Library Coq.Reals.Rgeom

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo.
Require Import R_sqrt. Open Local Scope R_scope.

Definition dist_euc (x0 y0 x1 y1:R) : R :=
  sqrt (Rsqr (x0 - x1) + Rsqr (y0 - y1)).

Lemma distance_refl : forall x0 y0:R, dist_euc x0 y0 x0 y0 = 0.
intros x0 y0; unfold dist_euc in |- *; apply Rsqr_inj;
 [ apply sqrt_positivity; apply Rplus_le_le_0_compat;
    [ apply Rle_0_sqr | apply Rle_0_sqr ]
 | right; reflexivity
 | rewrite Rsqr_0; rewrite Rsqr_sqrt;
    [ unfold Rsqr in |- *; ring
    | apply Rplus_le_le_0_compat; [ apply Rle_0_sqr | apply Rle_0_sqr ] ] ].
Qed.

Lemma distance_symm :
 forall x0 y0 x1 y1:R, dist_euc x0 y0 x1 y1 = dist_euc x1 y1 x0 y0.
intros x0 y0 x1 y1; unfold dist_euc in |- *; apply Rsqr_inj;
 [ apply sqrt_positivity; apply Rplus_le_le_0_compat
 | apply sqrt_positivity; apply Rplus_le_le_0_compat
 | repeat rewrite Rsqr_sqrt;
    [ unfold Rsqr in |- *; ring | apply Rplus_le_le_0_compat | apply Rplus_le_le_0_compat ] ]; apply Rle_0_sqr.
Qed.

Lemma law_cosines :
 forall x0 y0 x1 y1 x2 y2 ac:R,
   let a := dist_euc x1 y1 x0 y0 in
   let b := dist_euc x2 y2 x0 y0 in
   let c := dist_euc x2 y2 x1 y1 in
   a * c * cos ac = (x0 - x1) * (x2 - x1) + (y0 - y1) * (y2 - y1) ->
   Rsqr b = Rsqr c + Rsqr a - 2 * (a * c * cos ac).
unfold dist_euc in |- *; intros; repeat rewrite Rsqr_sqrt;
 [ rewrite H; unfold Rsqr in |- *; ring | apply Rplus_le_le_0_compat | apply Rplus_le_le_0_compat | apply Rplus_le_le_0_compat ]; apply Rle_0_sqr.
Qed.

Lemma triangle :
 forall x0 y0 x1 y1 x2 y2:R,
   dist_euc x0 y0 x1 y1 <= dist_euc x0 y0 x2 y2 + dist_euc x2 y2 x1 y1.
intros; unfold dist_euc in |- *; apply Rsqr_incr_0;
 [ rewrite Rsqr_plus; repeat rewrite Rsqr_sqrt;
    [ replace (Rsqr (x0 - x1)) with
       (Rsqr (x0 - x2) + Rsqr (x2 - x1) + 2 * (x0 - x2) * (x2 - x1));
       [ replace (Rsqr (y0 - y1)) with
          (Rsqr (y0 - y2) + Rsqr (y2 - y1) + 2 * (y0 - y2) * (y2 - y1));
          [ apply Rplus_le_reg_l with
             (- Rsqr (x0 - x2) - Rsqr (x2 - x1) - Rsqr (y0 - y2) -
              Rsqr (y2 - y1));
             replace
              (- Rsqr (x0 - x2) - Rsqr (x2 - x1) - Rsqr (y0 - y2) -
               Rsqr (y2 - y1) +
               (Rsqr (x0 - x2) + Rsqr (x2 - x1) + 2 * (x0 - x2) * (x2 - x1) +
                (Rsqr (y0 - y2) + Rsqr (y2 - y1) + 2 * (y0 - y2) * (y2 - y1))))
              with (2 * ((x0 - x2) * (x2 - x1) + (y0 - y2) * (y2 - y1)));
             [ replace
                (- Rsqr (x0 - x2) - Rsqr (x2 - x1) - Rsqr (y0 - y2) -
                 Rsqr (y2 - y1) +
                 (Rsqr (x0 - x2) + Rsqr (y0 - y2) +
                  (Rsqr (x2 - x1) + Rsqr (y2 - y1)) +
                  2 * sqrt (Rsqr (x0 - x2) + Rsqr (y0 - y2)) *
                  sqrt (Rsqr (x2 - x1) + Rsqr (y2 - y1)))) with
                (2 *
                 (sqrt (Rsqr (x0 - x2) + Rsqr (y0 - y2)) *
                  sqrt (Rsqr (x2 - x1) + Rsqr (y2 - y1))));
                [ apply Rmult_le_compat_l;
                   [ left; cut (0%nat <> 2%nat);
                      [ intros; generalize (lt_INR_0 2 (neq_O_lt 2 H)); intro H0; assumption | discriminate ]
                   | apply sqrt_cauchy ]
                | ring ]
             | ring ]
          | ring_Rsqr ]
       | ring_Rsqr ]
    | apply Rplus_le_le_0_compat; apply Rle_0_sqr
    | apply Rplus_le_le_0_compat; apply Rle_0_sqr
    | apply Rplus_le_le_0_compat; apply Rle_0_sqr ]
 | apply sqrt_positivity; apply Rplus_le_le_0_compat; apply Rle_0_sqr
 | apply Rplus_le_le_0_compat; apply sqrt_positivity;
    apply Rplus_le_le_0_compat; apply Rle_0_sqr ].
Qed.
Translation
Definition xt (x tx:R) : R := x + tx.
Definition yt (y ty:R) : R := y + ty.

