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Math Help - Does transformation P:(r,theta) -> (x=rcos(theta), y=rsin(theta)) preserve angles?

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    Does transformation P:(r,theta) -> (x=rcos(theta), y=rsin(theta)) preserve angles?

    Let P be a transformation from (r,theta) -> (x=rcos(theta), y=rsin(theta)) be a mapping from R^2 to R^2.
    1. Does P preserve angles at the point r=2, theta = pi/4?
    2. By what factor does the map P distort area at the point r=2, theta = pi/4?


    All I know so far is that P is a 2x2 matrix that maps a polar coordinate to it's Cartesian coordinate.

    I'm not sure how to proceed to answer the questions from here. I can't think of a way to construct the matrix of transformation because (r,theta) -> (rcos(theta), rsin(theta)) involves a substitution of theta.
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    Re: Does transformation P:(r,theta) -> (x=rcos(theta), y=rsin(theta)) preserve angles

    Hey Elusive1324.

    What exactly do you mean by preserve angles? Can you explain this using mathematics please?
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    Re: Does transformation P:(r,theta) -> (x=rcos(theta), y=rsin(theta)) preserve angles

    Quote Originally Posted by Elusive1324 View Post
    Let P be a transformation from (r,theta) -> (x=rcos(theta), y=rsin(theta)) be a mapping from R^2 to R^2.
    1. Does P preserve angles at the point r=2, theta = pi/4?
    2. By what factor does the map P distort area at the point r=2, theta = pi/4?


    All I know so far is that P is a 2x2 matrix that maps a polar coordinate to it's Cartesian coordinate.

    I'm not sure how to proceed to answer the questions from here. I can't think of a way to construct the matrix of transformation because (r,theta) -> (rcos(theta), rsin(theta)) involves a substitution of theta.
    I agree with chiro. I not sure I understand what you mean. First of all, a transformation is not a matrix. For any linear transformation T:\mathbb{R}^2 \to \mathbb{R}^2, there exists a 2x2 matrix of real numbers M_T such that for any vector \vec{v} \in \mathbb{R}^2, the transformation T\vec{v} = M_T\vec{v}. In that way, the matrix provides an alternate notation for the transformation, but the matrix itself is not a linear transformation.

    The transformation you are describing is not linear. Your transformation is a mapping from a nonlinear space to a linear space. So, suppose there is a 2x2 matrix that represents P, and let's call it K = \begin{pmatrix}a & b \\ c & d\end{pmatrix}. Then
    K \begin{pmatrix}r \\ \theta \end{pmatrix} = \begin{pmatrix}a & b \\ c & d\end{pmatrix}\begin{pmatrix}r \\ \theta \end{pmatrix} = \begin{pmatrix}ra + b\theta \\ rc + d\theta \end{pmatrix} = \begin{pmatrix}r\cos(\theta) \\ r\sin(\theta) \end{pmatrix} = P\begin{pmatrix}r \\ \theta \end{pmatrix}

    You now have a system of equations. For a set (r,\theta), you can solve for a,b,c,d. But, as you vary (r,\theta) across all of \mathbb{R}^2, you will find that there does not exist any fixed a,b,c,d that satisfies the system of equations for all values. Hence, there does not exist a 2x2 matrix that represents P.

    Transformations, however, do satisfy certain mathematical properties. They are supposed to preserve some algebraic or geometric structure. So, given two vectors (r_1,\theta_1) and (r_2,\theta_2), the angle between them is \theta_1 - \theta_2. Now, send those two vectors through your transformation. What is the angle between the vectors \left( r_1\cos(\theta_1), r_1\sin(\theta_1) \right) and \left( r_2\cos(\theta_2), r_2\sin(\theta_2) \right)? You can use what you know about dot product. You know that \vec{v_1}\cdot \vec{v_2} = \lVert{\vec{v_1}}\rVert \lVert{\vec{v_2}}\rVert \cos(\theta) where \theta is the angle between them. So, solve for \theta. Check your sum and difference of angles formulas for cosine. You will find that the angles are equal. So, you don't need to check at any particular point. It is true for any arbitrary points.

    Next, what vector operations give you area? Dot product gives the area of a parallelogram. The magnitude of the cross product also gives the area of a parallelogram. Between the two of those, you can check all cases. Note that the magnitude of the vector (r,\theta) in polar coordinates is r. Magnitude is calculated differently with polar coordinates than it is with linear coordinates.
    Last edited by SlipEternal; September 26th 2013 at 08:55 AM.
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