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Help with part of my Linear Algebra project - r-similitudes

  1. Mar 8, 2012 #1
    Help with part of my Linear Algebra project - "r-similitudes"

    1. The problem statement, all variables and given/known data

    Definition: An "r-similitude" on ℝ² is an affine mapping f:ℝ²→ℝ² such that, for all x and y in ℝ², ǁf(x)-f(y)ǁ = rǁx-yǁ (where ǁ·ǁ denotes the Euclidean distance in ℝ²)
    Let ABC be an equilateral triangle such that A=(0, 0) and B=(1, 0)
    Let D,E,F be the midpoints of AB,BC,CA respectively

    Question: Find r-similitudes of ℝ² mapping the triangular region ABC to the separate triangular regions ADF, DBE, FEC. What is the value of r?

    2. Relevant equations


    3. The attempt at a solution

    All points: A=(0, 0), B=(1, 0), C=(1/2, √3/2), D=(1/2, 0), E=(3/4, √3/4) F=(1/4, √3/4)

    For mappings from ABC to such triangular regions:
    the 1-dimensional measure in ℝ² is scaled by a factor r
    the 2-dimensional measure in ℝ² is scaled by a factor r²

    --

    There are some things I don't understand about this:
    (i) How will the mappings to ADF, DBE, FEC be different if these three triangles are the same? Is the direction of the mapping important?
    (ii) How do we use the definition of "r-similitude" in the mapping between regions?
    (iii) How are the scale factors used in the mappings (if at all)?

    Sorry if I seem kind of clueless about all this, but I'm pretty desperate here - I've been stuck with this all week. Any help will be very much appreciated.

    Thanks,
    Pete
     
  2. jcsd
  3. Mar 9, 2012 #2
    Re: Help with part of my Linear Algebra project - "r-similitudes"

    I think we are doing the same project. I've used this:
    http://ecademy.agnesscott.edu/~lriddle/ifs/siertri/siertri.htm

    I think the "r0-similitude f1" (that maps ABC to ADF) is the f1(x) mentioned roughly half way down the page. It is the matrix that is multiplied with the coordinate vector.

    e.g. f1(B) = {{0.5 , 0}, {0, 0.5}}.(1 , 0) = D

    Though the triangles have the same size and dimensions they are however in different places so f2(x) will map ABC to a similar triangle as in f1(x) but it will be in a different position.

    I am ALMOST certain this is correct..

    Tom
     
  4. Mar 12, 2012 #3

    micromass

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    Re: Help with part of my Linear Algebra project - "r-similitudes"

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