Definition 4.1.1. Transformation.
A function is a transformation if and only if it is one-to-one and onto.
Section 4.1 Transformation
If you see this statement, remind the instructor to revise this section to have more play before the proofs.
Subsection 4.1.1 Planar Transformations
Definition 4.1.2. Planar Transformation.
A transformation is a planar transformation if and only if it is from \(\R^2\) to \(\R^2.\)
For this course all transformations will be transformations of the Euclidean plane.
Checkpoint 4.1.3.
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.
(a)
\(T_1:(x,y) \mapsto (y,x)\text{.}\)
(b)
\(T_2:(x,y) \mapsto (x/2,y/2)\text{.}\)
(c)
\(T_3:(x,y) \mapsto (1-x^3,1-y^3)\text{.}\)
(d)
\(T_4:(x,y) \mapsto (\sqrt{x^2+y^2}x,\sqrt{x^2+y^2}y)\text{.}\)
Subsection 4.1.2 Isometry
Definition 4.1.4. Isometry.
A transformation is an isometry if and only if \(\|P-Q\|=\|T(P)-T(Q)\|.\)
Checkpoint 4.1.5.
Determine which of the following transformations are isometries.
(a)
\(T_1:(x,y) \mapsto (2x,2y)\)
(b)
\(T_2:(x,y) \mapsto (-y,-x)\)
(c)
\(T_3:(x,y) \mapsto \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \)
Lemma 4.1.6.
The composition of two isometries is an isometry.
Theorem 4.1.7. 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 4.1.8. Isometries preserve betweeness.
For any isometry \(T\) if \(A-B-C\) then \(T(A)-T(B)-T(C).\)
Theorem 4.1.9. Isometries preserve triangles.
For any isometry \(T\) and any three points \(\triangle ABC \cong \triangle T(A)T(B)T(C).\)
Theorem 4.1.10. Isometries preserve angles.
For any isometry \(T\) and any three points \(m\angle ABC = m\angle T(A)T(B)T(C).\)
Theorem 4.1.11. Isometries preserve parallelism.
For any isometry \(T\) \(\ell_1 \parallel \ell_2\) if and only if \(T(\ell_1) \parallel T(\ell_2).\)
Theorem 4.1.12. Isometries preserve circles.
For any isometry \(T\) circles are mapped to congruent circles.
Subsection 4.1.3 Dilations
Definition 4.1.13. 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 \(\|ZQ\|/\|ZP\|=k.\)
Definition 4.1.14. Similarity.
A transformation is a similarity if and only if it can be expressed as a composition of an isometry and a dilation.
Lemma 4.1.15. Similarity scales segments uniformly.
For any similarity \(T,\) \(\|T(A)T(B)\|/\|AB\| =k.\)
Theorem 4.1.16. 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 4.1.17. Similarity preserves betweeness.
For any similarity \(T\) if \(A-B-C\) then \(T(A)-T(B)-T(C).\)
Theorem 4.1.18. Similarity preserves triangle similarity.
For any similarity \(T\) and any three points \(\triangle ABC \sim \triangle T(A)T(B)T(C).\)
Theorem 4.1.19. Similarity preserves angles.
For any similarity \(T\) and any three points \(m\angle ABC = m\angle T(A)T(B)T(C).\)
Theorem 4.1.20. Similarity preserves parallelism.
For any similarity \(T\) \(\ell_1 \parallel \ell_2\) if and only if \(T(\ell_1) \parallel T(\ell_2).\)
Theorem 4.1.21.
For any similarity \(T\) circles are mapped to circles.
Lemma 4.1.22. Similarities are closed.
The composition of two similarities is a similarity.
