straight line is shortest curve between two points
Suppose and are two distinct points in , and is a rectifiable curve from to . Then every curve other than the straight line segment from to has a length greater than the Euclidean distance .
Proof.
Let
be the curve with length .
If it is not straight11If is a straight line segment but is not injective, that is, it moves and , then it is obvious that ., then there exists a point that does not lie on the line segment
from to .
We have
The first inequality comes from the definition of as the least upper bound of the length of any broken-line approximation to the curve .
The second inequality is the usual triangle inequality
,
but it is a strict inequality since lies outside the line segment between and ,
as shown in the following diagram.
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Title | straight line is shortest curve between two points |
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Canonical name | StraightLineIsShortestCurveBetweenTwoPoints |
Date of creation | 2013-03-22 15:39:43 |
Last modified on | 2013-03-22 15:39:43 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 11 |
Author | stevecheng (10074) |
Entry type | Result |
Classification | msc 51N05 |
Related topic | ArcLength |
Related topic | Rectifiable |