4.1: Transformation
- Page ID
- 89852
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)4.1.1 Planar Transformations
A function is a transformation if and only if it is one-to-one and onto.
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
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)→ \(\begin{bmatrix} cosθ & sinθ \\ -sinθ & cosθ \end{bmatrix}\)\(\begin{bmatrix} x \\ y \end{bmatrix}\)
The composition of two isometries is an isometry.
For any isometry T if A, B and C are colinear, then T(A), T(B) and T(C) are colinear.
For any isometry T if A-B-C then T(A)-T(B)-T(C).
For any isometry T and any three points △ABC ≌ △T(A)T(B)T(C).
For any isometry T m∠ABC = m∠T(A)T(B)T(C) and any three points.
For any isometry if T ℓ1 ll ℓ2 and only if T(ℓ1) ll T(ℓ2).
For any isometry T circles are mapped to congruent circles.
4.1.3 Dilations
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.
A transformation is a similarity if and only if it can be expressed as a composition of an isometry and a dilation.
For any similarity T, llT(A)T(B)ll/llABll=k.
For any similarity T if A, B, and C are colinear, then T(A), T(B), and T(C) are colinear.
For any similarity T if then A-B-C then T(A)-T(B)-T(C).
For any similarity T and any three points △ABC∽△T(A)T(B)T(C).
T For any similarity and any three points ∠mABC∽ ∠ mT(A)T(B)T(C).
For any similarity T ℓ1 ll ℓ2 if and only if T(ℓ1) ll T(ℓ2).
For any similarity T circles are mapped to circles.
The composition of two similarities is a similarity.