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TheRookie

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

## Homework Statement

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?

## Homework Equations

## 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

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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