Suppose we have $f(x)=\sin{(x)}$
The plot of $f(x)$ :
Suppose I want to modify the graph by :
- 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 $\theta$…
- 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]
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
Best Answer
Only addressing Q#2, italicizing:
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 $\phi=60^\circ$:
If you parametrize the sine-curve as $(t, \sin (t))$, then multiplying by the shear matrix results in $$ M \cdot (\; t,\; \sin (t) \;) = \left( \; t + \sin (t) \tan (\phi ), \; \sin(t) \; \right) \;. $$
This figure from Wikipedia (employing homogenous coordinates, and so $3 \times 3$ matrices) may help. I used the "Shear in the $x$-direction" matrix: