3-point' curveIf 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?

Question

3-point&#039; curveIf 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?

autarhie6i · Accepted Answer

Step 1Let length of sting be   &amp;gt;  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, thenthere are three possibilities to trace three ellipses and the locus is composed of the these three discontinuous ellipses.Step 2Center point B can be ignored when inter-focal distance is AC,Center point C can be ignored when inter-focal distance is BA, andCenter point A can be ignored when inter-focal distance is BC.So in effect each is only a 2-point curve.

Almintas2l · Accepted Answer

Step 1Assume that three non-collinear points A, B, C in the plane are given, and that you have a loop of string of length   ℓ  &amp;gt;      |    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 2The 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   ℓ  −      |    A  B      |   and the other to an ellipse with major axis   ℓ  −      |    B  C      |    −      |    C  A      |  .

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?

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