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rigid.v
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rigid.v
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(* coq-robot (c) 2017 AIST and INRIA. License: LGPL-2.1-or-later. *)
From mathcomp Require Import all_ssreflect ssralg ssrint ssrnum rat poly.
From mathcomp Require Import closed_field polyrcf matrix mxalgebra mxpoly zmodp.
From mathcomp Require Import realalg complex finset fingroup perm.
From mathcomp Require Import interval reals trigo.
Require Import ssr_ext euclidean skew vec_angle rot frame extra_trigo.
(******************************************************************************)
(* Rigid Body Transformations *)
(* *)
(* This file develops the theory of isometries, proving basic properties such *)
(* as the preservation of the cross-product by derivative maps, the facts *)
(* that a direct isometry preserves orientation and that an isometry that *)
(* preserves the cross-product of two non-colinear vectors is direct. Rigid *)
(* body transformations are represented by elements of the special Euclidean *)
(* group and are shown to preserve norms. *)
(* *)
(* 'Iso[T]_n == the type of isometries *)
(* 'CIso[T]_n == the type of central isometries, i.e., isometries f such *)
(* that f 0 = 0 *)
(* ortho_of_iso f == orthogonal part of the isometry f *)
(* trans_of_iso f == translation part of the isometry f *)
(* iso_sgn f == sign of the isometry f *)
(* f`* == derivative map of f, action on vectors induced by f *)
(* preserves_orientation f == the isometry f preserves orientation, i.e., f`* *)
(* preserves the cross-product *)
(* 'DIso_3[T] == the type of direct isometries *)
(* 'SE3[T] == the type of special Euclidean group *)
(* hom r v == homogeneous representation of the element of 'SE3[T] *)
(* where r is a rotation matrix and t a vector representing *)
(* a translation *)
(* hRx a == homogeneous representation of the rotation about the *)
(* x-axis of angle a (resp. hRy, hRz) *)
(* hTx d == homogeneous representation of the translation of length *)
(* d along the x-axis (resp. hTy, hTz) *)
(* inv_hom M == inverse of the rigid body transformation M in *)
(* homogeneous representation *)
(* Adjoint g == adjoint transformation associated with the homogeneous *)
(* matrix g *)
(* *)
(******************************************************************************)
Reserved Notation "''Iso[' T ]_ n"
(at level 8, n at level 2, format "''Iso[' T ]_ n").
Reserved Notation "''CIso[' T ]_ n"
(at level 8, n at level 2, format "''CIso[' T ]_ n").
Reserved Notation "''DIso_3[' T ]" (at level 8, format "''DIso_3[' T ]").
Reserved Notation "''SE3[' T ]" (at level 8, format "''SE3[' T ]").
Reserved Notation "f '`*'" (at level 5, format "f `*").
Reserved Notation "''hP[' T ]" (at level 8, format "''hP[' T ]").
Reserved Notation "''hV[' T ]" (at level 8, format "''hV[' T ]").
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GRing.Theory Num.Theory.
Local Open Scope ring_scope.
Module Iso.
Section isometry.
Variables (T : rcfType) (n : nat).
Record t := mk {
f :> 'rV[T]_n -> 'rV[T]_n ;
P : {mono f : a b / norm (a - b)} }.
End isometry.
End Iso.
Notation "''Iso[' T ]_ n" := (Iso.t T n).
Definition isometry_coercion := Iso.f.
Coercion isometry_coercion : Iso.t >-> Funclass.
Module CIso.
Section central_isometry.
Variable (T : rcfType) (n : nat).
Record t := mk {
f : 'Iso[T]_n ;
P : f 0 = 0 }.
End central_isometry.
End CIso.
Notation "''CIso[' T ]_ n" := (CIso.t T n).
Definition cisometry_coercion := CIso.f.
Coercion cisometry_coercion : CIso.t >-> Iso.t.
Section central_isometry_n.
Variable (T : rcfType) (n : nat).
Implicit Types f : 'CIso[T]_n.
Lemma central_isometry_preserves_norm f : {mono f : x / norm x}.
Proof. by case: f => f f0 p; rewrite -(subr0 (f p)) -f0 Iso.P subr0. Qed.
(* [oneill] first part of lemma 1.6, p.100 *)
Lemma central_isometry_preserves_dotmul f : {mono f : u v / u *d v}.
Proof.
case: f => f f0 a b.
have /eqP : norm (f a - f b) = norm (a - b) by rewrite (Iso.P f).
rewrite /norm eqr_sqrt ?le0dotmul // !dotmulDl !dotmulDr !dotmulvv !normN.
rewrite !(central_isometry_preserves_norm (CIso.mk f0)) !addrA.
rewrite 2!(addrC _ (norm b ^+ 2)) => /eqP/addrI.
rewrite -2!addrA => /addrI.
rewrite -(dotmulC (f a)) dotmulvN -(dotmulC a) dotmulvN -2!mulr2n => /eqP.
rewrite -mulr_natr -[in X in _ == X -> _]mulr_natr 2!mulNr eqr_opp.
by move/eqP/mulIr => -> //; rewrite unitfE pnatr_eq0.
Qed.
End central_isometry_n.
Section central_isometry_3.
Variable T : realType.
Implicit Types f : 'CIso[T]_3.
Local Open Scope frame_scope.
Definition frame_central_iso f (F : noframe T) : noframe T.
apply: (@NOFrame.mk _ (col_mx3 (f F~i) (f F~j) (f F~k))).
apply/orthogonal3P.
by rewrite !rowK /= 3!central_isometry_preserves_norm 3!noframe_norm
3!central_isometry_preserves_dotmul idotj noframe_idotk jdotk !eqxx.
Defined.
(* [oneill] second part of lemma 1.6, p.101 *)
Lemma central_isometry_is_linear f : linear f.
