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.
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.
Theorem 4.1.9. Isometries preserve triangles.
Theorem 4.1.10. Isometries preserve angles.
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.
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.
Theorem 4.1.18. Similarity preserves triangle similarity.
Theorem 4.1.19. Similarity preserves angles.
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.
