       Re: A New Scientist article verified with Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg107275] Re: [mg107238] A New Scientist article verified with Mathematica
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Sun, 7 Feb 2010 06:15:19 -0500 (EST)

```eqns = {x^2 + y^2 - 4 == 0, (x - 2)^2 + y^2 - 4 == 0};

pts = {x, y} /. Solve[eqns, {x, y}]

{{1, -Sqrt}, {1, Sqrt}}

Plot[Evaluate[Transpose[
(y /. Solve[#, y]) & /@ eqns]],
{x, -2.5, 4.5},
PlotStyle -> {Red, Directive[Red, Dashed]},
Epilog ->
{Blue, AbsolutePointSize, Point[pts]}]

pts2 = {x, Im[y]} /. Solve[
{x^2 + y^2 - 4 == 0, (x - 5)^2 + y^2 - 4 == 0},
{x, y}]

{{5/2, -(3/2)}, {5/2, 3/2}}

Graphics[{Red, Point[pts2]},
Axes -> True,
AxesLabel -> {"Re", "Im"}]

Bob Hanlon

---- sigismond kmiecik <sigismond.kmiecik at wanadoo.fr> wrote:

=============
Hello to everybody

In  the last Xmas issue of the New Scientist magazine there is on page
40 a small article about the continuity principle applied to
intersecting circles.
I used Mathematica to confirm its conclusions but some questions remain

These circles are represented by

Show[{Graphics[{Red, Circle[{0, 0}, 2]}], Graphics[Circle[{2, 0}, 2]],
Graphics[{Red, Dashed, Circle[{5, 0}, 2]}]}, AxesOrigin -> {0, 0},
PlotRange -> {{-3, 8}, {-3, 3}}, Axes -> True ]

The intersection coordinates of the red (non-dashed) and black circle is
found by:

Solve [{ x^2 + y^2 - 4 == 0, (x - 2)^2 + y^2  - 4  == 0 }, {x, y}=
]

And there is indeed an imaginary intersection of the red and red-dashed
circle even if they are not touching -  as found by:

Solve [{ x^2 + y^2 - 4 == 0, (x - 5)^2 + y^2  - 4  == 0 }, {x, y}=
]

My questions are:
- Is there a way to draw  with Mathematica these three circles using
their cartesian equations and not the Circle graphics =91primitive=92 ?
- How can I transform the list of rules solutions to the last equation
above  in order to represent them on the complex plane  (I thought about
a ListPlot [{Re[],Im[]}=85  but I know not  enough of Mathematica to
obtain that)
- And last is there a Mathematica notebook on the web dealing with the
intersection of  planes with cones?

Thanks

Sigismond Kmiecik

```

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