import Mathlib.Data.Real.Basic import Mathlib.Data.Fin.Basic import Mathlib.LinearAlgebra.Matrix.Notation import Mathlib.Analysis.InnerProductSpace.Basic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic namespace Graphics3D /-- A 3D point represented as three real coordinates -/ structure Point3D where x : ℝ y : ℝ z : ℝ /-- A 3x3 rotation matrix -/ structure RotationMatrix where m00 : ℝ m01 : ℝ m02 : ℝ m10 : ℝ m11 : ℝ m12 : ℝ m20 : ℝ m21 : ℝ m22 : ℝ /-- A translation vector -/ structure Translation where dx : ℝ dy : ℝ dz : ℝ /-- Apply a rotation matrix to a 3D point -/ def rotate (R : RotationMatrix) (p : Point3D) : Point3D := { x := R.m00 * p.x + R.m01 * p.y + R.m02 * p.z, y := R.m10 * p.x + R.m11 * p.y + R.m12 * p.z, z := R.m20 * p.x + R.m21 * p.y + R.m22 * p.z } /-- Apply a translation to a 3D point -/ def translate (t : Translation) (p : Point3D) : Point3D := { x := p.x + t.dx, y := p.y + t.dy, z := p.z + t.dz } /-- Compute the squared norm (length squared) of a 3D point -/ def normSquared (p : Point3D) : ℝ := p.x * p.x + p.y * p.y + p.z * p.z /-- Compute the norm (length) of a 3D point -/ noncomputable def norm (p : Point3D) : ℝ := Real.sqrt (normSquared p) /-- A rotation matrix is orthogonal if its columns form an orthonormal basis. This is a key property that ensures rotations preserve distances. -/ def isOrthogonal (R : RotationMatrix) : Prop := -- Column 1 · Column 1 = 1 R.m00 * R.m00 + R.m10 * R.m10 + R.m20 * R.m20 = 1 ∧ -- Column 2 · Column 2 = 1 R.m01 * R.m01 + R.m11 * R.m11 + R.m21 * R.m21 = 1 ∧ -- Column 3 · Column 3 = 1 R.m02 * R.m02 + R.m12 * R.m12 + R.m22 * R.m22 = 1 ∧ -- Column 1 · Column 2 = 0 R.m00 * R.m01 + R.m10 * R.m11 + R.m20 * R.m21 = 0 ∧ -- Column 1 · Column 3 = 0 R.m00 * R.m02 + R.m10 * R.m12 + R.m20 * R.m22 = 0 ∧ -- Column 2 · Column 3 = 0 R.m01 * R.m02 + R.m11 * R.m12 + R.m21 * R.m22 = 0 /-- The identity rotation (no rotation) -/ def identityRotation : RotationMatrix := { m00 := 1, m01 := 0, m02 := 0, m10 := 0, m11 := 1, m12 := 0, m20 := 0, m21 := 0, m22 := 1 } /-- Rotation around the Z-axis by angle θ (in the XY plane) -/ noncomputable def rotationZ (θ : ℝ) : RotationMatrix := { m00 := Real.cos θ, m01 := -Real.sin θ, m02 := 0, m10 := Real.sin θ, m11 := Real.cos θ, m12 := 0, m20 := 0, m21 := 0, m22 := 1 } /-- A transformation pipeline that first rotates, then translates -/ def transformPipeline (R : RotationMatrix) (t : Translation) (p : Point3D) : Point3D := translate t (rotate R p) /-- THEOREM: Orthogonal transformations preserve the squared norm. This is a fundamental property in computer graphics: rotations don't change the distance of a point from the origin. -/ theorem rotation_preserves_norm_squared (R : RotationMatrix) (p : Point3D) (h : isOrthogonal R) : normSquared (rotate R p) = normSquared p := by unfold normSquared rotate isOrthogonal at * obtain ⟨h1, h2, h3, h4, h5, h6⟩ := h -- The proof: expand the LHS and use orthogonality conditions calc (R.m00 * p.x + R.m01 * p.y + R.m02 * p.z) * (R.m00 * p.x + R.m01 * p.y + R.m02 * p.z) + (R.m10 * p.x + R.m11 * p.y + R.m12 * p.z) * (R.m10 * p.x + R.m11 * p.y + R.m12 * p.z) + (R.m20 * p.x + R.m21 * p.y + R.m22 * p.z) * (R.m20 * p.x + R.m21 * p.y + R.m22 * p.z) _ = p.x * p.x * (R.m00 * R.m00 + R.m10 * R.m10 + R.m20 * R.m20) + p.y * p.y * (R.m01 * R.m01 + R.m11 * R.m11 + R.m21 * R.m21) + p.z * p.z * (R.m02 * R.m02 + R.m12 * R.m12 + R.m22 * R.m22) + 2 * p.x * p.y * (R.m00 * R.m01 + R.m10 * R.m11 + R.m20 * R.m21) + 2 * p.x * p.z * (R.m00 * R.m02 + R.m10 * R.m12 + R.m20 * R.m22) + 2 * p.y * p.z * (R.m01 * R.m02 + R.m11 * R.m12 + R.m21 * R.m22) := by ring _ = p.x * p.x * 1 + p.y * p.y * 1 + p.z * p.z * 1 + 2 * p.x * p.y * 0 + 2 * p.x * p.z * 0 + 2 * p.y * p.z * 0 := by rw [h1, h2, h3, h4, h5, h6] _ = p.x * p.x + p.y * p.y + p.z * p.z := by ring /-- THEOREM: Translation changes position but preserves relative distances. The norm changes by a predictable amount based on the translation vector. -/ theorem translation_norm_relation (t : Translation) (p : Point3D) : normSquared (translate t p) = normSquared p + 2 * (t.dx * p.x + t.dy * p.y + t.dz * p.z) + normSquared ⟨t.dx, t.dy, t.dz⟩ := by unfold normSquared translate ring /-- THEOREM: The identity rotation leaves points unchanged. -/ theorem identity_rotation_is_identity (p : Point3D) : rotate identityRotation p = p := by unfold rotate identityRotation simp /-- THEOREM: The identity rotation is orthogonal. -/ theorem identity_is_orthogonal : isOrthogonal identityRotation := by unfold isOrthogonal identityRotation simp /-- THEOREM: Rotation around the Z-axis is orthogonal for any angle θ. This uses the fundamental trigonometric identity: cos²θ + sin²θ = 1 -/ theorem rotationZ_is_orthogonal (θ : ℝ) : isOrthogonal (rotationZ θ) := by unfold isOrthogonal rotationZ constructor · simp; ring_nf; exact Real.cos_sq_add_sin_sq θ constructor · simp; ring_nf; rw [Real.sin_sq, Real.cos_sq]; ring constructor · ring constructor · ring constructor · ring · ring /-! ## Example Usage Below we demonstrate the transformation pipeline with concrete values. -/ /-- Example: A point at (1, 0, 0) -/ def examplePoint : Point3D := ⟨1, 0, 0⟩ /-- Example: Rotate 90 degrees around Z-axis -/ noncomputable def rotate90Z : RotationMatrix := rotationZ (Real.pi / 2) /-- Example: Translate by (5, 3, 2) -/ def exampleTranslation : Translation := ⟨5, 3, 2⟩ /-- Apply the transformation pipeline -/ noncomputable def transformedPoint : Point3D := transformPipeline rotate90Z exampleTranslation examplePoint end Graphics3D -- Check that definitions type-check #check Graphics3D.Point3D #check Graphics3D.rotate #check Graphics3D.transformPipeline -- Verify theorems are proven #check Graphics3D.rotation_preserves_norm_squared #check Graphics3D.identity_is_orthogonal #check Graphics3D.rotationZ_is_orthogonal -- Test with concrete values open Graphics3D #check transformedPoint #eval examplePoint.x