Lemma translation_0 : forall x y:R, xt x 0 = x /\ yt y 0 = y.
intros x y; split; [ unfold xt in |- * | unfold yt in |- * ]; ring.
Qed.

Lemma isometric_translation :
 forall x1 x2 y1 y2 tx ty:R,
   Rsqr (x1 - x2) + Rsqr (y1 - y2) =
   Rsqr (xt x1 tx - xt x2 tx) + Rsqr (yt y1 ty - yt y2 ty).
intros; unfold Rsqr, xt, yt in |- *; ring.
Qed.
Rotation
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.

Lemma rotation_0 : forall x y:R, xr x y 0 = x /\ yr x y 0 = y.
intros x y; unfold xr, yr in |- *; split; rewrite cos_0; rewrite sin_0; ring.
Qed.

Lemma rotation_PI2 :
 forall x y:R, xr x y (PI / 2) = y /\ yr x y (PI / 2) = - x.
intros x y; unfold xr, yr in |- *; split; rewrite cos_PI2; rewrite sin_PI2;
 ring.
Qed.

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).
intros; unfold xr, yr in |- *;
 replace
  (x1 * cos theta + y1 * sin theta - (x2 * cos theta + y2 * sin theta)) with
  (cos theta * (x1 - x2) + sin theta * (y1 - y2));
 [ replace
    (- x1 * sin theta + y1 * cos theta - (- x2 * sin theta + y2 * cos theta))
    with (cos theta * (y1 - y2) + sin theta * (x2 - x1));
    [ repeat rewrite Rsqr_plus; repeat rewrite Rsqr_mult; repeat rewrite cos2;
       ring; replace (x2 - x1) with (- (x1 - x2));
       [ rewrite <- Rsqr_neg; ring | ring ]
    | ring ]
 | ring ].
Qed.

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).
unfold dist_euc in |- *; intros; apply Rsqr_inj;
 [ apply sqrt_positivity; apply Rplus_le_le_0_compat
 | apply sqrt_positivity; apply Rplus_le_le_0_compat
 | repeat rewrite Rsqr_sqrt;
    [ apply isometric_rotation_0 | apply Rplus_le_le_0_compat | apply Rplus_le_le_0_compat ] ]; apply Rle_0_sqr.
Qed.
Similarity
Lemma isometric_rot_trans :
 forall x1 y1 x2 y2 tx ty theta:R,
   Rsqr (x1 - x2) + Rsqr (y1 - y2) =
   Rsqr (xr (xt x1 tx) (yt y1 ty) theta - xr (xt x2 tx) (yt y2 ty) theta) +
   Rsqr (yr (xt x1 tx) (yt y1 ty) theta - yr (xt x2 tx) (yt y2 ty) theta).
intros; rewrite <- isometric_rotation_0; apply isometric_translation.
Qed.

Lemma isometric_trans_rot :
 forall x1 y1 x2 y2 tx ty theta:R,
   Rsqr (x1 - x2) + Rsqr (y1 - y2) =
   Rsqr (xt (xr x1 y1 theta) tx - xt (xr x2 y2 theta) tx) +
   Rsqr (yt (yr x1 y1 theta) ty - yt (yr x2 y2 theta) ty).
intros; rewrite <- isometric_translation; apply isometric_rotation_0.
Qed.
Index
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