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What's the general equation of a curve after a rotation about the origin? Specifically, what's the equation (either piecewise or implicit) of the unit parabola after a counterclockwise $45^{\circ}$ rotation?

I feel the generalised rotation matrix should be all I need but I can't figure out where to go from here :(

Michael Rozenberg
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Sup
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  • What do you do to a basis vector? Rotate it by 45 deg counterclockwise. So $(1,0)$ becomes $(1/\sqrt{2},1/\sqrt{2})$. What does $(0,1)$ become? – Roby5 Mar 25 '17 at 19:02
  • The question is not a duplicate. Sup is asking about and implicit equation describing the rotated graph, and particularily asking about conics! – Olivier Mar 26 '17 at 03:44

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The equation of the usual parabola takes the form $$ 0 = -y + ax^2 + bx +c. $$ Generally, you can describe a curve as the set of all points $(x,y)$ such that $$ F(x,y) = 0, $$ where $F(x,y)$ is a function such as $F(x,y) = -y+ax^2+bx+c$.

Now, suppose you want to rotate the curve by 45 degrees. Denote by $R$ this rotation. The new curve is given by the equation $$ F(R^{-1}(x,y)) = 0. $$ This is because a point $(a,b)$ is on the new curve if and only if $R^{-1}(a,b)$ was on the original curve. (Make sure you understand this).

The counter-clockwise rotation of angle $\theta$ is given by $$R_\theta(x,y) = (x\cos(\theta) -y\sin(\theta), x\sin(\theta) +y\cos(\theta)),$$ and $R^{-1}_\theta = R_{-\theta}$.

Olivier
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