Proof.
move=> k /= a b.
have Hp : forall p,
f p = p``_0 *: f 'e_0 + p``_1 *: f 'e_1 + p``_2%:R *: f 'e_2%:R.
move=> p.
have -> : f p = f p *d f 'e_0 *: f 'e_0 +
f p *d f 'e_1 *: f 'e_1 +
f p *d f 'e_2%:R *: f 'e_2%:R.
move: (orthogonal_expansion (frame_central_iso f (can_noframe T)) (f p)).
by rewrite !rowframeE !rowK /= !rowframeE /= !row1.
by rewrite 3!central_isometry_preserves_dotmul // 3!coorE.
rewrite Hp (Hp a) (Hp b) !mxE /= !(scalerDl, scalerDr).
rewrite !scalerA -!addrA; congr (_ + _).
rewrite addrC -!addrA; congr (_ + _).
rewrite addrC -!addrA; congr (_ + _).
by rewrite addrC -!addrA.
Qed.
End central_isometry_3.
Section isometry_3_properties.
Variable T : realType.
Let vector := 'rV[T]_3.
Let point := 'rV[T]_3.
Implicit Types f : 'Iso[T]_3.
(* [oneill] theorem 1.7, p.101 *)
(** every isometry of E^3 can be uniquely described as an orthogonal
transformation followed by a translation *)
Lemma trans_ortho_of_iso f :
{ trans : 'rV[T]_3 & { rot : 'M[T]_3 |
(forall x : 'rV[T]_3, f x == x *m rot + trans) /\
rot \is 'O[T]_3 /\
trans = f 0 } }.
Proof.
set m := f 0.
set Tm1f := fun x => f x - m.
have Tm1f_is_iso : {mono Tm1f : a b / norm (a - b)}.
move=> ? ?; by rewrite /Tm1f -addrA opprB 2!addrA subrK (Iso.P f).
have Tm1f0 : Tm1f 0 = 0 by rewrite /Tm1f subrr.
set c := @CIso.mk _ _ (Iso.mk Tm1f_is_iso) Tm1f0.
have /= linearTm1f := central_isometry_is_linear c.
have /= orthogonalTm1f := central_isometry_preserves_dotmul c.
exists m.
case: (lin1_mx' linearTm1f) => g Hg.
exists (lin1_mx g); split; last first.
split; last by [].
apply/orth_preserves_dotmul => u v /=.
by rewrite 2!mul_rV_lin1 -[in RHS]orthogonalTm1f 2!Hg.
move=> u; by rewrite mul_rV_lin1 -Hg subrK.
Qed.
Definition ortho_of_iso f : 'M[T]_3 := projT1 (projT2 (trans_ortho_of_iso f)).
Definition trans_of_iso f : 'rV[T]_3 := projT1 (trans_ortho_of_iso f).
Lemma trans_of_isoE f : trans_of_iso f = f 0.
Proof.
rewrite /trans_of_iso; by case: (trans_ortho_of_iso _) => ? [C [H1 [H2 H3]]].
Qed.
Lemma ortho_of_iso_is_O f : ortho_of_iso f \is 'O[T]_3.
Proof.
rewrite /ortho_of_iso; by case: (trans_ortho_of_iso _) => ? [C [H1 [H2 H3]]].
Qed.
Lemma trans_ortho_of_isoE f u : u *m ortho_of_iso f = f u - trans_of_iso f.
Proof.
rewrite /ortho_of_iso /trans_of_iso.
case: (trans_ortho_of_iso _) => ? [C [H1 [H2 H3]]] /=.
move: (H1 u) => /eqP ->; by rewrite addrK.
Qed.
Lemma ortho_of_iso_eq f1 f2 : (forall i, Iso.f f1 i = Iso.f f2 i) ->
ortho_of_iso f1 = ortho_of_iso f2.
Proof.
move=> f12.
apply/eqP/mulmxP => u.
rewrite 2!trans_ortho_of_isoE /= 2!trans_of_isoE /=.
case: f1 f2 f12 => [f1 Hf1] [f2 Hf2] /= f12; by rewrite !f12.
Qed.
Definition iso_sgn f : T := \det (ortho_of_iso f).
Lemma img_vec_iso f (a b : point) :
f b - f a = (b - a) *m ortho_of_iso f.
Proof.
move/esym/eqP: (trans_ortho_of_isoE f a).
move/esym/eqP: (trans_ortho_of_isoE f b).
rewrite mulmxBl => /eqP <- /eqP <-; by rewrite opprB addrA subrK.
Qed.
Definition displacement f p : vector := f p - p.
Definition relative_displacement f (p a : point) :=
(p - a) *m (ortho_of_iso f - 1).
(* NB: caused only by rotation *)
Lemma displacement_iso f p a :
displacement f p = displacement f a + relative_displacement f p a.
Proof.
rewrite /relative_displacement mulmxBr mulmx1 opprB addrA -(addrC a) 2!addrA.
rewrite subrK; congr (_ - _).
apply/eqP; by rewrite addrC -subr_eq img_vec_iso.
Qed.
End isometry_3_properties.
Module DIso.
Section direct_isometry.
Variable (T : realType).
Record t := mk {
f :> 'Iso[T]_3 ;
P : iso_sgn f == 1 }.
End direct_isometry.
End DIso.
Notation "''DIso_3[' T ]" := (DIso.t T).
Definition disometry_coercion := DIso.f.
Coercion disometry_coercion : DIso.t >-> Iso.t.
Section direct_isometry_3_properties.
Variable T : realType.
Lemma ortho_of_diso_is_SO (f : 'DIso_3[T]) : ortho_of_iso f \is 'SO[T]_3.
Proof.
case: f => f; rewrite /iso_sgn => Hf /=; by rewrite rotationE (ortho_of_iso_is_O f).
Qed.
End direct_isometry_3_properties.
Section derivative_map.
Variable T : realType.
Let vector := 'rV[T]_3.
Implicit Types f : 'Iso[T]_3.
Import rv3LieAlgebra.Exports.
(* [oneill] theorem 2.1, p. 104 *)
Definition dmap f (v : vector) : vector :=
let C := ortho_of_iso f in
(v *m C).
Local Notation "f '`*'" := (@dmap f).
Lemma dmap0 f : f `* 0 = 0.
Proof. by rewrite /dmap /= mul0mx. Qed.
Lemma dmapE f (u : vector) b a :
u = b - a :> vector ->
f `* u = f b - f a :> vector.
Proof. move=> uab; by rewrite /dmap /= uab img_vec_iso. Qed.
Lemma derivative_map_preserves_length f :
{mono (fun x : vector => f`* x) : u v / norm (u - v)}.
Proof.
move=> u v; rewrite /dmap /= -(mulmxBl u v (ortho_of_iso f)).
by rewrite orth_preserves_norm // ortho_of_iso_is_O.
Qed.
Local Open Scope frame_scope.
(* [oneill] lemma 3.2, p.108 *)
Lemma dmap_iso_sgnP (F : tframe T) f :
let i := F~i in let j := F~j in let k := F~k in
f`* i *d (f `* j *v f`* k) =
iso_sgn f * (i *d (j *v k)).
Proof.
move=> i j k.
move: (orthogonal_expansion (can_noframe T) i).
rewrite !rowframeE !row1.
set a11 := _ *d 'e_0. set a12 := _ *d 'e_1. set a13 := _ *d 'e_2%:R => He1.
move: (orthogonal_expansion (can_noframe T) j).
rewrite !rowframeE !row1.
set a21 := _ *d 'e_0. set a22 := _ *d 'e_1. set a23 := _ *d 'e_2%:R => He2.
move: (orthogonal_expansion (can_noframe T) k).
rewrite !rowframeE !row1.
set a31 := _ *d 'e_0. set a32 := _ *d 'e_1. set a33 := _ *d 'e_2%:R => He3.
have ia : i = row3 a11 a12 a13.
by rewrite (row3_proj i) !row3D !(add0r,addr0) !coorE.
have ja : j = row3 a21 a22 a23.
by rewrite (row3_proj j) !row3D !(add0r,addr0) !coorE.
have ka : k = row3 a31 a32 a33.
by rewrite (row3_proj k) !row3D !(add0r,addr0) !coorE.
transitivity (\det ((ortho_of_iso f)^T *m
(col_mx3 (row3 a11 a12 a13) (row3 a21 a22 a23) (row3 a31 a32 a33))^T)).
rewrite /= -det_tr trmx_mul mulmxE trmxK -col_mx3_mul.
by rewrite -crossmul_triple -ia -ja -ka trmxK.
rewrite det_mulmx det_tr; congr (_ * _).
by rewrite det_tr -crossmul_triple; congr (_ *d (_ *v _)).
Qed.
(* [oneill] theorem 3.6, p.110 *)
Lemma dmap_preserves_crossmul (u v : vector) f :
f`* (u *v v) = iso_sgn f *: (f`* u *v f`* v) :> vector.
Proof.
set tf := TFrame.trans (can_tframe T) 0.
set u1p := tf~i. set u2p := tf~j. set u3p := tf~k.
move: (orthogonal_expansion tf u).
rewrite !rowframeE !row1.
set u1 := _ *d 'e_0. set u2 := _ *d 'e_1. set u3 := _ *d 'e_2%:R => Hu.
move: (orthogonal_expansion tf v).
rewrite !rowframeE !row1.
set v1 := _ *d 'e_0. set v2 := _ *d 'e_1. set v3 := _ *d 'e_2%:R => Hv.
set e1 := f`* u1p. set e2 := f`* u2p. set e3 := f`* u3p.
have Ku : f`* u = u1 *: e1 + u2 *: e2 + u3 *: e3 :> vector.
rewrite [in LHS]/= Hu /dmap !mulmxDl.
rewrite !scalemxAl [in RHS]/=.
rewrite /u1p /u2p /u3p.
by rewrite 3!rowframeE 3!rowE !mulmx1.
have Kv : f`* v = v1 *: e1 + v2 *: e2 + v3 *: e3 :> vector.
rewrite [in LHS]/= Hv /dmap !mulmxDl.
rewrite !scalemxAl [in RHS]/=.
rewrite /u1p /u2p /u3p.
by rewrite 3!rowframeE 3!rowE !mulmx1.
have @f' : noframe T.
apply (@NOFrame.mk _ (col_mx3 e1 e2 e3)).
apply/orthogonal3P; rewrite !rowK /=.
do 3! rewrite orth_preserves_norm ?ortho_of_iso_is_O //.
rewrite /u1p /u2p /u3p.
rewrite !rowframeE !rowE !mulmx1 3!normeE !eqxx /=.
rewrite !(proj2 (orth_preserves_dotmul _)) ?ortho_of_iso_is_O //.
rewrite /u1p /u2p /u3p.
by rewrite !rowframeE /= !rowE ?mulmx1 !dote2 //= eqxx.
have -> : iso_sgn f = noframe_sgn f'.
(* TODO: move as a lemma? *)
rewrite noframe_sgnE.
have -> : f'~i = f`* u1p by rewrite rowframeE rowK.
have -> : f'~j = f`* u2p by rewrite rowframeE rowK.
have -> : f'~k = f`* u3p by rewrite rowframeE rowK.
by rewrite dmap_iso_sgnP /= !rowframeE !rowE !mulmx1 vecjk dote2 mulr1.
have : (f`* u) *v (f`* v) = noframe_sgn f' *: (f`* (u *v v)) :> vector.
rewrite /=.
rewrite (@crossmul_noframe_sgn _ f' (f`* u) u1 u2 u3 (f`* v) v1 v2 v3) //; last 2 first.
move: Ku.
by rewrite !rowframeE /= !rowK /=.
move: Kv.
by rewrite /= !rowframeE !rowK /=.
rewrite /=.
congr (_ *: _).
rewrite !rowframeE !rowK /= Hu Hv.
do 2 rewrite linearD [in RHS]/=.
rewrite /dmap.
rewrite 2!mulmxDl.
