ůk      !"#$%&'()*+,-./0123456789:;<=>? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j portable provisionalEdward Kmett <ekmett@gmail.com>None klmnopqrstu klmnopqrstuportable provisionalEdward Kmett <ekmett@gmail.com>None 9A vector is an additive group with additional structure. The zero vector Compute the sum of two vectors V2 1 2 ^+^ V2 3 4V2 4 6+Compute the difference between two vectors V2 4 5 - V2 3 1V2 1 4*Linearly interpolate between two vectors. !Compute the negation of a vector negated (V2 2 4) V2 (-2) (-4) Compute the left scalar product  2 *^ V2 3 4V2 6 8!Compute the right scalar product  V2 3 4 ^* 2V2 6 8+Compute division by a scalar on the right. BProduce a default basis for a vector space. If the dimensionality 2 of the vector space is not statically known, see  . :Produce a default basis for a vector space from which the  argument is drawn. v wxyz{   v wxyz{portable provisionalEdward Kmett <ekmett@gmail.com> Safe-Inferred EProvides a fairly subjective test to see if a quantity is near zero. nearZero (1e-11 :: Double)FalsenearZero (1e-17 :: Double)TruenearZero (1e-5 :: Float)FalsenearZero (1e-7 :: Float)True &Determine if a quantity is near zero. | } a ~ 1e-12 } a ~ 1e-6 |    | non-portable experimentalEdward Kmett <ekmett@gmail.com> Safe-Inferred A free inner product/ metric space ;Compute the inner product of two vectors or (equivalently)  convert a vector f a into a covector f a -> a. V2 1 2 `dot` V2 3 4119Compute the squared norm. The name quadrance arises from  Norman J. Wildberger's rational trigonometry. (Compute the quadrance of the difference ;Compute the distance between two vectors in a metric space /Compute the norm of a vector in a metric space *Convert a non-zero vector to unit vector.  Normalize a   functor to have unit . This function $ does not change the functor if its  is 0 or 1.      non-portable experimentalEdward Kmett <ekmett@gmail.com> Safe-InferredA  f+ is corepresentable if it is isomorphic to (x -> a) E for some x. Nearly all such functors can be represented by choosing x to be ? the set of lenses that are polymorphic in the contents of the ,  that is to say x = Rep f is a valid choice of x for (nearly) every   Representable .  non-portable experimentalEdward Kmett <ekmett@gmail.com>None6A space that distinguishes 2 orthogonal basis vectors  and , but may have more.  V2 1 2 ^._x1V2 1 2 & _x .~ 3V2 3 2    :: Lens' (t a) a  V2 1 2 ^._y2V2 1 2 & _y .~ 3V2 1 3    :: Lens' (t a) a     :: Lens' (t a) ( a) A 2-dimensional vector pure 1 :: V2 IntV2 1 1V2 1 2 + V2 3 4V2 4 6V2 1 2 * V2 3 4V2 3 8 sum (V2 1 2)3+the counter-clockwise perpendicular vector perp $ V2 10 20 V2 (-20) 10 non-portable experimentalEdward Kmett <ekmett@gmail.com>None7A space that distinguishes 3 orthogonal basis vectors: , , and . (It may have more)     :: Lens' (t a) a      :: Lens' (t a) (! a) !A 3-dimensional vector #cross product $scalar triple product  !"#$  !"#$ !"#$  !"#$ non-portable experimentalEdward Kmett <ekmett@gmail.com>None%4A space that distinguishes orthogonal basis vectors , , , &. (It may have more.) &   & :: Lens' (t a) a '   ' :: Lens' (t a) (( a) (A 4-dimensional vector. *OConvert a 3-dimensional affine vector into a 4-dimensional homogeneous vector. +NConvert a 3-dimensional affine point into a 4-dimensional homogeneous vector. %&'()*+ %&'()*+()*+ %&'%&'()*+ non-portable experimentalEdward Kmett <ekmett@gmail.com>None,8Plcker coordinates for lines in a 3-dimensional space. .