Geometry Dihedral Derivatives Has Deriv At Dihedral Angle3 Sq Along

The angle's rate of change

A dihedral angle is the angle between two faces of a solid shape, such as the angle at which two faces of a pyramid meet. In a tetrahedron, the simplest three-dimensional solid with four triangular faces, each of its six edges is shared by two faces, and the angle between those two faces is a dihedral angle. The declaration hasDerivAt_dihedralAngle3Sq_along concerns how one such angle changes when the tetrahedron's edges are stretched or compressed along a smooth one-parameter path.

Classically, the rate of change of an angle is a derivative, a concept from calculus that measures how fast a quantity changes. The theorem states that if the cosine of the dihedral angle has a derivative along the path, and if that cosine stays strictly between -1 and 1 at the point of interest, then the dihedral angle itself has a derivative along the path. The formula for that derivative is the standard chain rule for the arccosine function: the derivative of arccos(f) is -f' divided by the square root of 1 minus f squared. In plainer terms, the angle's rate of change is the cosine's rate of change, scaled by a factor that depends on the current angle, and the scaling factor blows up as the angle approaches 0 or 180 degrees, where the cosine approaches 1 or -1.

In Recognition Science, this theorem is a piece of the analytic machinery used to study how tetrahedra respond to deformation. The framework models physical structure through a discrete record of events, and the geometry of tetrahedra enters through Cayley-Menger determinants, which encode volumes and angles from edge lengths alone. The declaration isolates the calculus step: it takes as given that the cosine of the dihedral angle has a derivative, and it derives the derivative of the angle itself. The heavier algebraic work, computing the cosine's derivative from the edge coordinates, belongs to a separate layer of cofactor derivatives.

The theorem is proved in a machine-checked library of formal theorems, meaning the derivation is verified by a computer from first principles. It is a local statement: it applies at a single point along the path, not globally. It requires the cosine to stay away from the endpoints -1 and 1, which corresponds to the angle not being exactly 0 or 180 degrees, where the arccosine function is not differentiable. The theorem does not compute the cosine's derivative itself; it assumes that derivative is known. It also does not establish that any particular tetrahedron exists or that a path is physically realizable; it only relates the derivatives of the angle and its cosine along a given path.

What this means in practice is that downstream calculations about how tetrahedral angles respond to edge deformation have a firm foundation: one can differentiate the angle as long as one can differentiate its cosine and avoid the degenerate flat cases. The theorem turns a geometric question into a calculus one, and the calculus step is now machine-checked, so the remaining work is purely algebraic.

/-- Derivative of a dihedral angle along any one-parameter squared-edge path,
assuming the cofactor cosine derivative along that path is known and the
cosine stays away from the arccos endpoints at the base point. -/
theorem hasDerivAt_dihedralAngle3Sq_along
    {γ : ℝ → CayleyMengerPolynomial.SqEdges} {x cosDeriv : ℝ} (e : Fin 6)
    (hcos : HasDerivAt (fun t : ℝ => dihedralCos3Sq (γ t) e) cosDeriv x)
    (hm : dihedralCos3Sq (γ x) e ≠ -1)
    (hp : dihedralCos3Sq (γ x) e ≠ 1) :
    HasDerivAt (fun t : ℝ => dihedralAngle3Sq (γ t) e)
      (-(1 / Real.sqrt (1 - (dihedralCos3Sq (γ x) e) ^ 2)) * cosDeriv) x := by
  exact hasDerivAt_arccos_comp hcos hm hp

What this page does not claim

The theorem does not compute the derivative of the cosine of the dihedral angle; it assumes that derivative is known. The theorem does not assert that any particular tetrahedron exists or that a path is physically realizable. The theorem does not apply at angles of exactly 0 or 180 degrees, where the cosine is 1 or -1 and the arccosine is not differentiable.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Geometry/DihedralDerivatives.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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