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

• TheRookie
In summary, r-similitudes, or r-reflections, are a type of linear transformation in Linear Algebra that involve reflecting a vector or point across an r-dimensional subspace. They differ from other linear transformations in that they specifically involve reflection, making them useful for describing symmetries in higher-dimensional spaces. Real-life examples of r-similitudes include the reflection of light off of a mirror. In computer graphics, r-similitudes are commonly used to create realistic 3D images. They are significant in Linear Algebra as they allow for the description and understanding of symmetries in higher-dimensional spaces, and have practical applications in fields such as computer graphics and physics.
TheRookie
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?

## 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

--

(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

I think we are doing the same project. I've used this:

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

## 1. What are r-similitudes in Linear Algebra?

R-Similitudes, also known as r-reflections, are a type of linear transformation in Linear Algebra that involves reflecting a vector or point across an r-dimensional subspace. This transformation is defined by an r x r matrix and can be used to describe rotations, translations, and scaling in higher dimensions.

## 2. How do r-similitudes differ from other linear transformations?

R-similitudes are unique in that they involve reflecting a point or vector across an r-dimensional subspace, rather than just rotating, translating, or scaling it. This makes them useful for describing symmetries in higher-dimensional spaces.

## 3. Can you provide an example of r-similitudes in real life?

One example of r-similitudes in real life is the reflection of light off of a mirror. When light rays hit the mirror, they are reflected across the mirror's surface, which can be represented as an r-similitude transformation in Linear Algebra.

## 4. How are r-similitudes used in computer graphics?

R-similitudes are commonly used in computer graphics to create realistic 3D images. By applying r-similitude transformations to points or vectors representing objects in a 3D space, programmers can create accurate reflections and symmetries in their images.

## 5. What is the significance of r-similitudes in Linear Algebra?

R-similitudes are important in Linear Algebra because they allow us to describe and understand symmetries in higher-dimensional spaces. They also have practical applications in fields such as computer graphics and physics, where the concept of reflection is commonly used.

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