# Edit Create an Image Page: Change Of Coordinate Transformations

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 Image Title*: Upload a Math Image An example of various coordinate transformations applied to simple geometry. A Change Of Coordinate Transformation is a [[Transformation Matrix|transformation]] that converts coordinates from one coordinate system to another coordinate system. Transformations such as scaling, rotating, and translating are usually looked upon as changing or manipulating the geometry itself. However, with change of coordinate transformations, it is important to realize that the coordinate representation of the geometry is modified, rather than the geometry itself. The [[Change of Coordinate Systems]] in general is also common for converting coordinates from system to another, such as from Cartesian coordinates to Cylindrical coordinates. Change of coordinate transformations are different for [[Vector|vectors]] and points. ===Vectors=== [[Image:change_of_coords_xformation_vector.png|420px|thumb|Transformation of vector $\boldsymbol{\vec{p}}$ from coordinate system '''A''' to coordinate system '''B'''.]] :Consider a coordinate system '''A''', and a vector $\boldsymbol{\vec{p}}$. The coordinates of $\boldsymbol{\vec{p}}$ relative to coordinate system '''A''' is $\boldsymbol{\vec{p}}_A = (x, y)$. It is also apparent that ::$\boldsymbol{\vec{p}} = x\boldsymbol{\hat{u}} + y\boldsymbol{\hat{v}}$ :In which $\boldsymbol{\hat{u}}$ and $\boldsymbol{\hat{v}}$ are unit vectors along the x and y-axes of coordinate system '''A'''. Now consider a second coordinate system, '''B'''. In coordinate system '''B''', ::$\boldsymbol{\vec{p}}_B = x\boldsymbol{\hat{u}}_B + y\boldsymbol{\hat{v}}_B$ :More generally, given $\boldsymbol{\vec{p}}_A = (x, y)$ along with $\boldsymbol{\hat{u}}$ and $\boldsymbol{\hat{v}}$, $\boldsymbol{\vec{p}}_B = (x', y')$ may be found using the formula above. :In 3-dimensional space, given $\boldsymbol{\vec{p}}_A = (x, y, z)$ then ::$\boldsymbol{\vec{p}}_B = x\boldsymbol{\hat{u}}_B + y\boldsymbol{\hat{v}}_B + z\boldsymbol{\hat{w}}_B$ :In which $\boldsymbol{\hat{u}}$, $\boldsymbol{\hat{v}}$, and $\boldsymbol{\hat{w}}$ are unit vectors along the x, y, and z-axes of coordinate system '''A'''. :{{HideThis| 1=Mathematical Example|2= ::'''Given:''' :::$\boldsymbol{\vec{p}}_A = \begin{bmatrix} 5 & 7 & 13 \end{bmatrix}$, $\boldsymbol{\hat{u}}_B = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$, $\boldsymbol{\hat{v}}_B = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$, and $\boldsymbol{\hat{w}}_B = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$ ::'''Then:''' :::$\boldsymbol{\vec{p}}_B = \begin{bmatrix} 5 & 7 & 0 \end{bmatrix}$ ::'''Explanation:''' :::The formula, $\boldsymbol{\vec{p}}_B = x\boldsymbol{\hat{u}}_B + y\boldsymbol{\hat{v}}_B + z\boldsymbol{\hat{w}}_B$, was used for the change of coordinate transformation. The [[dot product]] of 5 and $\boldsymbol{\hat{u}}_B$ is 5, and the dot product of 7 and $\boldsymbol{\hat{v}}_B$ is 7. Since $\boldsymbol{\hat{w}}_B$ a zero vector, the dot product of 13 and $\boldsymbol{\hat{w}}_B$ is 0. }} ===Points=== [[Image:change_of_coords_xformation_point_final.png|420px|thumb|Transformation of point$\boldsymbol{q}$ from coordinate system '''A''' to coordinate system '''B'''.]] :Consider a coordinate system '''A''', and a point $\boldsymbol{q}$. Point $\boldsymbol{q}$ may be expressed as: ::$\boldsymbol{q} = x\boldsymbol{\hat{u}} + y\boldsymbol{\hat{v}} + \boldsymbol{O}$ :In which $\boldsymbol{\hat{u}}$ and $\boldsymbol{\hat{v}}$ are unit vectors along the x and y-axes of coordinate system '''A''', and $\boldsymbol{O}$ is the origin of coordinate system '''A'''. Now consider a second coordinate system, '''B'''. In coordinate system '''B''', ::$\boldsymbol{q}_B = x\boldsymbol{\hat{u}}_B + y\boldsymbol{\hat{v}}_B + \boldsymbol{O}_B$ :More generally, given $\boldsymbol{q}_A = (x, y)$ along with $\boldsymbol{\hat{u}}$, $\boldsymbol{\hat{v}}$, and $\boldsymbol{O}$ relative to coordinate system '''B''', then $\boldsymbol{q}_B = (x', y')$ may be found using the above formula. :In 3-dimensional space, given $\boldsymbol{q}_A = (x, y, z)$ then ::$\boldsymbol{q}_B = x\boldsymbol{\hat{u}}_B + y\boldsymbol{\hat{v}}_B + z\boldsymbol{\hat{w}}_B + \boldsymbol{O}_B$ :In which $\boldsymbol{\hat{u}}$, $\boldsymbol{\hat{v}}$, and $\boldsymbol{\hat{w}}$ are unit vectors along the x, y, and z-axes of coordinate system '''A''', and $\boldsymbol{O}$ is the origin of coordinate system '''A'''. :{{HideThis| 1=Mathematical Example|2= ::'''Given:''' :::$\boldsymbol{q}_A = \begin{bmatrix} 6 & 4 & 21\end{bmatrix}$, $\boldsymbol{\hat{u}}_B = \begin{bmatrix} 0 & 1 & 0 \end{bmatrix}$, $\boldsymbol{\hat{v}}_B = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$, $\boldsymbol{\hat{w}}_B = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$, and $\boldsymbol{O}_B = 17$ ::'''Then:''' :::$\boldsymbol{q}_B = \begin{bmatrix} 23 & 21 & 38 \end{bmatrix}$ ::'''Explanation:''' :::The formula, $\boldsymbol{q}_B = x\boldsymbol{\hat{u}}_B + y\boldsymbol{\hat{v}}_B + z\boldsymbol{\hat{w}}_B + \boldsymbol{O}_B$, was used for the change of coordinate transformation. The dot product of 6 and $\boldsymbol{\hat{u}}_B$ is 6, the dot product of 4 and $\boldsymbol{\hat{v}}_B$ is 4, and the dot product of 21 and $\boldsymbol{\hat{w}}_B$ is 21. $\boldsymbol{O}_B$, or 17, is then added to the resulting vector. }} ===Matrix Representation=== :The change of coordinate transformation varies for points and vectors and thus results in two different equations. However, by using homogeneous coordinates, both cases may be handled with the following equation: ::$(x', y', z', w) = x\boldsymbol{u}_B + y\boldsymbol{v}_B + z\boldsymbol{w}_B + w\boldsymbol{O}_B$ :When $w = 1$, the equation handles the change of coordinate transformation for points; when $w = 0$, the equation handles the transformation for vectors. As long as the $w$-coordinate is set correctly, there is no need to keep track of two different equations. Thus the change of coordinate matrix may be defined as: ::$\begin{bmatrix} x' & y' & z' & w \end{bmatrix} = \begin{bmatrix} x & y & z & w \end{bmatrix} \begin{bmatrix} u_x & u_y & u_z & 0 \\ v_x & v_y & v_z & 0 \\ w_x & w_y & w_z & 0 \\ O_x & O_y & O_z & 1 \\ \end{bmatrix} = x\boldsymbol{\hat{u}}_B + y\boldsymbol{\hat{v}}_B + z\boldsymbol{\hat{w}}_B + w\boldsymbol{O}_B$ :[[Matrix|Click here]] for an overview of matrix multiplication. Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Probability Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Probability Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Probability Topology Other :*[http://developer.apple.com/mac/library/documentation/Cocoa/Conceptual/CocoaDrawingGuide/Transforms/Transforms.html Coordinate Systems and Transforms] Luna, Frank D. Introduction to 3D game programming with DirectX 10. Plano, Tex: Wordware Pub., Inc., 2008. Print. Yes, it is.