(* on fait les remplacement veci *v vecj -> veck, veci *v veci -> 0, etc. *)
rewrite [in RHS]linearZ [in RHS]/=.
rewrite [in RHS]linearZ [in RHS]/=.
rewrite [in RHS]linearZ [in RHS]/=.
rewrite (lieC _ 'e_0) /= scalerN.
rewrite linearD [in RHS]/=.
rewrite [in X in _ = - (_ *: X) *m _ + _ + _]linearD.
rewrite [in RHS]/=.
rewrite (_ : 'e_0 *v (u1 *: _) = 0); last by rewrite linearZ /= liexx scaler0.
rewrite (_ : 'e_0 *v (u2 *: _) = u2 *: 'e_2%:R); last first.
rewrite linearZ /=.
by rewrite vecij.
rewrite (_ : 'e_0 *v (u3 *: _) = - u3 *: 'e_1); last first.
rewrite linearZ /=.
rewrite vecik.
by rewrite scalerN scaleNr.
rewrite add0r.
rewrite mulNmx -[in RHS]scalemxAl [in RHS]mulmxDl.
rewrite -![in RHS]scalemxAl.
rewrite [in RHS]scalerDr.
rewrite opprD.
rewrite (lieC _ 'e_1) /= [in X in _ = _ + X + _]linearD [in X in _ = _ + X + _]/=.
rewrite opprD.
rewrite [in X in _ = _ + X + _]linearD [in X in _ = _ + X + _]/=.
rewrite scaleNr scalerN opprK.
rewrite (_ : _ *v _ = - u1 *: 'e_2%:R); last first.
by rewrite linearZ /= lieC /= vecij scalerN scaleNr.
rewrite (_ : _ *v _ = 0); last by rewrite linearZ /= liexx scaler0.
rewrite addr0.
rewrite (_ : _ *v _ = u3 *: 'e_0); last by rewrite linearZ /= vecjk.
rewrite scaleNr opprK mulmxBl.
rewrite -![in RHS]scalemxAl.
rewrite scalerDr scalerN.
rewrite (lieC _ 'e_2%:R) /= [in X in _ = _ + _ + X]linearD [in X in _ = _ + _ + X]/=.
rewrite opprD.
rewrite [in X in _ = _ + _ + X]linearD [in X in _ = _ + _ + X]/=.
rewrite opprD.
rewrite (_ : _ *v _ = u1 *: 'e_1); last first.
by rewrite linearZ /= lieC /= vecik opprK.
rewrite (_ : _ *v _ = - u2 *: 'e_0); last first.
by rewrite linearZ /= lieC /= vecjk scalerN scaleNr.
rewrite (_ : _ *v _ = 0); last by rewrite linearZ /= liexx scaler0.
rewrite subr0 scaleNr opprK mulmxDl mulNmx.
rewrite -![in RHS]scalemxAl.
rewrite -![in RHS]addrA [in RHS]addrC -[in RHS]addrA [in RHS]addrCA -[in RHS]addrA [in RHS]addrC.
rewrite ![in RHS]addrA -[in RHS]addrA.
congr (_ + _); last first.
rewrite !scalerA -scaleNr -scalerDl addrC mulrC (mulrC u1).
by rewrite /e3 /dmap /u3p rowframeE rowE mulmx1.
rewrite scalerDr.
rewrite -![in RHS]addrA [in RHS]addrCA [in RHS]addrC ![in RHS]addrA -addrA; congr (_ + _).
rewrite /e1 /dmap /u1p.
rewrite rowframeE rowE mulmx1.
by rewrite !scalerA -scaleNr -scalerDl addrC mulrC (mulrC u2).
rewrite /e2 /dmap /u2p.
rewrite rowframeE rowE mulmx1.
by rewrite scalerN !scalerA -scalerBl -scaleNr opprB mulrC (mulrC u1).
move=> ->; by rewrite scalerA -expr2 /iso_sgn -sqr_normr abs_noframe_sgn expr1n scale1r.
Qed.
Definition preserves_orientation f :=
forall (u v : vector),
f`* (u *v v) = ((f`* u) *v (f`* v))
:> vector.
Lemma preserves_crossmul_is_diso f (u v : vector) :
~~ colinear u v ->
f`* (u *v v) = (f`* u) *v (f`* v) :> vector ->
iso_sgn f = 1.
Proof.
move=> uv0.
rewrite dmap_preserves_crossmul => H.
move: (orthogonal_det (ortho_of_iso_is_O f)).
rewrite -/(iso_sgn _).
case: (lerP 0 (iso_sgn f)) => K; first by rewrite ger0_norm.
rewrite ltr0_norm // => /eqP.
rewrite eqr_oppLR => /eqP {}K.
exfalso.
move: H.
rewrite K scaleN1r => /eqP; rewrite eqmxNxx.
move: (mulmxr_crossmulr u v (ortho_of_iso_is_O f)).
rewrite -/(iso_sgn f) K scaleN1r => /esym/eqP.
rewrite eqr_oppLR => /eqP ->.
rewrite oppr_eq0 mul_mx_rowfree_eq0; last first.
apply/row_freeP.
exists (ortho_of_iso f)^T.
apply/eqP; by rewrite -orthogonalE ortho_of_iso_is_O.
move: uv0.
rewrite /colinear; by move/negbTE => ->.
Qed.
Lemma diso_preserves_orientation (df : 'DIso_3[T]) : preserves_orientation df.
Proof. move=> u v; by rewrite dmap_preserves_crossmul (eqP (DIso.P df)) scale1r. Qed.
End derivative_map.
Section homogeneous_points_and_vectors.
Variable T : ringType.
Let point := 'rV[T]_3.
Let vector := 'rV[T]_3.
Let homogeneous := 'rV[T]_4.
Lemma rsubmx_coor3 (x : homogeneous) : @rsubmx _ 1 3 1 x = x``_3%:R%:M.
Proof.
apply/rowP => i; rewrite {i}(ord1 i) !mxE eqxx.
rewrite (_ : (rshift _ _) = 3%:R :> 'I_(3 + 1) ) //; by apply val_inj.
Qed.
Definition from_h (x : homogeneous) : 'rV[T]_3 := @lsubmx _ 1 3 1 x.
Lemma from_hD (x y : homogeneous) : from_h (x + y) = from_h x + from_h y.
Proof. apply/rowP => i; by rewrite !mxE. Qed.
Lemma from_hZ k (x : homogeneous) : from_h (k *: x) = k *: from_h x.
Proof. apply/rowP => i; by rewrite !mxE. Qed.
Lemma from_hB (x y : homogeneous) : from_h (x - y) = from_h x - from_h y.
Proof. apply/rowP => i; by rewrite !mxE. Qed.
Lemma from_hE (x : homogeneous) : from_h x = \row_(i < 3) x 0 (inord i).
