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)
- Reply-to: hanlonr at cox.net
eqns = {x^2 + y^2 - 4 == 0, (x - 2)^2 + y^2 - 4 == 0};
pts = {x, y} /. Solve[eqns, {x, y}]
{{1, -Sqrt[3]}, {1, Sqrt[3]}}
Plot[Evaluate[Transpose[
(y /. Solve[#, y]) & /@ eqns]],
{x, -2.5, 4.5},
PlotStyle -> {Red, Directive[Red, Dashed]},
Epilog ->
{Blue, AbsolutePointSize[4], 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
to be answered.
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