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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import OAI.Combinatorics.Crossing.PhysicalCircleAveraging
namespace OAI
noncomputable section
namespace Zarankiewicz
theorem similarity_circleOffset (o u v : Plane) (δ : ℝ) :
similarityPoint o u (circleOffset v δ)-o =
Real.sqrt (1-δ^2) • (similarityPoint o u v-o) +
δ • rotateQuarter (similarityPoint o u v-o) := by
apply Prod.ext <;> dsimp [similarityPoint,circleOffset,rotateQuarter] <;> ring
theorem rotated_parallel_det_ne_zero (d w : Plane) (hd : d ≠ 0) (hw : w ≠ 0)
(hpar : det2 d w=0) : det2 d (rotateQuarter w) ≠ 0 := by
have hp := vector_parallel_of_det_zero d w hd hpar
have hc : dot2 d w/squaredLength d ≠ 0 := by
intro h
rw [h,zero_smul] at hp
exact hw hp
have he : det2 d (rotateQuarter w)=(dot2 d w/squaredLength d)*squaredLength d := by
conv_lhs => rw [hp]
dsimp [det2,rotateQuarter,squaredLength]
ring
rw [he]
exact mul_ne_zero hc (ne_of_gt (squaredLength_pos d hd))
def circleTrackScale (d w : Plane) : ℝ := (det2 d (rotateQuarter w))⁻¹
theorem circleTrackScale_ne_zero (d w : Plane) (hd : d ≠ 0) (hw : w ≠ 0)
(hpar : det2 d w=0) : circleTrackScale d w ≠ 0 :=
inv_ne_zero (rotated_parallel_det_ne_zero d w hd hw hpar)
theorem circleOffset_track_level (o u v d : Plane) (hd : d ≠ 0)
(hw : similarityPoint o u v-o ≠ 0)
(hpar : det2 d (similarityPoint o u v-o)=0) (ε : ℝ) :
det2 d (similarityPoint o u (circleOffset v (circleTrackScale d (similarityPoint o u v-o)*ε))-
similarityPoint o u v)=ε := by
let w := similarityPoint o u v-o
let δ := circleTrackScale d w*ε
have he : similarityPoint o u (circleOffset v δ)-similarityPoint o u v =
(Real.sqrt (1-δ^2)-1) • w+δ • rotateQuarter w := by
have hh := similarity_circleOffset o u v δ
calc
_ = (similarityPoint o u (circleOffset v δ)-o)-(similarityPoint o u v-o) := by abel
_ = Real.sqrt (1-δ^2) • w+δ • rotateQuarter w-w := by rw [hh]
_ = _ := by rw [sub_smul,one_smul]; abel
change det2 d (similarityPoint o u (circleOffset v δ)-similarityPoint o u v)=ε
rw [he,det2_add_right,det2_smul_right,det2_smul_right,hpar,mul_zero,zero_add]
have hn := rotated_parallel_det_ne_zero d w hd hw hpar
dsimp [δ,circleTrackScale]
field_simp
end Zarankiewicz
end
end OAI
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