Proof.
apply/rowP => i; rewrite !mxE; congr (x 0 _).
apply val_inj => /=; by rewrite inordK // (ltn_trans (ltn_ord i)).
Qed.
Definition hpoint := [qualify x : homogeneous | x``_3%:R == 1].
Fact hpoint_key : pred_key hpoint. Proof. by []. Qed.
Canonical hpoint_keyed := KeyedQualifier hpoint_key.
Lemma hpointE (x : homogeneous) : (x \is hpoint) = (x``_3%:R == 1).
Proof. by []. Qed.
Definition to_hpoint (p : point) : homogeneous := row_mx p 1.
Lemma to_hpointK (p : point) : from_h (to_hpoint p) = p.
Proof. by rewrite /from_h row_mxKl. Qed.
Lemma hpoint_from_h x : (x \is hpoint) = (x == row_mx (from_h x) 1).
Proof.
rewrite hpointE -{2}(@hsubmxK _ 1 3 1 x) rsubmx_coor3.
apply/idP/idP => [/eqP -> // | /eqP/(@eq_row_mx _ 1 3 1) [_ /rowP/(_ ord0)]].
by rewrite !mxE eqxx /= => /eqP.
Qed.
Lemma to_hpointP (p : point) : to_hpoint p \is hpoint.
Proof. by rewrite hpoint_from_h to_hpointK. Qed.
Definition hvector := [qualify x : homogeneous | x``_3%:R == 0].
Fact hvector_key : pred_key hvector. Proof. by []. Qed.
Canonical hvector_keyed := KeyedQualifier hvector_key.
Lemma hvectorE (x : homogeneous) : (x \is hvector) = (x``_3%:R == 0).
Proof. by []. Qed.
Definition to_hvector (v : vector) : homogeneous := row_mx v 0.
Lemma to_hvectorK (v : vector) : from_h (to_hvector v) = v.
Proof. by rewrite /from_h row_mxKl. Qed.
Lemma hvector_from_h (x : homogeneous) : (x \is hvector) = (x == row_mx (from_h x) 0).
Proof.
rewrite hvectorE -{2}(@hsubmxK _ 1 3 1 x) rsubmx_coor3.
apply/idP/idP => [/eqP -> //| /eqP/(@eq_row_mx _ 1 3 1) [_ /rowP/(_ ord0)]].
by rewrite (_ : 0%:M = 0) //; apply/rowP => i; rewrite {i}(ord1 i) !mxE eqxx.
by rewrite !mxE eqxx /= => /eqP.
Qed.
Lemma to_hvectorP (v : vector) : to_hvector v \is hvector.
Proof. by rewrite hvector_from_h to_hvectorK. Qed.
Lemma hpointB (x y : homogeneous) : x \is hpoint -> y \is hpoint -> x - y \is hvector.
Proof.
rewrite 2!hpoint_from_h hvector_from_h => /eqP Hp /eqP Hq.
by rewrite {1}Hp {1}Hq (opp_row_mx (from_h y)) (add_row_mx (from_h x)) subrr -from_hB.
Qed.
Lemma to_hpointB (p q : point) : to_hpoint p - to_hpoint q = to_hvector (p - q).
Proof. by rewrite /to_hpoint (opp_row_mx q) (add_row_mx p) subrr. Qed.
End homogeneous_points_and_vectors.
Notation "''hP[' T ]" := (hpoint T).
Notation "''hV[' T ]" := (hvector T).
Section homogeneous_matrices.
Variable T : ringType.
Let homogeneous := 'M[T]_4.
Implicit Types M : homogeneous.
Implicit Types r : 'M[T]_3.
Definition hom r (v : 'rV[T]_3) : homogeneous := block_mx r 0 v 1.
Definition rot_of_hom M : 'M[T]_3 := @ulsubmx _ 3 1 3 1 M.
Definition trans_of_hom M : 'rV[T]_3 := @dlsubmx _ 3 1 3 1 M.
Lemma hom10 : hom 1 0 = 1 :> homogeneous.
Proof.
rewrite /hom -[in RHS](@submxK _ 3 1 3 1 1).
congr (@block_mx _ 3 1 3 1); apply/matrixP => i j; rewrite !mxE -val_eqE //.
rewrite {j}(ord1 j) /= addn0; by case: i => -[] // [] // [].
rewrite {i}(ord1 i) /= addn0; by case: j => -[] // [] // [].
Qed.
Lemma rot_of_hom_hom t r : rot_of_hom (hom r t) = r.
Proof. by rewrite /rot_of_hom /hom block_mxKul. Qed.
Lemma rot_of_hom1 : rot_of_hom 1 = 1 :> 'M[T]__.
Proof. by rewrite -hom10 rot_of_hom_hom. Qed.
Lemma rot_of_homN M : rot_of_hom (- M) = - rot_of_hom M.
Proof. apply/matrixP => i j; by rewrite !mxE. Qed.
Lemma tr_rot_of_hom M : (rot_of_hom M)^T = rot_of_hom M^T.
Proof. by rewrite /rot_of_hom trmx_ulsub. Qed.
Lemma trans_of_hom_hom r t : trans_of_hom (hom r t) = t.
Proof. by rewrite /trans_of_hom /hom block_mxKdl. Qed.
Lemma trans_of_hom1 : trans_of_hom 1 = 0 :> 'rV[T]__.
Proof. by rewrite -hom10 trans_of_hom_hom. Qed.
Lemma homM r r' t t' : hom r t * hom r' t' = hom (r * r') (t *m r' + t').
Proof.
rewrite /hom -mulmxE (mulmx_block r _ _ _ r').
by rewrite !(mulmx0,mul0mx,addr0,add0r,mulmx1) mulmxE mul1mx.
Qed.
Definition inv_hom M := hom (rot_of_hom M)^T (- trans_of_hom M *m (rot_of_hom M)^T).
Lemma trmx_hom (r : 'M[T]_3) t : (hom r t)^T = block_mx r^T t^T 0 1.
Proof. by rewrite /hom (tr_block_mx r) trmx1 trmx0. Qed.
End homogeneous_matrices.
Lemma det_hom (T : comRingType) (r : 'M[T]_3) t : \det (hom r t) = \det r.
Proof. by rewrite /hom (det_lblock r) det1 mulr1. Qed.
