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Definition of *straight* lines

  1. May 27, 2010 #1
    How is a straight line/straight curve defined (in contexts where it is meaningful) ?

    I am trying to wrap my mind around this concept, but I can't understand it at all : how is straight-ness defined? is it defined ? or should I take it as an axiom, or just think about it at an intuitive level and say "a straight line is something that looks like this _____________" ?

    To clear things up a bit : is there a concrete synthetic definition of straightness that precisely describes something which looks like this ____ (beyond the circular attempts of algebraic/analytic geometry) ?
     
  2. jcsd
  3. May 27, 2010 #2
    If you just care about Euclidean space, you can define a straight line to be the set of all points u+tv where u and v are vectors and t ranges over some interval on the real line.

    Another description (which generalizes arbitrary spaces with a metric) is that a curve between two points is "straight" if its length is minimal among all curves that connect the two points.
     
  4. May 28, 2010 #3
    but going that way, that vector equation "is" a straight line if we draw it in a system formed by some perpendicular "straight" lines... like I said, using algebraic geometry seems a little like going in circles to me... that equation doesn't characterize straightness, only some abstract 1-dimensional affine plane corresponding to a certain vector space...

    yeah, but which metric of which space... and on this side we again return to the same issue as above: you need to represent that curve inside a system formed by perpendicular straight lines to get what we want... not to mention that it's again a pretty detached and abstract notion.

    I'm interested about this thing at an elementary level of sorts... we use "straight" lines all the time in math(geometry) but I never read anywhere about a definition for "straight" line...

    sorry if I'm being a pain in the ***.
     
  5. May 28, 2010 #4
    I think your issue is in going from mathematics to a pictorial representation. Mathematically, I don't think there is anything vague in saying that a straight line in R^n (and we understand R^n as a topological vector space) is a set of points of the form u + tv. This agrees with our intuitive understanding: if we think of R^n as having coordinate axes formed by straight lines (they don't even have to be perpendicular), and plot the points accordingly, we get something that looks like _____ . Of course, we could draw the coordinate axes in some weird way and what we would get wouldn't look like ____. But topologically, their the same, and if we introduce the metric dx_1^2+...+dx_n^2, curves of the form u + tv minimize distance.
     
  6. May 28, 2010 #5

    lavinia

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    I think historically the concept of straightness was taken to be intuitively obvious, something that we can see with our mind's eye. The same would be true of circular or any other geometric shape.

    Mathematically, is was finally realized that a path "looks straight" if an object that moves along it at constant speed has no acceleration or what is the same physically, feels no forces.
    This is the idea of a geodesic. The key insight is that geodesics may not actually be Euclidean straight lines but they still "look straight."
     
  7. May 29, 2010 #6
    I think eok20 hit the nail on the head with this:

    This notion of "straightness" relies only on a prior notion of "length". So, if you can assign a real number to each one-dimensional connected set of points, then you have enough to define straightness.
     
  8. May 29, 2010 #7
    Curvature of metric spaces is defined: given a curve (a continuous function from an interval to a metric space X) [tex]\gamma : [a, b] \rightarrow X[/tex], choose some arbitrary [tex]p \in X[/tex] and choose two points [tex]x, y \in \mathbb{R}^n[/tex] that are the same distance from one another as [tex]\gamma(a)[/tex] and [tex]\gamma(b)[/tex].

    Next choose a point [tex]p_0 \in \mathbb{R}^n[/tex] that is the same distance from each of [tex]x, y[/tex] as [tex]p[/tex] is from each of [tex]\gamma(a),\ \gamma(b)[/tex] respectively (i.e.,

    [tex]\Vert x - p_0) \Vert = distance(\gamma(a),\ p)[/tex], and
    [tex]\Vert y - p_0 \Vert = distance(\gamma(b),\ p)[/tex] ).

    Then you can consider points along the curve in the metric space and compare them to points along the straight line between the two points in Euclidean space to see how your curve compares.

    A few more assumptions I left out are required (in particular about how [tex]\gamma[/tex] is parameterized, and the existence of the other points), but the gist of it is you can measure curvature in a general setting by comparing paths (particularly "shortest paths", but again that requires more assumptions) in said space to "similar" paths in Euclidean space. Try googling/looking up info about length spaces, comparison geometry and geodesics.
     
    Last edited: May 29, 2010
  9. May 31, 2010 #8

    lavinia

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    not on the head. The reason is that a straight line does not need to be the shortest line. The idea of straightness is different - it is the idea of a geodesic.

    Further, in some spaces there may be no shortest line between two points.

    Mathematically, it is possible to define straightness without any notion of distance though in Levi Cevita connections, the connection is compatible with a Riemannian metric on the tangent space.
     
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