mathlib documentation

linear_algebra.affine_space.midpoint

Midpoint of a segment

Main definitions

Main theorems

We do not mark most lemmas as @[simp] because it is hard to tell which side is simpler.

Tags

midpoint, add_monoid_hom

def midpoint (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (x y : P) :
P

midpoint x y is the midpoint of the segment [x, y].

Equations
@[simp]
theorem affine_map.map_midpoint {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] [add_comm_group V'] [semimodule R V'] [affine_space V' P'] (f : P →ᵃ[R] P') (a b : P) :
f (midpoint R a b) = midpoint R (f a) (f b)

@[simp]
theorem affine_equiv.map_midpoint {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] [add_comm_group V'] [semimodule R V'] [affine_space V' P'] (f : P ≃ᵃ[R] P') (a b : P) :
f (midpoint R a b) = midpoint R (f a) (f b)

@[simp]
theorem affine_equiv.point_reflection_midpoint_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (x y : P) :

theorem midpoint_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (x y : P) :
midpoint R x y = midpoint R y x

@[simp]
theorem affine_equiv.point_reflection_midpoint_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (x y : P) :

theorem midpoint_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (p₁ p₂ p₃ p₄ : P) :
midpoint R p₁ p₂ -ᵥ midpoint R p₃ p₄ = midpoint R (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)

theorem midpoint_vadd_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (v v' : V) (p p' : P) :
midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p')

theorem midpoint_eq_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] {x y z : P} :

theorem midpoint_eq_midpoint_iff_vsub_eq_vsub (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] {x x' y y' : P} :
midpoint R x y = midpoint R x' y' x -ᵥ x' = y' -ᵥ y

theorem midpoint_eq_iff' (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] {x y z : P} :

theorem midpoint_unique (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (R' : Type u_3) [ring R'] [invertible 2] [semimodule R' V] (x y : P) :
midpoint R x y = midpoint R' x y

midpoint does not depend on the ring R.

@[simp]
theorem midpoint_self (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (x : P) :
midpoint R x x = x

@[simp]
theorem midpoint_add_self (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] (x y : V) :
midpoint R x y + midpoint R x y = x + y

theorem midpoint_zero_add (R : Type u_1) {V : Type u_2} [ring R] [invertible 2] [add_comm_group V] [semimodule R V] (x y : V) :
midpoint R 0 (x + y) = midpoint R x y

theorem line_map_inv_two {R : Type u_1} {V : Type u_2} {P : Type u_3} [division_ring R] [char_zero R] [add_comm_group V] [semimodule R V] [affine_space V P] (a b : P) :

theorem line_map_one_half {R : Type u_1} {V : Type u_2} {P : Type u_3} [division_ring R] [char_zero R] [add_comm_group V] [semimodule R V] [affine_space V P] (a b : P) :

theorem homothety_inv_of_two {R : Type u_1} {V : Type u_2} {P : Type u_3} [comm_ring R] [invertible 2] [add_comm_group V] [semimodule R V] [affine_space V P] (a b : P) :

theorem homothety_inv_two {k : Type u_1} {V : Type u_2} {P : Type u_3} [field k] [char_zero k] [add_comm_group V] [semimodule k V] [affine_space V P] (a b : P) :

theorem homothety_one_half {k : Type u_1} {V : Type u_2} {P : Type u_3} [field k] [char_zero k] [add_comm_group V] [semimodule k V] [affine_space V P] (a b : P) :

@[simp]
theorem pi_midpoint_apply {k : Type u_1} {ι : Type u_2} {V : ι → Type u_3} {P : ι → Type u_4} [field k] [invertible 2] [Π (i : ι), add_comm_group (V i)] [Π (i : ι), semimodule k (V i)] [Π (i : ι), affine_space (V i) (P i)] (f g : Π (i : ι), P i) (i : ι) :
midpoint k f g i = midpoint k (f i) (g i)

def add_monoid_hom.of_map_midpoint (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [ring R] [invertible 2] [add_comm_group E] [semimodule R E] [ring R'] [invertible 2] [add_comm_group F] [semimodule R' F] (f : E → F) (h0 : f 0 = 0) (hm : ∀ (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) :
E →+ F

A map f : E → F sending zero to zero and midpoints to midpoints is an add_monoid_hom.

Equations
@[simp]
theorem add_monoid_hom.coe_of_map_midpoint (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [ring R] [invertible 2] [add_comm_group E] [semimodule R E] [ring R'] [invertible 2] [add_comm_group F] [semimodule R' F] (f : E → F) (h0 : f 0 = 0) (hm : ∀ (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) :