Then construct the locus of the point of intersection as a function of A's motion around the circle. This will produce a new line, which is likely to be almost parallel to the original line. Once you've constructed the point A + delta A, repeat your original construction using the new point (A + delta A) instead of the old one (A). Drag A to verify that both A and A + delta A move together around the circle. On a circle, you can construct this point by rotating A by a small quantity. So a point "arbitrarily close" to A can be thought of as a point just a little bit farther around the circle than A (for any position of A). ![]() In this example, A travels around a circle. How you construct this second point A + delta A depends upon the nature of your locus. This means finding a new point, A + delta A, which is arbitrarily close to A, and repeating your construction (of the line determined by A) on this new point (to get the line determined by f(A + delta A)). (In our example, this is the perpendicular bisector through C.) So you need to construct a new line, f(A + delta A), which is "arbitrarily close" to the line determined by A. You have already constructed the first line f(A)-the line of which you've taken the locus. Thus, to construct an envelope, you'll need two lines, f(A) and f(A + delta A). The envelope of the lines f(A) is the locus of the intersection of f(A) and f(A + delta A) as delta A approaches zero. For each possible position A, think of the corresponding line in the locus as f(A). For this discussion, let's assume (as in the example) that you have a locus of lines determined by the motion of a point A. You can construct an arbitrarily accurate approximation of an envelope by constructing the locus of the point of intersection of two arbitrarily closely spaced lines (or circles) in your locus. The envelope of the line is the hyperbola itself: the "edge" of the blue locus. The locus of the line as point A travels about a circle clearly suggests a hyperbola. For instance, in the sketch at right, the line through C is the perpendicular bisector AB. ![]() FAQ: Envelope Constructions How do I construct an envelope?Ī geometric envelope can be thought of as the limit or edge of the locus of a line or a circle.
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