what is the parametric function of the new Bezier curve?

Question

The cubic Bezier curve can be given in matrix form as

If a cubic Bezier curve is rotated by an angle 30 around x-axis what is the parametric function of the new Bezier curve?

Answer 1

It seems that you are starting with the curve parameterized by $$ c(t) = \pmatrix{t\\ P(t)} $$ In order to rotate a parameterized curve about the origin, apply the rotation matrix. That is, we multiply $c(t)$ on the left by the matrix $$ R_{30^\circ} = \pmatrix{ \cos(30^\circ) & -\sin(30^\circ)\\ \sin (30^\circ) & \cos(30^\circ) } = \frac 12 \pmatrix{ \sqrt 3 & -1\\ 1 & \sqrt 3 } $$

Answer 2

To get the parametric equation of the rotated curve, you just use the rotated control points. So, you rotate the original four control points using the rotation matrix given by omnomnomnom, to get four new control points. Plugging these four rotated control points into your matrix equation will give the equation of the rotated curve.

If this is a homework question, I suspect that it's meant to illustrate this idea -- to rotate a Bezier curve (or transform it using any affine transformation), you simply have to apply this transformation to the control points. This is a very nice property of Bezier curves.