 1,
and y^{2} + z^{2} ≤ 1.
As a fun exercise in integral calculus, find the volume of this region.
The answer is given at the end of this chapter.

In an engineering context, an envelope is often more than just abstract geometry. Consider the field of aviation. A typical envelope comprises two dimensions: air speed and altitude. (Other variables are possible, such as the payload of the aircraft, but let's stick with these two for now.) The envelope is the region wherein the plane operates safely and effectively. It depends, naturally, on the airplane. The Wright Flyer, built by the Wright brothers and flown successfully in 1903, never rose more than 50 feet in the air. If altitude is measured in feet, the flight envelope of this aircraft is bounded below by y = 0, and above by y = 50. The Flyer attained level flight in air speeds of 30 to 40 mph. Any modern aircraft would stall and crash at such a low speed, but the Wright brothers didn't have powerful engines, and thus their plane simply had to fly at low speeds. They took advantage of a 20 mph headwind that was blowing on December 17, 1903. By driving their plane at 10 mph into the wind, it attained an air speed of 30 mph, and rose into the air. Thus the first flight envelope was a rectangle bounded by 30 and 40 mph on the x axis, and 0 and 50 feet on the y axis. It's a tiny rectangle.

The Fokker Triplane, flown by the Red Baron in World War I, was quite a different story. In just 15 years, the flight envelope had increased by a factor of a thousand. The Triplane could fly between 75 and 105 mph, up to an altitude of 19,685 feet. Flights above 15,000 feet were not recommended however, because cockpits were open, and the pilot could enter a state of hypoxia, often without realizing it. If he lost consciousness, that was the end of the ballgame. This was described in an earlier chapter. Other than rapid combat maneuvers, the ceiling should be placed at 15,000 feet, the maximum altitude that provides sufficient oxygen for the pilot.

So far, the flight envelopes were rectangles, though they were not generally drawn on graph paper, and would not be referred to as envelopes until World War II.

In the 1930's, as planes flew higher, with closed cockpits to protect the pilot, they encountered thinner air, which allowed them to fly faster. The flight region shifts to the right as you rise in altitude. The plane stalls more easily in thin air, thus increasing the minimum air speed, and there is less drag in thin air, thus increasing the maximum air speed. The envelope tilts, from a rectangle to a parallelogram.

Consider the Concorde, built and flown in the 1970's. At low altitudes, below 30,000 feet, it flies subsonic, from 150 to 500 mph, like a commercial aircraft - but at high altitudes, 60,000 feet, it can reach speeds of Mach 2.0, 1,350 mph. In fact the Concorde has to fly high, 25,000 feet higher than other passenger aircraft, to streak through the thin air at Mach 2. I'm sure military aircraft have even larger flight envelopes, but that is classified.

The term "flight envelope" was used only in military circles during World War II,
but it entered the general lexicon after Tom Wolf published his book *The Right Stuff* in 1979,
which was made into a popular movie in 1983.
A pilot would "push the envelope" if he pushed his plane near the limits of its flight envelope,
thus risking his own life.
In the same way, a new airplane would push the envelope if it could fly higher or faster than previous models.
Chuck Yeager did just that in 1947,
when he flew the experimental Bell X-1 at Mach 1, at 45,000 feet.
Six years later, Yeager
flew the X-1A to Mach 2.44, setting another record.
He was a pilot who truly pushed the envelope.

When someone says you are pushing the envelope, you are going up to or beyond the boundaries that have been set by engineering limits, or perhaps by social norms and conventions.

And now, as promised, here is the volume of the three intersecting cylinders. The shape contains the unit sphere, and thus has a volume greater than 4π/3, or 4.188. It is also contained in any one of the 3 cylinders, having radius 1 and height 2, giving a volume below 2π, or 6.283. The volume is bracketed between 4.188 and 6.283.

Start with two intersecting cylinders, sheathing the x and y axes respectively.
Focus on one eighth of the envelope, with x y and z ≥ 0.
For each level z, the cross section of this shape is a square.
The square has sides of length 1 at z = 0,
the unit square to be precise,
and it shrinks to a square of side 0 at z = 1.
The length of the square is sqrt(1-z^{2}), and its area is 1-z^{2}.
Integrate this as z runs from 0 to 1 and get a volume of 2/3.
Multiply by 8 to get a volume of 16/3, or 5.333.
Thus the shape that we seek, the three intersecting cylinders, has a volume bounded between 4.188 and 5.333.

Now bring in the third cylinder surrounding the z axis.
As the square rises off the floor and shrinks to a point at z = 1,
intersect it with the unit circle at each level.
For notational convenience, let t = sqrt(½), approximately 0.707.
When z = t, the square has sides of length t, and it just touches the z cylinder at (t,t,t).
At higher levels, the shrinking square is not cut by the circle at all.
The volume, for z ≥ t, can be found by the same formula as above.
Evaluate z - z^{3}/3 at 1 and at t to get a volume of 2/3 - t + t/6.
This shape appears three times in the first octant, for z ≥ t, x ≥ t, and y ≥ t.
Multiply by 3 for a volume of 2 - 5t/2.
Finally add in the volume of the cube of side t, which is ½t.
The result is 2-2t.
This is just the first octant, so multiply by 8 and get 16×(1-t), or 4.686.