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From: Kent Holing on 11 Aug 2010 06:01
By drawing a circle on a map, can I always claim that there are at least two diametrically opposite points on the circle with the same height above the sea level?
From: bert on 11 Aug 2010 10:25
On 11 Aug, 15:01, Kent Holing <K...(a)statoil.com> wrote:
> By drawing a circle on a map, can I always claim that there are at least two diametrically opposite points on the circle with the same height above the sea level?

Only when the height varies continuously around
the circular path (the proof would be like that
of the "ham sandwich" theorem). But if there
are vertical discontinuities, then no. Imagine
the circle passing through three flat regions
of different heights, joining at vertical steps,
such that each region occupies less than half of
the circumference.
--
From: Sherman Forte on 11 Aug 2010 11:16

"Kent Holing" <KHO(a)statoil.com> wrote in message
news:578672460.93238.1281535318830.JavaMail.root(a)gallium.mathforum.org...
> By drawing a circle on a map, can I always claim that there are at least
> two diametrically opposite points on the circle with the same height above
> the sea level?


no. trivial.


From: quasi on 11 Aug 2010 16:52
On Wed, 11 Aug 2010 10:01:28 EDT, Kent Holing <KHO(a)statoil.com> wrote:

>By drawing a circle on a map, can I always claim that there are at least two diametrically opposite points on the circle with the same height above the sea level?

For theta in the closed interval [0, 2*Pi], define P(theta) to be the
point on the circle which is at an angle theta counterclockwise from
due East of the center. Note that P(0) = P(2*Pi). Next define h(theta)
to be the height of P(theta) above sea level. In the context of this
problem, it makes sense to assume h is continuous. Now consider the
graph of h on the interval [0, 2*Pi].

quasi
From: Gerry Myerson on 11 Aug 2010 23:31
In article <6d26665msdnjtclhk7g4pfcnltqrjckcte(a)4ax.com>,
quasi <quasi(a)null.set> wrote:

> On Wed, 11 Aug 2010 10:01:28 EDT, Kent Holing <KHO(a)statoil.com> wrote:
>
> >By drawing a circle on a map, can I always claim that there are at least two
> >diametrically opposite points on the circle with the same height above the
> >sea level?
>
> For theta in the closed interval [0, 2*Pi], define P(theta) to be the
> point on the circle which is at an angle theta counterclockwise from
> due East of the center. Note that P(0) = P(2*Pi). Next define h(theta)
> to be the height of P(theta) above sea level. In the context of this
> problem, it makes sense to assume h is continuous. Now consider the
> graph of h on the interval [0, 2*Pi].

Or consider h(theta) - h(theta + pi), and the intermediate value
theorem.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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