Transformations#
in \(\mathbb{R}^2\)
Revised
02 Apr 2023
Transformation#
A transformation \(T\) is just a parameterization for a region.
A function \(f\) on the region is also transformed.
Given a region that is difficult to integrate, find a transformation \(T\) to get a region that is easier to integrate and keep track of how transformation \(T\) affects the size of the (infinitesimal) pixels in \(\mathbb{R}^2\), voxels in \(\mathbb{R}^3\), or, in general, gridboxes (i.e., keep track of the Jacobian!).
Polar#
Cylindrical#
Spherical#
Affine Transformations#
Translation#
\(2\times2\) Translation#
\( \boxed{ \begin{aligned} T&:(u,v)\mapsto(x,y) \\ T^{-1}&:(x,y)\mapsto(u,v) \\ T(u,v)&=(u+2,v+2) \\ &=(x(u,v),y(u,v)) &&\text{transformation} \\ \hline \\ x(u,v)&=u+2 \\ y(u,v)&=v+2 &&\text{transformation equations} \\ \hline \\ dx\,dy&=du\,dv &&\text{Jacobian} \end{aligned} } \)
\( \begin{aligned} T_{\begin{bmatrix}2\\2\end{bmatrix}}\left(\begin{bmatrix}u\\ v\\1\end{bmatrix}\right)= \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} u\\ v\\1 \end{bmatrix} =\begin{bmatrix} u+2\\v+2\\1 \end{bmatrix} =\begin{bmatrix}x\\ y\\1\end{bmatrix} \end{aligned} \)
Pythagorean Identity#
Under Construction
\( \begin{aligned} x^2+y^2 &=(u+2)^2+(v+2)^2 \\ &=u^2+4u+4+v^2+4v+4 \\ &=u^2+v^2+4(u+v+2) \sqrt{x^2+y^2} \\ &=\sqrt{u^2+v^2+4(u+v+2)} \end{aligned} \)
Jacobian#
\( \begin{aligned} \frac{\partial(x,y)}{\partial(u,v)} =\begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\\\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{vmatrix} =\begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} =1 \end{aligned} \)
\(dx\,dy=du\,dv\)
Translations don’t distort pixels.
Example#
\( \begin{aligned} \underset{R}{\iint}(3x+2y)\,dA \,\,\,\text{where}\,\,\, R \,\,\,\text{is the region bounded by the triangle whose vertices are}\,\,\, (2,1),(4,2),(2,3) \end{aligned} \)
Find the new bounds.
\( \begin{aligned} v=1-\frac{1}{2}u \\ v=\frac{1}{2}u-1 \end{aligned} \)
\( \begin{aligned} \underset{R}{\iint}(3x+2y)\,dx\,dy &=\underset{R}{\iint}3(u+2)+2(v+2)\,du\,dv =\underset{R}{\iint}3u+2v+10\,du\,dv \\ &=\int_{u=0}^{u=2}\int_{v=\frac{1}{2}u-1}^{v=1-\frac{1}{2}u}3u+2v+10\,dv\,du \end{aligned} \)
Linear Transformations#
Shear#
Squeeze (Hyperbolic Rotation)#
\( \begin{aligned} T\left(\begin{bmatrix}u\\ v\end{bmatrix}\right)= \begin{bmatrix} k & 0 \\ 0 & \frac{1}{k} \end{bmatrix} \begin{bmatrix}u\\ v\end{bmatrix} =\begin{bmatrix}ku\\ \frac{1}{k}v\end{bmatrix} \end{aligned} \)
\( \begin{aligned} T(\mathbf{u})=\mathbf{Au=x} \end{aligned} \)
Stretch by a factor \(k\) along the \(x\)-axis#
\( \begin{aligned} T\left(\begin{bmatrix}u\\ v\end{bmatrix}\right)= \begin{bmatrix} k & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix}u\\ v\end{bmatrix} =\begin{bmatrix}ku\\ v\end{bmatrix} \end{aligned} \)
Stretch by a factor \(k\) along the \(y\)-axis#
\( \begin{aligned} T\left(\begin{bmatrix}u\\ v\end{bmatrix}\right)= \begin{bmatrix} 1 & 0 \\ 0 & k \end{bmatrix} \begin{bmatrix}u\\ v\end{bmatrix} =\begin{bmatrix}u\\ kv\end{bmatrix} \end{aligned} \)
Stretch by a factor \(k_1\) along the \(x\)-axis and a factor \(k_2\) along the \(y\)-axis#
\( \begin{aligned} T\left(\begin{bmatrix}u\\ v\end{bmatrix}\right)= \begin{bmatrix} k_1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & k_2 \end{bmatrix} \begin{bmatrix}u\\ v\end{bmatrix} =\begin{bmatrix} k_1 & 0 \\ 0 & k_2 \end{bmatrix} \begin{bmatrix}u\\ v\end{bmatrix} =\begin{bmatrix}k_1u\\ k_2v\end{bmatrix} \end{aligned} \)
Rotation#
Terms#
[W] Affine Transformation
[W] Geometric Transformation
[W] Linear Transformation
[W] Reflection
[W] Rotation
[W] Scaling
[W] Shearing
[W] Squeezing
[W] Transformation
[W] Translation