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Re: Mars lander crash

From David Brown <david.brown@hesbynett.no>
Newsgroups comp.lang.c
Subject Re: Mars lander crash
Date 2016-12-15 13:26 +0100
Organization A noiseless patient Spider
Message-ID <o2u230$2ri$1@dont-email.me> (permalink)
References (17 earlier) <o2ph6u$aec$1@dont-email.me> <874m27a0ic.fsf@bsb.me.uk> <o2r2mj$n50$1@dont-email.me> <87twa67klp.fsf@bsb.me.uk> <kfnh965hi6h.fsf@x-alumni2.alumni.caltech.edu>

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On 15/12/16 06:31, Tim Rentsch wrote:
> Ben Bacarisse <ben.usenet@bsb.me.uk> writes:
> 
>> David Brown <david.brown@hesbynett.no> writes:
>>
>>> On 14/12/16 00:10, Ben Bacarisse wrote:
>>>> David Brown <david.brown@hesbynett.no> writes:
>>>>
>>>>> On 13/12/16 18:01, Ben Bacarisse wrote:
>>>>>> Keith Thompson <kst-u@mib.org> writes:
>>>>>>
>>>>>>> Malcolm McLean <malcolm.mclean5@btinternet.com> writes:
>>>>>>> [...]
>>>>>>>> One way of solving the curve intersection problem is to build
>>>>>>>> a root finder which will solve equations with nine roots.
>>>>>>>> If the root finder works for all representable values of
>>>>>>>> coefficients, then great.  But what if it doesn't?  Is to going
>>>>>>>> to be good enough, (2 cubics can intersect nine times, but
>>>>>>>> almost always they intersect only once or twice on the interval
>>>>>>>> you are interested in)?
>>>>>>>
>>>>>>> Two cubics can intersect nine times?  I don't think so.
>>>>>>>
>>>>>>> A cubic is an equation of the form ax^3 + bx^2 + cx + d = 0.
>>>>>>> It can have up to three roots (values of x for which it's true).
>>>>>>> The intersections of two cubics are the solutions of the difference
>>>>>>> between the two cubics -- which is itself a cubic, which have no
>>>>>>> more than three solutions.
>>>>>>>
>>>>>>> Is there a confusion about terminology?
>>>>>>
>>>>>> I think so.  Malcolm is almost certainly talking about a cubic curve
>>>>>> rather than a cubic equation in one variable.  A cubic curve is a
>>>>>> polynomial equation in /two/ variables p(x, y) = 0 where the maximum
>>>>>> degree of any term in x or y is 3.
>>>>>
>>>>> I think you mean that the sum of the powers is at most 3,
>>>>
>>>> Yes (the sum of the powers in any term is at most three).  Actually I
>>>> meant to say that the degree of any term is at most 3 which I think is
>>>> right but maybe the degree of a term is not defined as I think it is.  I
>>>> don't know where my "degree of any term in x or y" came from!  Anyway,
>>>> your
>>>>
>>>>> a.x**3 + b.x**2y + c.xy**2 + d.y**3 + e.x**2
>>>>> + f.xy + g.y**2 + h.x + i.y + j
>>>>
>>>> makes it clear we are talking about the same equations.
>>>>
>>>>> 10 coefficients, so 9 roots (counting doubles as necessary, and
>>>>> including complex numbers).  I don't think you you need a point at
>>>>> infinity.
>>>>
>>>> Well it was <mumble> decades ago but I think the key theorem here is
>>>> Bezout?s Theorem.  This is a theorem about curves in projective space
>>>> rather than affine space (again from hazy memeory).  But, as I say it's
>>>> all rather dim so maybe I should back out now...
>>>>
>>>> <snip>
>>>
>>> If you are working in projective space, then your roots will be in the
>>> projective space too, I think.
>>
>> Yes.  I think (but I am now way outside my comfort zone) to always get 9
>> roots you need to be in a projective space because in some cases the
>> curves only "meet at infinity".  In affine space, you get up to 9
>> roots rather than always 9 roots.
> 
> Yes, in general points at infinity are needed.  It's easy to
> see that they are by considering the two cubics
> 
>     ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy = 1
>     ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy = 2
> 
> which obviously have no finite points in common.  But they do
> meet on one or more points on the line at infinity (perhaps
> with a requirement that not all of a,b,c,d,e,f,g,h,i are
> zero).
> 

Good point.  It is even easier with the examples:

	x = 1
and
	x = 2

They are parallel lines, and will meet at a point of infinity in a
projective geometry, but not in a plain Euclidian geometry.

To take this a step further, is it possible for pairs of equations like
this to have some finite and some infinite meeting points?  I suspect
that you would have /either/ finite /or/ infinite meeting points, but
not both (assuming the equations are not coincidental, of course).  But
this is not something I can figure out in my head.

