Diagrams.Core.Transform

Diagrams defines the core library of primitives forming the basis of an embedded domain-specific language for describing and rendering diagrams.

The Transform module defines generic transformations parameterized by any vector space.

Transformations

Invertible linear transformations

data u :-: v infixr 7

(v1 :-: v2) is a linear map paired with its inverse.

Constructors

(u :-* v) :-: (v :-* u) infixr 7

Instances

HasLinearMap v => Monoid ((:-:) v v)

Invertible linear maps from a vector space to itself form a monoid under composition.

HasLinearMap v => Semigroup ((:-:) v v)

(<->) :: (HasLinearMap u, HasLinearMap v) => (u -> v) -> (v -> u) -> u :-: v

Create an invertible linear map from two functions which are assumed to be linear inverses.

linv :: (u :-: v) -> v :-: u

Invert a linear map.

lapp :: (VectorSpace v, Scalar u ~ Scalar v, HasLinearMap u) => (u :-: v) -> u -> v

Apply a linear map to a vector.

General transformations

data Transformation v

General (affine) transformations, represented by an invertible linear map, its transpose, and a vector representing a translation component.

By the transpose of a linear map we mean simply the linear map corresponding to the transpose of the map's matrix representation. For example, any scale is its own transpose, since scales are represented by matrices with zeros everywhere except the diagonal. The transpose of a rotation is the same as its inverse.

The reason we need to keep track of transposes is because it turns out that when transforming a shape according to some linear map L, the shape's normal vectors transform according to L's inverse transpose. This is exactly what we need when transforming bounding functions, which are defined in terms of perpendicular (i.e. normal) hyperplanes.

For more general, non-invertable transformations, see Diagrams.Deform (in diagrams-lib).

Constructors

Transformation (v :-: v) (v :-: v) v

Instances

HasLinearMap v => Monoid (Transformation v)

HasLinearMap v => Semigroup (Transformation v)

Transformations are closed under composition; t1 <> t2 is the transformation which performs first t2, then t1.

HasLinearMap v => HasOrigin (Transformation v)

HasLinearMap v => Transformable (Transformation v)

(HasLinearMap v, (~) * v (V a), Transformable a) => Action (Transformation v) a

Transformations can act on transformable things.

type V (Transformation v) = v

inv :: HasLinearMap v => Transformation v -> Transformation v

Invert a transformation.

transp :: Transformation v -> v :-: v

Get the transpose of a transformation (ignoring the translation component).

transl :: Transformation v -> v

Get the translational component of a transformation.

apply :: HasLinearMap v => Transformation v -> v -> v

Apply a transformation to a vector. Note that any translational component of the transformation will not affect the vector, since vectors are invariant under translation.

papply :: HasLinearMap v => Transformation v -> Point v -> Point v

Apply a transformation to a point.

fromLinear :: AdditiveGroup v => (v :-: v) -> (v :-: v) -> Transformation v

Create a general affine transformation from an invertible linear transformation and its transpose. The translational component is assumed to be zero.

basis :: forall v. HasLinearMap v => [v]

Get the matrix equivalent of the basis of the vector space v as a list of columns.

onBasis :: forall v. HasLinearMap v => Transformation v -> ([v], v)

Get the matrix equivalent of the linear transform, (as a list of columns) and the translation vector. This is mostly useful for implementing backends.

matrixRep :: HasLinearMap v => Transformation v -> [[Scalar v]]

Convert a `Transformation v` to a matrix representation as a list of column vectors which are also lists.

determinant :: (HasLinearMap v, Num (Scalar v)) => Transformation v -> Scalar v

The determinant of a Transformation.

The Transformable class

class (HasBasis v, HasTrie (Basis v), VectorSpace v) => HasLinearMap v

HasLinearMap is a poor man's class constraint synonym, just to help shorten some of the ridiculously long constraint sets.

Instances

(HasBasis v, HasTrie (Basis v), VectorSpace v) => HasLinearMap v

class HasLinearMap (V t) => Transformable t where

Type class for things t which can be transformed.

Methods

transform :: Transformation (V t) -> t -> t

Apply a transformation to an object.

Instances

Transformable Double

Transformable Rational

Transformable t => Transformable [t]

(Transformable t, Ord t) => Transformable (Set t)

Transformable m => Transformable (Deletable m)

HasLinearMap v => Transformable (Point v)

Transformable t => Transformable (TransInv t)

HasLinearMap v => Transformable (Transformation v)

HasLinearMap v => Transformable (Style v)

HasLinearMap v => Transformable (Attribute v)

HasLinearMap v => Transformable (Trace v)

(HasLinearMap v, InnerSpace v, Floating (Scalar v)) => Transformable (Envelope v)

(HasBasis (V b), HasTrie (Basis (V b)), Transformable a, Transformable b, (~) * (V b) (V a)) => Transformable (a -> b)

(Transformable a, Transformable b, (~) * (V a) (V b)) => Transformable (a, b)

Transformable t => Transformable (Map k t)

HasLinearMap v => Transformable (Query v m)

HasLinearMap v => Transformable (Prim b v)

The Transformable instance for Prim just pushes calls to transform down through the Prim constructor.

(Transformable a, Transformable b, Transformable c, (~) * (V a) (V b), (~) * (V a) (V c)) => Transformable (a, b, c)

(InnerSpace v, Floating (Scalar v), HasLinearMap v) => Transformable (SubMap b v m)

(HasLinearMap v, InnerSpace v, Floating (Scalar v)) => Transformable (Subdiagram b v m)

(HasLinearMap v, OrderedField (Scalar v), InnerSpace v, Semigroup m) => Transformable (QDiagram b v m)

Diagrams can be transformed by transforming each of their components appropriately.

Translational invariance

newtype TransInv t

TransInv is a wrapper which makes a transformable type translationally invariant; the translational component of transformations will no longer affect things wrapped in TransInv.

Constructors

TransInv t

Instances

Eq t => Eq (TransInv t)

Ord t => Ord (TransInv t)

Show t => Show (TransInv t)

Monoid t => Monoid (TransInv t)

Semigroup t => Semigroup (TransInv t)

Wrapped (TransInv t)

VectorSpace (V t) => HasOrigin (TransInv t)

Transformable t => Transformable (TransInv t)

Qualifiable a => Qualifiable (TransInv a)

Traced t => Traced (TransInv t)

Enveloped t => Enveloped (TransInv t)

Rewrapped (TransInv t) (TransInv t')

type Unwrapped (TransInv t) = t

type V (TransInv t) = V t

Vector space independent transformations

Most transformations are specific to a particular vector space, but a few can be defined generically over any vector space.

translation :: HasLinearMap v => v -> Transformation v

Create a translation.

translate :: (Transformable t, HasLinearMap (V t)) => V t -> t -> t

Translate by a vector.

scaling :: (HasLinearMap v, Fractional (Scalar v)) => Scalar v -> Transformation v

Create a uniform scaling transformation.

scale :: (Transformable t, Fractional (Scalar (V t)), Eq (Scalar (V t))) => Scalar (V t) -> t -> t

Scale uniformly in every dimension by the given scalar.