Wumpus.Core.Geometry

2D geometry

Type family

type family DUnit a :: *

Some unit of dimension usually double.

Data types

data Vec2 a

Constructors

V2 !a !a

Instances

Functor Vec2

Num a => MatrixMult Matrix3'3 (Vec2 a)

Eq a => Eq (Vec2 a)

Show a => Show (Vec2 a)

Num a => Monoid (Vec2 a)

Num a => VectorSpace (Vec2 a)

(Scalar a ~ a, Num a, InnerSpace a) => InnerSpace (Vec2 a)

Num a => AdditiveGroup (Vec2 a)

Pretty a => Pretty (Vec2 a)

Pointwise (Vec2 a)

Num u => Translate (Vec2 u)

Num u => Scale (Vec2 u)

(Floating a, Real a) => RotateAbout (Vec2 a)

(Floating a, Real a) => Rotate (Vec2 a)

type DVec2 = Vec2 Double

data Point2 a

Constructors

P2 !a !a

Instances

Functor Point2

Num a => MatrixMult Matrix3'3 (Point2 a)

Eq a => Eq (Point2 a)

Show a => Show (Point2 a)

Num a => AffineSpace (Point2 a)

Pretty a => Pretty (Point2 a)

Ord a => CMinMax (Point2 a)

Pointwise (Point2 a)

Num u => Translate (Point2 u)

translate x y.

Num u => Scale (Point2 u)

(Floating a, Real a) => RotateAbout (Point2 a)

(Floating a, Real a) => Rotate (Point2 a)

type DPoint2 = Point2 Double

data Frame2 a

A two dimensional frame.

The components are the two basis vectors e0 and e1 and the origin o.

Typically these names for the elements will be used:

 Frame2 (V2 e0x e0y) (V2 e1x e1y) (P2 ox oy)

Constructors

Frame2 (Vec2 a) (Vec2 a) (Point2 a)

Instances

Eq a => Eq (Frame2 a)

Show a => Show (Frame2 a)

(Num a, InnerSpace (Vec2 a)) => Monoid (Frame2 a)

Pretty a => Pretty (Frame2 a)

ToCTM (Frame2 a)

type DFrame2 = Frame2 Double

data Matrix3'3 a

3x3 matrix, considered to be in row-major form.

 (M3'3 a b c
       d e f
       g h i)

For instance the rotation matrix is represented as

  ( cos(a) -sin(a) 0
    sin(a)  cos(a) 0  
      0         0  1 )

This is congruent with the form presented in Santos - Example 45, page 17 extended to 3x3.

ref. David A. Santos Multivariable and Vector Calculus, July 17, 2008 Version.

The right-most column is considered to represent a coordinate:

  ( 1 0 x
    0 1 y  
    0 0 1 ) 

So a translation matrix representing the displacement in x of 40 and in y of 10 would be:

  ( 1 0 40
    0 1 10  
    0 0 1  ) 

Constructors

M3'3 !a !a !a !a !a !a !a !a !a

Instances

Functor Matrix3'3

Num a => MatrixMult Matrix3'3 (Point2 a)

Num a => MatrixMult Matrix3'3 (Vec2 a)

Eq a => Eq (Matrix3'3 a)

Num a => Num (Matrix3'3 a)

Show a => Show (Matrix3'3 a)

Num a => VectorSpace (Matrix3'3 a)

Num a => AdditiveGroup (Matrix3'3 a)

PSUnit a => Pretty (Matrix3'3 a)

ToCTM (Matrix3'3 a)

type DMatrix3'3 = Matrix3'3 Double

data Radian

Radian is represented with a distinct type. Equality and ordering are approximate where the epsilon is 0.0001.

Instances

Eq Radian

Floating Radian

Fractional Radian

Num Radian

Ord Radian

Real Radian

RealFloat Radian

RealFrac Radian

Show Radian

Pretty Radian

Pointwise type class

class Pointwise sh where

Pointwise is a Functor like type class, except that the container/element relationship is defined by a type family rather than a type parameter. This means that applied function must be type preserving.

Associated Types

type Pt sh :: *

Methods

pointwise :: (Pt sh -> Pt sh) -> sh -> sh

Instances

Pointwise a => Pointwise [a]

Pointwise (Point2 a)

Pointwise (Vec2 a)

Pointwise (BoundingBox a)

Pointwise (PathSegment u)

Pointwise (Path u)

Pointwise (a -> a)

Matrix multiply type class

class MatrixMult mat t where

Associated Types

type MatrixParam t :: *

Methods

(*#) :: MatrixParam t ~ a => mat a -> t -> t

Instances

Num a => MatrixMult Matrix3'3 (Point2 a)

Num a => MatrixMult Matrix3'3 (Vec2 a)

Vector operations

hvec :: Num a => a -> Vec2 a

Construct a vector with horizontal displacement.

vvec :: Num a => a -> Vec2 a

Construct a vector with vertical displacement.

avec :: Floating a => Radian -> a -> Vec2 a

Construct a vector from an angle and magnitude.

Point operations

zeroPt :: Num a => Point2 a

langle :: (Floating u, Real u) => Point2 u -> Point2 u -> Radian

Frame operations

ortho :: Num a => Point2 a -> Frame2 a

displaceOrigin :: Num a => Vec2 a -> Frame2 a -> Frame2 a

pointInFrame :: Num a => Point2 a -> Frame2 a -> Point2 a

frame2Matrix :: Num a => Frame2 a -> Matrix3'3 a

Concatenate the elements of the frame as columns forming a 3x3 matrix. Points and vectors are considered homogeneous coordinates - triples where the least element is either 0 indicating a vector or 1 indicating a point:

 Frame (V2 e0x e0y) (V2 e1x e1y) (P2 ox oy)

becomes

 (M3'3 e0x e1x ox
       e0y e1y oy
        0   0   1  )

matrix2Frame :: Matrix3'3 a -> Frame2 a

Interpret the matrix as columns forming a frame.

 (M3'3 e0x e1x ox
       e0y e1y oy
        0   0   1  )

becomes

 Frame (V2 e0x e0y) (V2 e1x e1y) (P2 ox oy)

frameProduct :: (Num a, InnerSpace (Vec2 a)) => Frame2 a -> Frame2 a -> Frame2 a

Multiplication of frames to form their product.

standardFrame :: Num a => Frame2 a -> Bool

Is the origin at (0,0) and are the basis vectors orthogonal with unit length?

Matrix contruction

identityMatrix :: Num a => Matrix3'3 a

Construct the identity matrix:

 (M3'3 1 0 0
       0 1 0
       0 0 1 )

scalingMatrix :: Num a => a -> a -> Matrix3'3 a

Construct a scaling matrix:

 (M3'3 sx 0  0
       0  sy 0
       0  0  1 )

translationMatrix :: Num a => a -> a -> Matrix3'3 a

rotationMatrix :: (Floating a, Real a) => Radian -> Matrix3'3 a

originatedRotationMatrix :: (Floating a, Real a) => Radian -> Point2 a -> Matrix3'3 a

matrix operations

invert :: Fractional a => Matrix3'3 a -> Matrix3'3 a

determinant :: Num a => Matrix3'3 a -> a

transpose :: Matrix3'3 a -> Matrix3'3 a

Radian operations

req :: Radian -> Radian -> Bool

toRadian :: Real a => a -> Radian

fromRadian :: Fractional a => Radian -> a

d2r :: (Floating a, Real a) => a -> Radian

Degrees to radians.

r2d :: (Floating a, Real a) => Radian -> a

Radians to degrees.