# Library Coq.Reals.Rgeom

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo1.
Require Import R_sqrt.
Local Open Scope R_scope.

# Distance

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.

Lemma distance_symm :
forall x0 y0 x1 y1:R, dist_euc x0 y0 x1 y1 = dist_euc x1 y1 x0 y0.

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).

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.

# 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.

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).

# 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.

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).

# 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).

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).