Re: Mars lander crash

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From	Ben Bacarisse <ben.usenet@bsb.me.uk>
Newsgroups	comp.lang.c
Subject	Re: Mars lander crash
Date	Wed, 14 Dec 2016 12:37:06 +0000
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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.x³ + b.x²y + c.xy² + d.y³ + e.x² + f.xy + g.y² + 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
>> Bézout’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.

<snip>
-- 
Ben.

Thread

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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