Section homogeneous_transformations.
Variable T : realType.
Let homogeneous := 'M[T]_4.
Implicit Types M : homogeneous.
Implicit Types r : 'M[T]_3.
(* elementary rotations in homogeneous form *)
Definition hRx a : homogeneous := hom (Rx a) 0.
Lemma hRx_correct a (p : 'rV[T]_3) : from_h ((to_hpoint p) *m hRx a) = p *m Rx a.
Proof.
rewrite {1}/to_hpoint /hRx /hom (mul_row_block p 1 (Rx a)).
by rewrite !(mulmx0,addr0,add0r,mulmx1) -/(to_hpoint (p *m Rx a)) to_hpointK.
Qed.
Definition hRz a : homogeneous := hom (Rz a) 0.
Definition hRy a : homogeneous := hom (Ry a) 0.
(* elementary translations in homogeneous form *)
Definition hTx d : homogeneous := hom 1 (row3 d 0 0).
Definition hTy d : homogeneous := hom 1 (row3 0 d 0).
Definition hTz d : homogeneous := hom 1 (row3 0 0 d).
Lemma hTxRz (d : T) (a : T) :
hTx d * hRz a = hom (Rz a) (row3 (d * cos a) (d * sin a) 0).
Proof.
by rewrite homM mul1r addr0 mulmx_row3_col3 2!scale0r !addr0 row3Z mulr0.
Qed.
Lemma hTzRz (d : T) (a : T) : hTz d * hRz a = hom (Rz a) (row3 0 0 d).
Proof.
rewrite homM mul1r mulmx_row3_col3 2!scale0r !(add0r,addr0) e2row row3Z.
by rewrite !(mulr0,mulr1).
Qed.
End homogeneous_transformations.
Section SE3_qualifier.
Variable T : ringType.
Implicit Types M : 'M[T]_4.
Definition SE3 := [qualify M |
[&& rot_of_hom M \is 'SO[T]_3,
@ursubmx _ 3 1 3 1 M == 0 &
@drsubmx _ 3 1 3 1 M == 1%:M] ].
Fact SE3_key : pred_key SE3. Proof. by []. Qed.
Canonical SE3_keyed := KeyedQualifier SE3_key.
End SE3_qualifier.
Notation "''SE3[' T ]" := (SE3 T) : ring_scope.
Section SE3_hom.
Lemma rot_of_hom_is_SO (T : ringType) M :
M \is 'SE3[T] -> rot_of_hom M \is 'SO[T]_3.
Proof. by case/and3P. Qed.
Lemma hom_is_SE (T : ringType) r t : r \is 'SO[T]_3 -> hom r t \is 'SE3[T].
Proof.
move=> Hr; apply/and3P; rewrite rot_of_hom_hom Hr; split => //.
- by rewrite /hom block_mxKur.
- by rewrite /hom block_mxKdr.
Qed.
Lemma SE3E (T : ringType) M :
M \is 'SE3[T] -> M = hom (rot_of_hom M) (trans_of_hom M).
Proof.
case/and3P => T1 /eqP T2 /eqP T3.
by rewrite /hom -[in LHS](@submxK _ 3 1 3 1 M) T2 T3.
Qed.
Lemma SE31 (T : comUnitRingType) : 1 \is 'SE3[T].
Proof.
apply/and3P; split; first by rewrite rot_of_hom1 rotation1.
- apply/eqP/matrixP => i j; rewrite !mxE -val_eqE /= {j}(ord1 j) addn0.
by case: i => -[] // [] // [].
- by apply/eqP/rowP => i; rewrite {i}(ord1 i) !mxE -val_eqE.
Qed.
Lemma SE3_in_unitmx (T : comUnitRingType) M : M \is 'SE3[T] -> M \in unitmx.
Proof.
move=> H; rewrite (SE3E H).
by rewrite unitmxE /= det_hom rotation_det // ?unitr1 // rot_of_hom_is_SO.
Qed.
Lemma rot_of_homM (T : ringType) M1 M2 : M1 \is 'SE3[T] -> M2 \is 'SE3[T] ->
rot_of_hom (M1 * M2) = rot_of_hom M1 * rot_of_hom M2.
Proof. move/SE3E => -> /SE3E ->; by rewrite homM !rot_of_hom_hom. Qed.
Lemma trans_of_homM (T : ringType) M1 M2 : M1 \is 'SE3[T] -> M2 \is 'SE3[T] ->
trans_of_hom (M1 * M2) = trans_of_hom M1 *m rot_of_hom M2 + trans_of_hom M2.
Proof.
move/SE3E => -> /SE3E tmp; rewrite [in LHS]tmp; by rewrite homM 2!trans_of_hom_hom.
Qed.
Lemma homV (T : comUnitRingType) M : M \is 'SE3[T] -> M * inv_hom M = 1.
Proof.
move=> HM.
rewrite (SE3E HM) /= /inv_hom rot_of_hom_hom trans_of_hom_hom.
rewrite homM -rotation_inv ?rot_of_hom_is_SO // divrr; last first.
by apply/orthogonal_unit/rotation_sub/rot_of_hom_is_SO.
by rewrite mulNmx subrr hom10.
Qed.
Lemma Vhom (T : fieldType) M : M \is 'SE3[T] -> inv_hom M * M = 1.
Proof.
move=> HM.
rewrite (SE3E HM) /= /inv_hom rot_of_hom_hom trans_of_hom_hom.
rewrite homM -rotation_inv ?rot_of_hom_is_SO // mulVr; last first.
by apply/orthogonal_unit/rotation_sub/rot_of_hom_is_SO.
rewrite -mulmxA mulVmx ?mulmx1 1?addrC ?subrr ?hom10 // .
by rewrite unitmxE unitfE rotation_det ?oner_eq0 // rot_of_hom_is_SO.
Qed.
Lemma inv_homE (T : fieldType) M : M \is 'SE3[T] -> inv_hom M = M^-1.
Proof.
move=> HM.
rewrite -[RHS]mul1mx -[X in _ = X *m _ ](Vhom HM) -mulmxA.
by rewrite mulmxV ?mulmx1 // SE3_in_unitmx.
Qed.
