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(** *                         Rotation                            *)
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Definition xr (x y theta:R) : R := x * cos theta + y * sin theta.
Definition yr (x y theta:R) : R := - x * sin theta + y * cos theta.

(* Specifications: *)
Lemma rotation_0 : forall x y:R, xr x y 0 = x /\ yr x y 0 = y.
Lemma rotation_PI2 :
forall x y:R, xr x y (PI / 2) = y /\ yr x y (PI / 2) = - x.
Lemma isometric_rotation_0 :
forall x1 y1 x2 y2 theta:R,
Rsqr (x1 - x2) + Rsqr (y1 - y2) =
Rsqr (xr x1 y1 theta - xr x2 y2 theta) +
Rsqr (yr x1 y1 theta - yr x2 y2 theta).
Lemma isometric_rotation :
forall x1 y1 x2 y2 theta:R,
dist_euc x1 y1 x2 y2 =
dist_euc (xr x1 y1 theta) (yr x1 y1 theta) (xr x2 y2 theta)
(yr x2 y2 theta).