diff --git a/engine-proto/engine.jl b/engine-proto/engine.jl
index 8b82b47..df75fbe 100644
--- a/engine-proto/engine.jl
+++ b/engine-proto/engine.jl
@@ -1,34 +1,105 @@
 module Engine
 
-export Construction, Sphere, mprod, point
+export Construction, mprod
 
+import Subscripts
 using LinearAlgebra
+using AbstractAlgebra
 using Groebner
 
-mutable struct Construction
-  nextid::Int64
+# --- primitve elements ---
+
+mutable struct Point{T}
+  coords::Union{Vector{MPolyRingElem{T}}, Nothing}
+  vec::Union{Vector{MPolyRingElem{T}}, Nothing}
   
-  Construction(; nextid = 0) = new(nextid)
+  ## [to do] constructor argument never needed?
+  Point{T}(vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing) where T = new(vec)
 end
 
-struct Sphere{T<:Number}
-  vec::Vector{T}
-  id
+coordnames(_::Point) = [:xₚ, :yₚ, :zₚ]
+
+function buildvec(pt::Point, coordqueue)
+  pt.coords = splice!(coordqueue, 1:3)
+  coordring = parent(coordqueue[1])
+  pt.vec = [one(coordring), dot(pt.coords, pt.coords), pt.coords...]
+end
+
+mutable struct Sphere{T}
+  coords::Union{Vector{MPolyRingElem{T}}, Nothing}
+  vec::Union{Vector{MPolyRingElem{T}}, Nothing}
   
-  function Sphere(vec::Vector{T}, ctx::Construction) where T <: Number
-    id = ctx.nextid
-    ctx.nextid += 1
-    new{T}(vec, id)
+  Sphere{T}(vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing) where T = new(vec)
+end
+
+coordnames(_::Sphere) = [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
+
+function buildvec(sph::Sphere, coordqueue)
+  sph.coords = splice!(coordqueue, 1:5)
+  sph.vec = sph.coords
+end
+
+# --- primitive relations ---
+
+abstract type Relation{T} end
+
+mprod(v, w) = v[1]*w[2] + w[1]*v[2] - dot(v[3:end], w[3:end])
+
+struct LiesOn{T} <: Relation{T}
+  pt::Point{T}
+  sph::Sphere{T}
+end
+
+struct AlignsWithBy{T} <: Relation{T}
+  sph_v::Sphere{T}
+  sph_w::Sphere{T}
+  cos_angle::T
+end
+
+# --- constructions ---
+
+mutable struct Construction{T}
+  points::Vector{Point{T}}
+  spheres::Vector{Sphere{T}}
+  
+  Construction{T}(; points = Point{T}[], spheres = Sphere{T}[]) where T = new{T}(points, spheres)
+end
+
+function Base.push!(ctx::Construction{T}, elem::Point{T}) where T
+  push!(ctx.points, elem)
+end
+
+function Base.push!(ctx::Construction{T}, elem::Sphere{T}) where T
+  push!(ctx.spheres, elem)
+end
+
+function realize(ctx::Construction{T}) where T
+  # collect variable names
+  allcoordnames = Symbol[]
+  elements = vcat(ctx.points, ctx.spheres)
+  for (index, elem) in enumerate(elements)
+    subscript = Subscripts.sub(string(index))
+    append!(allcoordnames,
+      [Symbol(name, subscript) for name in coordnames(elem)]
+    )
+  end
+  
+  # construct coordinate ring
+  coordring, coordqueue = polynomial_ring(parent_type(T)(), allcoordnames)
+  
+  # construct coordinate vectors
+  for elem in elements
+    buildvec(elem, coordqueue)
   end
 end
 
-function mprod(sv::Sphere, sw::Sphere)
-  v = sv.vec
-  w = sw.vec
-  v[1]*w[2] + v[2]*w[1] - dot(v[3:end], w[3:end])
 end
 
-point(pt::Vector{<:Number}, ctx::Construction) =
-  Sphere([one(eltype(pt)), dot(pt, pt), pt...], ctx)
+# ~~~ sandbox setup ~~~
 
-end
\ No newline at end of file
+a = Engine.Point{Rational{Int64}}()
+b = Engine.Point{Rational{Int64}}()
+s = Engine.Sphere{Rational{Int64}}()
+ctx = Engine.Construction{Rational{Int64}}(points = [a])
+Engine.push!(ctx, b)
+Engine.push!(ctx, s)
\ No newline at end of file