3-point' curveIf you have a loop of string, a fixed point and a pencil, and...

PoentWeptgj

PoentWeptgj

Answered

2022-07-22

3-point' curve
If you have a loop of string, a fixed point and a pencil, and stretch the string as much as possible, you draw a circle. With 2 fixed points you draw an ellipse. What do you draw with 3 fixed points?

Answer & Explanation

autarhie6i

autarhie6i

Expert

2022-07-23Added 18 answers

Step 1
Let length of sting be > m a x ( A B , B C , C A ). If the given three points are A,B and C are not in a straight line, then
there are three possibilities to trace three ellipses and the locus is composed of the these three discontinuous ellipses.
Step 2
Center point B can be ignored when inter-focal distance is AC,
Center point C can be ignored when inter-focal distance is BA, and
Center point A can be ignored when inter-focal distance is BC.
So in effect each is only a 2-point curve.
Almintas2l

Almintas2l

Expert

2022-07-24Added 6 answers

Step 1
Assume that three non-collinear points A, B, C in the plane are given, and that you have a loop of string of length > | A B | + | B C | + | C A | . Slinging this string around the three points and a pencil you can draw a loop γ around ( A B C ) in the obvious way, keeping the string tight at all times. This loop will be a continuous curve. In order to describe γ more precisely we draw the lines A B, B C, C A in full. In this way the plane is divided into 7 compartments, one of them the triangle ( A B C ).
Step 2
The loop γ traverses the 6 unbounded compartments, and at each intersection with one of the above lines it has a corner. Within a given compartment γ is an arc of an ellipse, whereby two of the three vertices A, B, C act as foci. The points A, B are foci for two such arcs. One of these arcs belongs to an ellipse with major axis | A B | and the other to an ellipse with major axis | B C | | C A | .

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