Lemma inv_hom_is_SE3 (T : comUnitRingType) M :
M \is 'SE3[T] -> inv_hom M \is 'SE3[T].
Proof.
case/and3P=> ? ? ?; apply/and3P; split.
- by rewrite /inv_hom rot_of_hom_hom rotationV.
- by rewrite /inv_hom /hom block_mxKur.
- by rewrite /inv_hom /hom block_mxKdr.
Qed.
Lemma SE3_invr_closed (T : fieldType) : invr_closed 'SE3[T].
Proof. move=> M HM; by rewrite -inv_homE // inv_hom_is_SE3. Qed.
Lemma SE3_mulr_closed (T : comUnitRingType) : mulr_closed 'SE3[T].
Proof.
split; first exact: SE31.
move=> /= A B HA HB.
rewrite (SE3E HA) (SE3E HB) /= homM.
apply/and3P; split.
- rewrite /rot_of_hom /hom block_mxKul.
case/and3P : HA; rewrite /rot_of_hom => HA _ _.
case/and3P : HB; rewrite /rot_of_hom => HB _ _.
by rewrite rpredM.
- by rewrite /hom block_mxKur.
- by rewrite /hom block_mxKdr.
Qed.
Canonical SE3_is_mulr_closed (T : comUnitRingType) :=
MulrPred (SE3_mulr_closed T).
Lemma SE3_divr_closed (T : fieldType) : divr_closed 'SE3[T].
Proof.
split; first by rewrite SE31.
move=> A B HA HB.
by rewrite rpredM // SE3_invr_closed.
Qed.
Canonical SE3_is_divr_closed (T : fieldType) := DivrPred (SE3_divr_closed T).
End SE3_hom.
Section Adjoint.
Variables (T : comRingType).
Implicit Types g : 'M[T]_4.
Definition Adjoint g : 'M_6 :=
let r := rot_of_hom g in
let t := trans_of_hom g in
block_mx r 0 (r * \S(t)) r.
Lemma Adjoint1 : Adjoint 1 = 1 :> 'M[T]_6.
Proof.
by rewrite /Adjoint rot_of_hom1 mul1r trans_of_hom1 spin0 -scalar_mx_block.
Qed.
Definition inv_Adjoint g : 'M_6 :=
let r := rot_of_hom g in
let t := trans_of_hom g in
block_mx r^T 0 (- r^T * \S(t *m r^T)) r^T.
End Adjoint.
Section adjoint_theory.
Lemma Adjoint_in_unitmx (T : comUnitRingType) M :
M \is 'SE3[T] -> Adjoint M \in unitmx.
Proof.
move=> ?; rewrite unitmxE /Adjoint (det_lblock (rot_of_hom M)).
by rewrite rotation_det ?mulr1 ?unitr1 // rot_of_hom_is_SO.
Qed.
Variable (T : realFieldType).
Lemma VAdjoint g : rot_of_hom g \is 'SO[T]_3 -> inv_Adjoint g * Adjoint g = 1.
Proof.
set r := rot_of_hom g. set t := trans_of_hom g. move=> rSO.
rewrite /inv_Adjoint /Adjoint -mulmxE -/r -/t (mulmx_block r^T _ _ _ r).
rewrite !(mul0mx,addr0,mulmx0,add0r) mulmxA orthogonal_tr_mul ?rotation_sub //.
rewrite mul1mx mulmxE 2!mulNr spin_similarity // -mulmxA.
by rewrite orthogonal_tr_mul ?rotation_sub // mulmx1 addrC subrr -scalar_mx_block.
Qed.
Lemma inv_AdjointE g : g \is 'SE3[T] -> inv_Adjoint g = (Adjoint g)^-1.
Proof.
move=> ?; rewrite -[RHS]mul1r -[X in _ = X *m _](@VAdjoint g) ?rot_of_hom_is_SO //.
by rewrite mulmxE -mulrA mulrV ?mulr1 // Adjoint_in_unitmx.
Qed.
(* [murray] exercise 14 (a), p.77 *)
Lemma inv_Adjoint_inv g : g \is 'SE3[T] -> (Adjoint g)^-1 = Adjoint g^-1.
Proof.
move=> ?; rewrite -inv_AdjointE // /inv_Adjoint /Adjoint -inv_homE // /inv_hom.
by rewrite !(rot_of_hom_hom,trans_of_hom_hom) mulNmx mulNr spinN !mulrN.
Qed.
Lemma AdjointM_helper g1 g2 : g1 \is 'SE3[T] -> g2 \is 'SE3[T] ->
let r1 := rot_of_hom g1 in let r2 := rot_of_hom g2 in
let t1 := trans_of_hom g1 in let t2 := trans_of_hom g2 in
Adjoint (g1 * g2) = block_mx (r1 * r2) 0
((r1 * r2) * (\S(t2) + r2^T * \S(t1) * r2)) (r1 * r2).
Proof.
move=> Hg1 Hg2 r1 r2 t1 t2.
rewrite /Adjoint -rot_of_homM // trans_of_homM // spinD.
set a := rot_of_hom (_ * _) * _. set b := rot_of_hom (_ * _) * _.
suff : a = b by move=> ->.
rewrite {}/a {}/b; congr (_ * _).
rewrite spin_similarity; last by rewrite rot_of_hom_is_SO.
by rewrite -/t1 -/t2 -/r2 addrC.
Qed.
(* [murray] exercise 14 (b), p. 77 *)
Lemma AdjointM g1 g2 : g1 \is 'SE3[T] -> g2 \is 'SE3[T] ->
Adjoint (g1 * g2) = Adjoint g1 * Adjoint g2.
Proof.
move=> Hg1 Hg2.
rewrite [in RHS]/Adjoint -[in RHS]mulmxE.
set r1 := rot_of_hom g1. set r2 := rot_of_hom g2.
set t1 := trans_of_hom g1. set t2 := trans_of_hom g2.
rewrite (mulmx_block r1 _ _ _ r2) !(mul0mx,mulmx0,addr0,add0r).
rewrite AdjointM_helper // -/r1 -/r2 -/t1 -/t2; f_equal.
rewrite mulmxE mulrDr [in RHS]addrC -mulrA; congr (_ + _).
rewrite !mulrA; congr (_ * _ * _); rewrite -mulrA.
move: (rot_of_hom_is_SO Hg2).
rewrite rotationE orthogonalE => /andP[/eqP -> _]; by rewrite mulr1.
