Transformation

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.

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(a)

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T_{1} : (x, y) \mapsto (y, x) .

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(b)

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T_{2} : (x, y) \mapsto (x / 2, y / 2) .

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(c)

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T_{3} : (x, y) \mapsto (1 - x^{3}, 1 - y^{3}) .

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(d)

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T_{4} : (x, y) \mapsto (\sqrt{x^{2} + y^{2}} x, \sqrt{x^{2} + y^{2}} y) .

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Subsection 4.1.2 Isometry

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Definition 4.1.4. Isometry.

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A transformation is an isometry if and only if

‖ P - Q ‖ = ‖ T (P) - T (Q) ‖ .

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Checkpoint 4.1.5.

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Determine which of the following transformations are isometries.

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(a)

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T_{1} : (x, y) \mapsto (2 x, 2 y)

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(b)

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T_{2} : (x, y) \mapsto (- y, - x)

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(c)

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T_{3} : (x, y) \mapsto [\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]

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Lemma 4.1.6.

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The composition of two isometries is an isometry.

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Theorem 4.1.7. Isometries preserve colinearity.

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For any isometry

T

A,

B,

and

C

are colinear, then

T (A),

T (B),

and

T (C)

are colinear.

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Corollary 4.1.8. Isometries preserve betweeness.

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For any isometry

T

A - B - C

then

T (A) - T (B) - T (C) .

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Theorem 4.1.9. Isometries preserve triangles.

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For any isometry

T

and any three points

△ A B C ≅ △ T (A) T (B) T (C) .

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Theorem 4.1.10. Isometries preserve angles.

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For any isometry

T

and any three points

m ∠ A B C = m ∠ T (A) T (B) T (C) .

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Theorem 4.1.11. Isometries preserve parallelism.

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For any isometry

T

ℓ_{1} ∥ ℓ_{2}

if and only if

T (ℓ_{1}) ∥ T (ℓ_{2}) .

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Theorem 4.1.12. Isometries preserve circles.

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For any isometry

T

circles are mapped to congruent circles.

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Subsection 4.1.3 Dilations

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Definition 4.1.13. Dilation.

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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

‖ Z Q ‖ / ‖ Z P ‖ = k .

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Definition 4.1.14. Similarity.

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

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Lemma 4.1.15. Similarity scales segments uniformly.

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For any similarity

T,

‖ T (A) T (B) ‖ / ‖ A B ‖ = k .

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Theorem 4.1.16. Similarity preserves colinearity.

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For any similarity

T

A,

B,

and

C

are colinear, then

T (A),

T (B),

and

T (C)

are colinear.

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Corollary 4.1.17. Similarity preserves betweeness.

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For any similarity

T

A - B - C

then

T (A) - T (B) - T (C) .

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Theorem 4.1.18. Similarity preserves triangle similarity.

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For any similarity

T

and any three points

△ A B C \sim △ T (A) T (B) T (C) .

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Theorem 4.1.19. Similarity preserves angles.

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For any similarity

T

and any three points

m ∠ A B C = m ∠ T (A) T (B) T (C) .

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Theorem 4.1.20. Similarity preserves parallelism.

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For any similarity

T

ℓ_{1} ∥ ℓ_{2}

if and only if

T (ℓ_{1}) ∥ T (ℓ_{2}) .

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Theorem 4.1.21.

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For any similarity

T

circles are mapped to circles.

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Lemma 4.1.22. Similarities are closed.

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The composition of two similarities is a similarity.

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