Parallels
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From: William Elliot on 5 May 2010 06:10 What's the definition of parallel lines in Euclidean three space? Two coplanar lines that don't intersect? In Euclidean three space, let the line j be parallel to the line k and k parallel to the line l. Prove j is parallel to l.
From: Greg Neill on 5 May 2010 11:14 William Elliot wrote: > What's the definition of parallel lines in Euclidean three space? > Two coplanar lines that don't intersect? That would include skew lines, no? I would think that there would be several ways to define parallel lines operationally. For example, lines with contant perpendicular distance between each other over their entire lengths, or lines with the same direction vectors, or lines that are linear translations (one can laid atop the other by a displacement that does not involve rotation). > > In Euclidean three space, let the line j be parallel to the line k > and k parallel to the line l. Prove j is parallel to l.
From: Hero on 5 May 2010 15:10 On 5 Mai, 12:10, William Elliot <ma...(a)rdrop.remove.com> wrote: > What's the definition of parallel lines in Euclidean three space? > Two coplanar lines that don't intersect? > > In Euclidean three space, let the line j be parallel to the line k > and k parallel to the line l. Prove j is parallel to l. Euclid Book I, Definition 23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html Some translate "infinitely" in place of "indefinitely". Anyhow, this is 3d, isn't it? When You choose straight lines, which are not in the same plane, You don't get parallels. With friendly greetings Hero
From: Hero on 5 May 2010 15:17 On 5 Mai, 12:10, William Elliot <ma...(a)rdrop.remove.com> wrote: > What's the definition of parallel lines in Euclidean three space? > Two coplanar lines that don't intersect? > > In Euclidean three space, let the line j be parallel to the line k > and k parallel to the line l. Prove j is parallel to l. Euclid, Book XI, Proposition 9 Proposition 9 Straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other. That will cover half of the answer. With friendly greetings Hero
From: Gerry Myerson on 5 May 2010 19:42
In article <bXfEn.41958$nI2.13354(a)unlimited.newshosting.com>, "Greg Neill" <gneillRE(a)MOVEsympatico.ca> wrote: > William Elliot wrote: > > What's the definition of parallel lines in Euclidean three space? > > Two coplanar lines that don't intersect? > > That would include skew lines, no? No. Skew lines are (by definition) not coplanar. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |