Documentation

This is the reified form of the property that the interior of a semimanifold is a manifold. These constraints would ideally be expressed directly as superclass constraints, but that would require the UndecidableSuperclasses extension, which is not reliable yet.

Also, if all those equality constraints are in scope, GHC tends to infer needlessly complicated types like Needle (Needle (Needle x)), which is the same as just Needle x.

This is the class underlying what we understand as manifolds.

The interface is almost identical to the better-known AffineSpace class, but we don't require associativity of .+~^ with ^+^ – except in an asymptotic sense for small vectors.

That innocent-looking change makes the class applicable to vastly more general types: while an affine space is basically nothing but a vector space without particularly designated origin, a pseudo-affine space can have nontrivial topology on the global scale, and yet be used in practically the same way as an affine space. At least the usual spheres and tori make good instances, perhaps the class is in fact equivalent to manifolds in their usual maths definition (with an atlas of charts: a family of overlapping regions of the topological space, each homeomorphic to the Needle vector space or some simply-connected subset thereof).

The Semimanifold and PseudoAffine classes can be anyclass-derived or empty-instantiated based on Generic for product types (including newtypes) of existing PseudoAffine instances. For example, the definition

data Cylinder = CylinderPolar { zCyl :: !D¹, φCyl :: !S¹ }
  deriving (Generic, Semimanifold, PseudoAffine)

is equivalent to

data Cylinder = CylinderPolar { zCyl :: !D¹, φCyl :: !S¹ }

data CylinderNeedle = CylinderPolarNeedle { δzCyl :: !(Needle D¹), δφCyl :: !(Needle S¹) }

instance Semimanifold Cylinder where
  type Needle Cylinder = CylinderNeedle
  CylinderPolar z φ .+~^ CylinderPolarNeedle δz δφ
       = CylinderPolar (z.+~^δz) (φ.+~^δφ)

instance PseudoAffine Cylinder where
  CylinderPolar z₁ φ₁ .-~. CylinderPolar z₀ φ₀
       = CylinderPolarNeedle $ z₁.-~.z₀ * φ₁.-~.φ₀
  CylinderPolar z₁ φ₁ .-~! CylinderPolar z₀ φ₀
       = CylinderPolarNeedle (z₁.-~!z₀) (φ₁.-~.φ₀)

A fibre bundle combines points in the base space b with points in the fibre f. The type FibreBundle b f is thus isomorphic to the tuple space (b,f), but it can have a different topology, the prime example being TangentBundle, where nearby points may have differently-oriented tangent spaces.

Points on a manifold, combined with vectors in the respective tangent space.

Interpolate between points, approximately linearly. For points that aren't close neighbours (i.e. lie in an almost flat region), the pathway is basically undefined – save for its end points.

A proper, really well-defined (on global scales) interpolation only makes sense on a Riemannian manifold, as Geodesic.

Like palerp, but actually restricted to the interval between the points, with a signature like geodesicBetween rather than alerp.

Like alerp, but actually restricted to the interval between the points.

A (closed) cone over a space x is the product of x with the closed interval D¹ of “heights”, except on its “tip”: here, x is smashed to a single point.

This construct becomes (homeomorphic-to-) an actual geometric cone (and to D²) in the special case x = S¹.

An open cone is homeomorphic to a closed cone without the “lid”, i.e. without the “last copy” of x, at the far end of the height interval. Since that means the height does not include its supremum, it is actually more natural to express it as the entire real ray, hence the name.