Take a coordinate system, make it orthogonal for simplicity. Mark both axes with… for instance one through seven. Draw straight lines between the axes pairing off the numbers that add up to eight. This creates the impression of a nice curve, and I expect the lines would be tangents to the curve.
If you add 0.5, 1.5, … 7.5 to both axes you get 8 more lines, bringing you closer to the implicit curve. But how do I work out a function for the curve? Does one even exist?
You’re looking for a hyperbola of the form y = a / x. Choose a based on your coordinate system and which points you’re connecting.
I know this from experience, but to derive from first principles is hard. Since you’re defining the curve by its tangents, you’re basically creating it’s derivative. Integrate that and you’ll have the curve you want. That’s means calculus.
Weird. Cabbage, I looked at the citation, and I wish I was a better mathematician, but that form looks like a hyperbola to me. I agree that the writer says it’s a parabola, but it looks like a hyperbola, doesn’t it?
And the illustration makes it obvious that the curve approaches two lines asymptotically, as one branch of a hyperbola would. The slopes of both ends of the parabola are both infinite in the limits, but the slopes of both ends of a hyperbola would be different, and at least one would be finite.
The general equation for a conic section is ax[sup]2[/sup] + bxy + cy[sup]2[/sup] + dx + ey + f =0, and the relationship between b[sup]2[/sup] and 4ac determines exactly what you have. Assuming you have a non-degenerate conic section, when b[sup]2[/sup] = 4ac, you have a parabola; when b[sup]2[/sup] < 4ac, you have an ellipse; and when b[sup]2[/sup] > 4ac you have a hyperbola.
In the case of the envelope, a = c = 1 and b = -2, so b[sup]2[/sup] = 4ac and it’s a parabola.
It’s not the entire parabola that you’re getting by this technique, just a segment of one. Besides, in the OP’s example, the curve would obviously intersect the x-axis at (8,0) and the y-axis at (0,8).
The form in the Wiki article just looks confusing because the “principal axes” of the parabola are at 45° angles to the x- and y- axes. If you define two new coordinates, u = x + y and v = x - y, then you can rewrite the equation given in the article as
2 k u = k[sup]2[/sup] + v[sup]2[/sup]
which looks a lot more like a parabola. The “u-axis” would then point “northeast” and the v-axis “northwest.”
My geometry software wouldn’t let me create the required parabola, so I used an ordinary quadratic function and then rotated the drawing 45 degrees. I’m fairly happy with the result considering I wasted the whole evening on it.
Interesting aside. If we redefine the problem so that the tangent drawn is a prescribed length (think of a ladder sliding down a wall) we get a shape called an astroid. The full astroid has four cusps - i.e. enters the negative regions of the axes originally drawn. And the locus of a point a fixed distance from the end of teh tangent line segment is an ellipse. This is my favourite construction of an ellipse because it is so counter-intuitive.
My most immediate need to understand this (for the sake of my mental health :D) has been satisfied, but I’d still like to understand what the approach in the wikipedia article on envelopes actually means.
What does the rate of change for t really signify, and what does it mean that it’s zero? Anyone able to make it clearer?