signature VECTOR =
  sig
    type vector
    val zero : vector
    val scale : real * vector -> vector
    val add : vector * vector -> vector
    (* scalar product: dot ((x1,...,xn),(y1,...,yn)) = x1*y1 + ... + xn*yn *)
    val dot : vector * vector -> real
  end

signature POINT =
  sig
    structure Vector : VECTOR
    type point
    (* move a point along a vector *)
    val translate : point * Vector.vector -> point
    (* the vector from a to b *)
    val ray : point * point -> Vector.vector
  end

signature SPHERE =
  sig
    structure Vector : VECTOR
    structure Point : POINT
    sharing Point.Vector = Vector
    type sphere
    val sphere : Point.point * Vector.vector -> sphere
  end

signature GEOMETRY =
  sig
    structure Point : POINT
    structure Sphere : SPHERE
    sharing Point = Sphere.Point
        and Point.Vector = Sphere.Vector
  end