Qed.
End adjoint_theory.
Module EuclideanMotion.
Section euclidean_motion.
Variable T : comUnitRingType.
Record t : Type := mk {
trans_rot : 'rV[T]_3 * 'M[T]_3;
_ : trans_rot.2 \is 'SO[T]_3 }.
Arguments mk _ _ : clear implicits.
Implicit Types m : t.
Let homogeneous := 'M[T]_4.
Canonical t_subType := [subType for trans_rot].
Definition trans m := (trans_rot m).1.
Definition rot m := (trans_rot m).2.
Lemma rotP m : rot m \is 'SO[T]_3.
Proof. by case: m. Qed.
Local Hint Resolve rotP : core.
Let vector := 'rV[T]_3.
Let point := 'rV[T]_3.
Definition one := mk (0, _) (@rotation1 _ T).
Definition hom_of m := hom (rot m) (trans m).
Definition from_hom (M : homogeneous) (MSE : M \is 'SE3[T]) : t :=
mk (trans_of_hom M, _) (rot_of_hom_is_SO MSE).
Definition hrot m := hom (rot m) 0.
Definition htrans m := hom 1 (trans m).
Lemma hom_ofE m : hom_of m = hrot m *m htrans m.
Proof. by rewrite /trans /rot mulmxE homM mulr1 mul0mx add0r. Qed.
Lemma hom_of_is_SE3 m : hom_of m \is 'SE3[T].
Proof.
case: m => -[t r] rSO /=; apply/and3P; split.
- by rewrite /rot_of_hom /hom block_mxKul.
- by rewrite /hom block_mxKur.
- by rewrite /hom block_mxKdr.
Qed.
Definition inv' m : homogeneous := hom (rot m)^T (- trans m *m (rot m)^T).
Definition inv (m : t) : t := insubd one (- trans m *m (rot m)^T, (rot m)^T).
Lemma inv'V m : hom_of m *m inv' m = 1.
Proof.
rewrite /inv' mulmxE homM -rotation_inv // divrr; last first.
exact/orthogonal_unit/rotation_sub.
by rewrite mulNmx subrr hom10.
Qed.
Lemma invV m : hom_of m *m hom_of (inv m) = 1.
Proof.
case: m => -[t r] /= Hr.
rewrite /hom_of /inv /rot /trans /= insubdK; first exact: (inv'V (mk (t, r) Hr)).
by rewrite -rotationV in Hr.
Qed.
Lemma rot_inv m : rot (inv m) = (rot m)^T.
Proof.
case: m => -[t r] /= Hr.
rewrite /rot /trans /inv /= insubdK //=.
move: (Hr) => Hr'; by rewrite -rotationV in Hr.
Qed.
Lemma trans_inv m : trans (inv m) = - trans m *m (rot m)^T.
Proof.
case: m => -[t r] /= Hr.
rewrite /rot /trans /inv /= insubdK //=.
move: (Hr) => Hr'; by rewrite -rotationV in Hr.
Qed.
(* NB: not used, does not seem interesting *)
(*Definition inv_trans (T : t R) := hom 1 (- SE.trans T).
Lemma inv_transP (T : t R) : trans T *m inv_trans T = 1.
Proof.
by rewrite /trans /inv_trans mulmxE homM mulr1 trmx1 mulmx1 addrC subrr hom10.
Qed.*)
Definition hom_motion (m : t) (x : 'rV[T]_4) : 'rV[T]_4 := x *m hom_of m.
Lemma hom_motion_point (p : 'rV[T]_4) m : p \is 'hP[T] ->
hom_motion m p = from_h p *m row_mx (rot m) 0 + row_mx (trans m) 1.
Proof.
rewrite hpoint_from_h => /eqP Hp.
rewrite /hom_motion /= {1}Hp (mul_row_block (from_h p) 1 (rot m)).
by rewrite mulmx0 mulmx1 -add_row_mx mul1mx mul_mx_row mulmx0.
Qed.
Lemma hom_motion_vector (u : 'rV[T]_4) m : u \is 'hV[T] ->
hom_motion m u = from_h u *m row_mx (rot m) 0.
Proof.
rewrite hvector_from_h => /eqP Hu.
rewrite /hom_motion /= /hom {1}Hu (mul_row_block (from_h u) 0 (rot m)).
by rewrite mulmx0 mulmx1 -add_row_mx mul0mx mul_mx_row mulmx0 row_mx0 addr0.
Qed.
Lemma hom_motionB p q m : p \is 'hP[T] -> q \is 'hP[T] ->
hom_motion m p - hom_motion m q = hom_motion m (p - q).
Proof.
move=> Hu Hv.
rewrite hom_motion_point // hom_motion_point // opprD -addrCA -addrA subrr addr0 addrC.
by rewrite hom_motion_vector ?hpointB // from_hB mulmxBl.
Qed.
Definition motion_point m p := from_h (hom_motion m (to_hpoint p)).
Lemma motion_pointE u m : motion_point m u =
from_h (u *m row_mx (rot m) 0 + row_mx (trans m) 1).
Proof. by rewrite /motion_point hom_motion_point ?to_hpointP // to_hpointK. Qed.
Definition motion_vector m v := from_h (hom_motion m (to_hvector v)).
Lemma motion_vectorE u m : motion_vector m u = u *m rot m.
Proof.
rewrite /motion_vector hom_motion_vector ?to_hvectorP //.
by rewrite to_hvectorK mul_mx_row mulmx0 to_hvectorK.
Qed.
Lemma motion_pointB u v m :
motion_point m u - motion_point m v = motion_vector m (u - v).
Proof.
by rewrite /motion_point -from_hB hom_motionB ?to_hpointP // to_hpointB.
Qed.
End euclidean_motion.
Lemma motion_vector_preserves_norm (T : realType) (m : t T) :
{mono (motion_vector m) : u / norm u}.
Proof.
move=> ?; rewrite motion_vectorE orth_preserves_norm // rotation_sub //.
exact: rotP.
Qed.