"""Exact computable placements use rational centers and orthonormal two-column
frames. The universal theorem in P03 applies to arbitrary real linear isometric
embeddings, including normed ambient spaces; this finite matrix API is a
realization subclass, not a replacement hypothesis for that theorem.
"""
struct ExactPlacement
    center::Vector{Rat}
    frame::Matrix{Rat}
    function ExactPlacement(center::AbstractVector{<:Union{Integer,Rational}},
                            frame::AbstractMatrix{<:Union{Integer,Rational}})
        size(frame,1) == length(center) || throw(DimensionMismatch("center/frame"))
        size(frame,2) == 2 || throw(DimensionMismatch("frame must have two columns"))
        C=rational.(center);E=rational.(frame)
        for i in 1:2,j in 1:2
            sum(E[k,i]*E[k,j] for k in eachindex(C)) == rational(i==j) ||
                throw(ArgumentError("frame columns must be exactly orthonormal"))
        end
        new(C,E)
    end
end

function symbolic_heights()
    phi=variable(:phi); s=variable(:s)
    [-(phi^2+s),-(1+s),-s,s,1+s,phi^2+s]
end
function symbolic_marks(a::Poly=variable(:a))
    c=variable(:c); hs=symbolic_heights()
    [ [a*c,a*z] for z in hs ]
end
function placed_marks(a::Union{Integer,Rational},p::ExactPlacement)
    points=symbolic_marks(Poly(a))
    [[Poly(p.center[i])+p.frame[i,1]*z[1]+p.frame[i,2]*z[2]
      for i in eachindex(p.center)] for z in points]
end

"""Parametric descriptor of the entire curve, not a sampled polygon.
Arc endpoints are rational multiples of pi; segments are affine parameter maps.
Zero radius is a termination descriptor. Invoke geometric provenance only at a>0.
"""
function outline_spec(a::Union{Integer,Rational},p::ExactPlacement)
    a=rational(a)
    (radius=a,center=copy(p.center),frame=copy(p.frame),
     major_angle_pi=(rational(1,1296000),2-rational(1,1296000)),
     circle_rule="center + frame * [a*cos(t), a*sin(t)]",
     upper_levels=(variable(:s),variable(:phi)^2+variable(:s)),
     lower_levels=(-(variable(:phi)^2+variable(:s)),-variable(:s)),
     segment_rule="center + frame * [a*c, a*z]",marks=placed_marks(a,p))
end
function reconstruct_outline(h::Union{Integer,Rational},p::ExactPlacement,
                             terminal::Integer,remainders::AbstractVector{<:Integer})
    h=rational(h)
    h > 0 || throw(DomainError(h,"retained grid scale must be positive"))
    d=decode_checked(Q,S,terminal,remainders)
    outline_spec(h*d,p)
end

"""Optional BigFloat illustration of formulas; no approximate value certifies a theorem."""
function numerical_geometry(a::Rational; bits::Int=256)
    a > 0 || throw(DomainError(a,"positive geometric radius required"))
    setprecision(BigFloat,bits) do
        av=BigFloat(a);pv=(1+sqrt(BigFloat(5)))/2;πv=BigFloat(pi)
        δ=πv/1296000;sv=sin(δ);cv=cos(δ)
        hs=[-(pv^2+sv),-(1+sv),-sv,sv,1+sv,pv^2+sv]
        marks=[complex(av*cv,av*z) for z in hs]
        G=(abs(marks[1]-marks[2])+abs(marks[6]-marks[5]))/2
        C=av*(3*πv/2-πv/2)
        (marks=marks,G=G,C=C,lambda=G/(G+C),phi=pv,pi=πv,s=sv,c=cv,delta=δ)
    end
end