# Non-Linear Reflections? {moved from calculus for assumed better relevance}

• Jun 9th 2010, 07:34 AM
mfetch22
Non-Linear Reflections? {moved from calculus for assumed better relevance}
Would this be considered a pluasible definition of a non-linear reflection? :

Say I have the line $g(x)$ and I want to reflect it over the parabola $f (x)$, and call the resulting graph $R(x)$. At an arbitrary point $x$ on $f (x)$ I construct a line normal to $f (x)$ and call it $N(x)$. I determine the point on $P$ on $g(x)$ at which $N(x)$ intersects $g(x)$. Then I construct a line tangent to $f (x)$ at $x$ and call it $T(x)$. Finnally, I reflect every point $( P, g(P) )$ over every coresponding line $T(x)$ to arrive at $R(x)$. This is just a thought I had, I dont think it has any applications, although the one thing it reminds me of is looking into the curved parabolic mirrors at the grocery store that help you see around corners, it takes a straight line frame of referance and "squeezes" is outward. Anyway I picked two random equations for $f (x)$ and $g(x)$ and graphed the reflection defined above. I'll attach an image of it to this post.
• Jun 9th 2010, 04:59 PM
math2009
fantastic problem, distorting mirror
• Jun 9th 2010, 08:35 PM
math2009
fantastic problem, distorting mirror

if g(x) is straight line, reflection image is rotation

$T(\begin{bmatrix}x\\f(x)\end{bmatrix})=\begin{bmat rix}cos\theta &-sin\theta \\sin\theta &cos\theta \end{bmatrix}\begin{bmatrix}x\\f(x)\end{bmatrix}$

$tan\frac{\theta}{2}=\frac{d}{dx}g(x)$

Otherwise, apply project , see examples in attchments
• Jun 10th 2010, 05:30 PM
math2009
It updated
• Jun 10th 2010, 08:06 PM
mfetch22
Quote:

Originally Posted by math2009
fantastic problem, distorting mirror

if g(x) is straight line, reflection image is rotation

$T(\begin{bmatrix}x\\f(x)\end{bmatrix})=\begin{bmat rix}cos\theta &-sin\theta \\sin\theta &cos\theta \end{bmatrix}\begin{bmatrix}x\\f(x)\end{bmatrix}$

$tan\frac{\theta}{2}=\frac{d}{dx}g(x)$

Otherwise, apply project , see examples in attchments

Did you generate these reflections? They are absolutely fantastic! I considered looking at reflecting equations other then lines but I never took the time to generate the reflections! I love the trig function ones. Does the transformation matrix you posted above, when multiplied by the vector of $f(x)$ and $x$ return these reflections for curved equations of $g(x)$ also? Its so much simpler then my lengthy method. I knew transformation matricies could be used to rotate lines about the origin but I didnt know it could be applied to an explicit function. Great images in your attachments.
• Jun 10th 2010, 08:20 PM
math2009
For arbitrary function g(x), it doesn't work by simple matrix transformation.
As above mention, it need projection transformation