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Diffstat (limited to 'theories/Reals/Rgeom.v')
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diff --git a/theories/Reals/Rgeom.v b/theories/Reals/Rgeom.v new file mode 100644 index 00000000..a01e7b52 --- /dev/null +++ b/theories/Reals/Rgeom.v @@ -0,0 +1,187 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rgeom.v,v 1.13.2.1 2004/07/16 19:31:13 herbelin Exp $ i*) + +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.
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