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Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-09 13:26 +0100
  Re: Mars lander crash Richard Heathfield <rjh@cpax.org.uk> - 2016-12-09 12:44 +0000
    Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 14:53 +0100
      Re: Mars lander crash already5chosen@yahoo.com - 2016-12-09 05:58 -0800
        Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 15:11 +0100
        Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-09 16:56 +0100
          Re: Mars lander crash supercat@casperkitty.com - 2016-12-09 08:51 -0800
          Re: Mars lander crash already5chosen@yahoo.com - 2016-12-11 02:23 -0800
            Re: Mars lander crash Gareth Owen <gwowen@gmail.com> - 2016-12-11 10:57 +0000
          Re: Mars lander crash "James R. Kuyper" <jameskuyper@verizon.net> - 2016-12-11 12:34 -0500
            Re: Mars lander crash Robert Wessel <robertwessel2@yahoo.com> - 2016-12-11 23:33 -0600
      Re: Mars lander crash Richard Heathfield <rjh@cpax.org.uk> - 2016-12-09 14:22 +0000
        Re: Mars lander crash asetofsymbols@gmail.com - 2016-12-09 06:30 -0800
          Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 15:48 +0100
        Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 15:46 +0100
          Re: Mars lander crash Richard Heathfield <rjh@cpax.org.uk> - 2016-12-09 14:55 +0000
            Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 16:08 +0100
              Re: Mars lander crash Richard Heathfield <rjh@cpax.org.uk> - 2016-12-09 15:52 +0000
          Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-09 17:35 +0100
            Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 18:35 +0100
              Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-09 19:10 +0100
                Re: Mars lander crash "James R. Kuyper" <jameskuyper@verizon.net> - 2016-12-09 19:18 -0500
                Re: Mars lander crash Keith Thompson <kst-u@mib.org> - 2016-12-09 16:49 -0800
                Re: Mars lander crash "J. Clarke" <j.clarke.873638@gmail.com> - 2016-12-20 20:03 -0500
                Re: Mars lander crash <william@wilbur.25thandClement.com> - 2016-12-09 22:04 -0800
                Re: Mars lander crash supercat@casperkitty.com - 2016-12-10 10:59 -0800
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-11 13:22 +0100
                Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-11 14:18 +0100
                Re: Mars lander crash already5chosen@yahoo.com - 2016-12-11 06:03 -0800
                Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-11 15:08 +0100
                Re: Mars lander crash already5chosen@yahoo.com - 2016-12-11 06:20 -0800
                Re: Mars lander crash luser droog <luser.droog@gmail.com> - 2016-12-11 07:08 -0800
                Re: Mars lander crash Keith Thompson <kst-u@mib.org> - 2016-12-11 11:56 -0800
                Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-11 21:05 +0100
                Re: Mars lander crash jameskuyper@verizon.net - 2016-12-11 12:57 -0800
                Re: Mars lander crash Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-12-11 21:26 +0000
                Re: Mars lander crash supercat@casperkitty.com - 2016-12-11 09:23 -0800
                Re: Mars lander crash Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-12-11 14:01 +0000
                Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-11 15:11 +0100
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-11 18:57 +0100
                Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-11 19:23 +0100
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-11 19:55 +0100
                Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-11 20:52 +0100
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-11 23:37 +0100
                Re: Mars lander crash supercat@casperkitty.com - 2016-12-11 16:10 -0800
                Re: Mars lander crash BartC <bc@freeuk.com> - 2016-12-12 01:13 +0000
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-12 09:53 +0100
                Re: Mars lander crash supercat@casperkitty.com - 2016-12-12 08:30 -0800
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-12 09:25 +0100
                Re: Mars lander crash supercat@casperkitty.com - 2016-12-11 10:36 -0800
                Re: Mars lander crash Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-12-11 18:46 +0000
                Re: Mars lander crash supercat@casperkitty.com - 2016-12-11 09:12 -0800
                Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-11 18:53 +0100
                Re: Mars lander crash Keith Thompson <kst-u@mib.org> - 2016-12-11 12:09 -0800
                Re: Mars lander crash Jean-Marc Bourguet <jm@bourguet.org> - 2016-12-11 21:30 +0100
                Re: Mars lander crash Lew Pitcher <lew.pitcher@digitalfreehold.ca> - 2016-12-11 15:50 -0500
                Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-11 23:02 +0100
                Re: Mars lander crash Keith Thompson <kst-u@mib.org> - 2016-12-11 18:29 -0800
                Re: Mars lander crash Keith Thompson <kst-u@mib.org> - 2016-12-11 12:03 -0800
                Re: Mars lander crash Keith Thompson <kst-u@mib.org> - 2016-12-11 11:55 -0800
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-11 23:42 +0100
                Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-11 14:53 -0800
                Re: Mars lander crash Ian Collins <ian-news@hotmail.com> - 2016-12-12 17:09 +1300
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-12 10:00 +0100
                Re: Mars lander crash Ian Collins <ian-news@hotmail.com> - 2016-12-12 22:20 +1300
                Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-12 01:28 -0800
                Re: Mars lander crash Ian Collins <ian-news@hotmail.com> - 2016-12-12 22:38 +1300
                Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-12 02:43 -0800
                Re: Mars lander crash Ian Collins <ian-news@hotmail.com> - 2016-12-13 08:20 +1300
                Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-12 12:57 -0800
                Re: Mars lander crash Ian Collins <ian-news@hotmail.com> - 2016-12-13 14:14 +1300
                Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-13 01:19 -0800
                Re: Mars lander crash Ian Collins <ian-news@hotmail.com> - 2016-12-13 22:56 +1300
                Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-13 05:16 -0800
                Re: Mars lander crash Ian Collins <ian-news@hotmail.com> - 2016-12-14 18:05 +1300
                Re: Mars lander crash Keith Thompson <kst-u@mib.org> - 2016-12-13 08:19 -0800
                Re: Mars lander crash Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-12-13 17:01 +0000
                Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-13 10:48 -0800
                Re: Mars lander crash Geoff <geoff@invalid.invalid> - 2016-12-13 11:56 -0800
                Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-13 13:37 -0800
                Re: Mars lander crash Geoff <geoff@invalid.invalid> - 2016-12-13 19:11 -0800
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-13 20:13 +0100
                Re: Mars lander crash supercat@casperkitty.com - 2016-12-13 11:45 -0800
                Re: Mars lander crash Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-12-13 23:10 +0000
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-14 10:18 +0100
                Re: Mars lander crash Ben Bacarisse <ben.usenet@bsb.me.uk> - 2016-12-14 12:37 +0000
                Re: Mars lander crash Tim Rentsch <txr@alumni.caltech.edu> - 2016-12-14 21:31 -0800
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-15 13:26 +0100
                Re: Mars lander crash Ian Collins <ian-news@hotmail.com> - 2016-12-12 22:40 +1300
                Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-12 13:22 +0100
                Re: Mars lander crash supercat@casperkitty.com - 2016-12-12 08:35 -0800
        Re: Mars lander crash supercat@casperkitty.com - 2016-12-09 08:43 -0800
      Re: Mars lander crash Jerry Stuckle <jstucklex@attglobal.net> - 2016-12-09 09:47 -0500
        Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 16:11 +0100
          Re: Mars lander crash Jerry Stuckle <jstucklex@attglobal.net> - 2016-12-09 10:49 -0500
  Re: Mars lander crash already5chosen@yahoo.com - 2016-12-09 05:53 -0800
    Re: Mars lander crash asetofsymbols@gmail.com - 2016-12-09 06:04 -0800
      Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 15:09 +0100
        Re: Mars lander crash Jerry Stuckle <jstucklex@attglobal.net> - 2016-12-09 10:06 -0500
          Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 16:32 +0100
    Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-09 17:30 +0100
      Re: Mars lander crash BartC <bc@freeuk.com> - 2016-12-09 16:42 +0000
      Re: Mars lander crash Lew Pitcher <lew.pitcher@digitalfreehold.ca> - 2016-12-09 11:43 -0500
      Re: Mars lander crash Keith Thompson <kst-u@mib.org> - 2016-12-09 08:59 -0800
      Re: Mars lander crash "J. Clarke" <j.clarke.873638@gmail.com> - 2016-12-20 19:36 -0500
  Re: Mars lander crash "Rick C. Hodgin" <rick.c.hodgin@gmail.com> - 2016-12-09 06:46 -0800
    Re: Mars lander crash Richard Heathfield <rjh@cpax.org.uk> - 2016-12-09 14:57 +0000
      Re: Mars lander crash Noob <root@127.0.0.1> - 2016-12-09 16:07 +0100
    Re: Mars lander crash David Brown <david.brown@hesbynett.no> - 2016-12-09 16:03 +0100
      Re: Mars lander crash "James R. Kuyper" <jameskuyper@verizon.net> - 2016-12-09 12:41 -0500
    Re: Mars lander crash Jerry Stuckle <jstucklex@attglobal.net> - 2016-12-09 10:52 -0500
      Re: Mars lander crash jameskuyper@verizon.net - 2016-12-09 12:13 -0800
        Re: Mars lander crash jacobnavia <jacob@jacob.remcomp.fr> - 2016-12-09 21:56 +0100
          Re: Mars lander crash jameskuyper@verizon.net - 2016-12-09 14:11 -0800
        Re: Mars lander crash Jerry Stuckle <jstucklex@attglobal.net> - 2016-12-09 22:46 -0500
          Re: Mars lander crash <william@wilbur.25thandClement.com> - 2016-12-09 22:34 -0800
          Re: Mars lander crash James Kuyper <jameskuyper@verizon.net> - 2016-12-10 18:38 -0500
      Re: Mars lander crash Malcolm McLean <malcolm.mclean5@btinternet.com> - 2016-12-11 06:06 -0800
  Re: Mars lander crash "Mr. Man-wai Chang" <toylet.toylet@gmail.com> - 2016-12-10 01:17 +0800
  Re: Mars lander crash Ken Brody <kenbrody@spamcop.net> - 2016-12-13 14:28 -0500
  Re: Mars lander crash Ken Brody <kenbrody@spamcop.net> - 2016-12-13 14:30 -0500

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