4.1: Transformation

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4.1.1 Planar Transformations

Definition: Transformation

A function is a transformation if and only if it is one-to-one and onto.

Definition: Planar Transformation

A transformation is a planar transformation if and only if it is from ℝ² to ℝ².

For this course all transformations will be transformations of the Euclidean plane.

Describe the effect of each of the following transformations by considering the effects on the region (area) with the following vertices. (0,0), (2,0), (3,5), (0,4). Hint: map the vertices, then map the lines connecting the vertices.

T₁: (x, y) → (y, x).
T₂: (x, y) → (x/2, y/2).
T₃: (x, y) → (1-x³, 1-y³)
T₄: (x, y) → ((x²+y²)^1/2x, (x²+y²)^1/2y).

4.1.2 Isometry

Definition: Isometry

A transformation is an isometry if and only if ll P - Q ll = ll T(P) - T(Q) ll.

Determine which of the following transformations are isometries.

T₁: (x, y)→ (2x, 2y)
T₁: (x, y)→ (-y, -x)
T₁: (x, y)→ \(\begin{bmatrix} cosθ & sinθ \\ -sinθ & cosθ \end{bmatrix}\)\(\begin{bmatrix} x \\ y \end{bmatrix}\)

Lemma

The composition of two isometries is an isometry.

Theorem: Isometries preserve colinearity

For any isometry T if A, B and C are colinear, then T(A), T(B) and T(C) are colinear.

Corollary: Isometries preserve betweeness

For any isometry T if A-B-C then T(A)-T(B)-T(C).

Theorem: Isometries preserve triangles

For any isometry T and any three points △ABC ≌ △T(A)T(B)T(C).

Theorem: Isometries preserve angles

For any isometry T m∠ABC = m∠T(A)T(B)T(C) and any three points.

Theorem: Isometries preserve parallelism

For any isometry if T ℓ₁ ll ℓ₂ and only if T(ℓ₁) ll T(ℓ₂).

Theorem: Isometries preserve circles

For any isometry T circles are mapped to congruent circles.

4.1.3 Dilations

Definition: Dilation

A transformation is a dilation if and only if it can be defined by a point Z and a ratio k such that T(P)=Q where Z-P-Q and llZQll/llZPll=k.

Definition: Similarity

A transformation is a similarity if and only if it can be expressed as a composition of an isometry and a dilation.

Lemma: Similarity scales segments uniformly

For any similarity T, llT(A)T(B)ll/llABll=k.

Theorem: Similarity preserves colinearity

For any similarity T if A, B, and C are colinear, then T(A), T(B), and T(C) are colinear.

Corollary: Similarity preserves betweeness

For any similarity T if then A-B-C then T(A)-T(B)-T(C).

Theorem: Similarity preserves triangle similarity

For any similarity T and any three points △ABC∽△T(A)T(B)T(C).

Theorem: Similarity preserves angles

T For any similarity and any three points ∠mABC∽ ∠ mT(A)T(B)T(C).

Theorem: Similarity preserves parallelism

For any similarity T ℓ₁ ll ℓ₂ if and only if T(ℓ₁) ll T(ℓ₂).

Theorem

For any similarity T circles are mapped to circles.

Lemma: Similarities are closed

The composition of two similarities is a similarity.