Re: Question:
- To: mathgroup at smc.vnet.net
- Subject: [mg34178] Re: [mg34165] Question:
- From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
- Date: Wed, 8 May 2002 01:57:47 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
First of all, looking at the graph of
ParametricPlot[{(Cos[6 t])^2, 2Sin[7 t]}, {t, 0, 2Pi/7}]
I can see no points of self-intersection at all! Did you not mean 3Pi/7?
Assuming the letter:
this is the sort of thing that is easier done exactly and by hand (with
just a bit of help form Mathematica, maybe) than with accuracy 1x10^-7 !
Basically, you want to find values s and t satisfying
Cos[6s] == +Cos[6t] or -Cos[6t]
Sin[7s]==Sin[7t]
the first condition will be satisfied if either
(s + t) == k Pi/6
or
(s-t) == k Pi/6 (k integer) while the second if
s - t == 2Pi k/7 (k integer) or
(s + t) == (2k + 1)Pi/7 k integer
it is now not hard to find the following four pairs of solutions (with
t<=3Pi/7)
In[5]:=
sols = {{{t -> Pi/21}, {t -> (8*Pi)/21}},
{{t -> (3*Pi)/28}, {t -> (11*Pi)/28}},
{{t -> Pi/42}, {t -> (13*Pi)/42}}, {{t -> (11*Pi)/84},
{t -> (25*Pi)/84}}};
Checking
In[6]:=
N[{(Cos[6 t])^2,2Sin[7 t]}/.sols]
Out[6]=
{{{0.38874,1.73205},{0.38874,1.73205}},{{0.188255,1.41421},{0.188255,
1.41421}},{{0.811745,1.},{0.811745,1.}},{{0.61126,0.517638},{0.61126,
0.517638}}}
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
On Tuesday, May 7, 2002, at 04:54 PM, Per Lundgren wrote:
>
> Hi,
>
> Here is my question: How do I calculate the t-values for the four
> points (x,y) where the curve below intersects itself with an accuracy
> of 1x10^-7
>
> x==(Cos[6 t])^2
>
> y==2Sin[7 t]
>
> in the intervall: [0,2Pi/7]
>
> Thank you in advance
>
> Per Lundgren, Sweden
>
> (Plot the parametric curve and you will understand what I am asking for)
>
>
>
>