Changing the angle of the vertical and horizontal axis of Sine curve

Question

Suppose we have $f(x)=\sin{(x)}$

The plot of $f(x)$ :

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Suppose I want to modify the graph by :

1. Changing the angle of horizontal axis: (Plane rotation...so both horizontal and the vertical axis changes)

How to achieve this? I mean the whole graph will look tilted say at an angle $\theta$...

2. Changing the angle of vertical axis: (I guess it is called vertical axis!)

Again, how to achieve this? I mean the whole graph will kind of look italicized at an angle $\phi$...[Keeping the horizontal axis same]

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Hence, my question is how to achieve this rotation and tilt of $\sin(x)$.By 'achieve' I mean the new equations. I guess there will be a general formula too for almost all $f(x)$.

EDIT

Also what will happen to the 1st question if the $z$-axis is involved (Revolution of that graph). That is how can i also revolve the curve along z axis?

BTW please give the new "equations" too along with "how-to-do" coz some examples help also along with the hints

P.S. - I know that after these transformations the curve may not remain a function but i still want the equations please!

Thanks

Answer 1

Only addressing Q#2, italicizing:

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The result above was achieved by multiplying by the shear matrix below, with $\phi=30^\circ$: $$ M = \left( \begin{array}{ccc} 1 & \tan (\phi ) \\ 0 & 1 \\ \end{array} \right) $$ For $\phi=60^\circ$:

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If you parametrize the sine-curve as $(t, \sin (t))$, then multiplying by the shear matrix results in $$ M \cdot (\; t,\; \sin (t) \;) = \left( \; t + \sin (t) \tan (\phi ), \; \sin(t) \; \right) \;. $$

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This figure from Wikipedia (employing homogenous coordinates, and so $3 \times 3$ matrices) may help. I used the "Shear in the $x$-direction" matrix:

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Answer 2

There is a method to rotate a graph, that is, using the general rotation matrix, however, there is no general method, lets say $M$, such that $M(f(x))=f(x)_\phi$, since a counterexample holds, assuming $f(x)$ is not monotonic on interval $[a,b]$:

Say that $f(x_1)=f(x_2) = y_1$ when $x_1 < x_2$, a tilt of 90° will guarantee that $f(y_1)_\phi = x_1 = x_2$, so $f_\phi$ is not a function.