[Given a pair of points represented by homogeneous coordinates generate Plcker coordinates  for the line through them. /OThese elements form a basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   / :: Lens' (, a) a  0 :: Lens' (, a) a  1 :: Lens' (, a) a  2 :: Lens' (, a) a  3 :: Lens' (, a) a  4 :: Lens' (, a) a 0OThese elements form a basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   / :: Lens' (, a) a  0 :: Lens' (, a) a  1 :: Lens' (, a) a  2 :: Lens' (, a) a  3 :: Lens' (, a) a  4 :: Lens' (, a) a 1OThese elements form a basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   / :: Lens' (, a) a  0 :: Lens' (, a) a  1 :: Lens' (, a) a  2 :: Lens' (, a) a  3 :: Lens' (, a) a  4 :: Lens' (, a) a 2OThese elements form a basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   / :: Lens' (, a) a  0 :: Lens' (, a) a  1 :: Lens' (, a) a  2 :: Lens' (, a) a  3 :: Lens' (, a) a  4 :: Lens' (, a) a 3OThese elements form a basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   / :: Lens' (, a) a  0 :: Lens' (, a) a  1 :: Lens' (, a) a  2 :: Lens' (, a) a  3 :: Lens' (, a) a  4 :: Lens' (, a) a 4OThese elements form a basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   / :: Lens' (, a) a  0 :: Lens' (, a) a  1 :: Lens' (, a) a  2 :: Lens' (, a) a  3 :: Lens' (, a) a  4 :: Lens' (, a) a 5ZThese elements form an alternate basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   5 ::  a => Lens' (, a) a  6 ::  a => Lens' (, a) a  7 ::  a => Lens' (, a) a  8 ::  a => Lens' (, a) a  9 ::  a => Lens' (, a) a  : ::  a => Lens' (, a) a 6ZThese elements form an alternate basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   5 ::  a => Lens' (, a) a  6 ::  a => Lens' (, a) a  7 ::  a => Lens' (, a) a  8 ::  a => Lens' (, a) a  9 ::  a => Lens' (, a) a  : ::  a => Lens' (, a) a 7ZThese elements form an alternate basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   5 ::  a => Lens' (, a) a  6 ::  a => Lens' (, a) a  7 ::  a => Lens' (, a) a  8 ::  a => Lens' (, a) a  9 ::  a => Lens' (, a) a  : ::  a => Lens' (, a) a 8ZThese elements form an alternate basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   5 ::  a => Lens' (, a) a  6 ::  a => Lens' (, a) a  7 ::  a => Lens' (, a) a  8 ::  a => Lens' (, a) a  9 ::  a => Lens' (, a) a  : ::  a => Lens' (, a) a 9ZThese elements form an alternate basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   5 ::  a => Lens' (, a) a  6 ::  a => Lens' (, a) a  7 ::  a => Lens' (, a) a  8 ::  a => Lens' (, a) a  9 ::  a => Lens' (, a) a  : ::  a => Lens' (, a) a :ZThese elements form an alternate basis for the Plcker space, or the Grassmanian manifold Gr(2,V4).   5 ::  a => Lens' (, a) a  6 ::  a => Lens' (, a) a  7 ::  a => Lens' (, a) a  8 ::  a => Lens' (, a) a  9 ::  a => Lens' (, a) a  : ::  a => Lens' (, a) a ;Valid Plcker coordinates p will have ; p  0 PThat said, floating point makes a mockery of this claim, so you may want to use  . <This isn'Yt th actual metric because this bilinear form gives rise to an isotropic quadratic space =rChecks if the line is near-isotropic (isotropic vectors in this quadratic space represent lines in real 3d space) >:Checks if the two vectors intersect (or nearly intersect) &,-./0123456789:;<=>,-./0123456789:;<=>,-;=<.>/015496:2738%,-./0123456789:;<=>  non-portable experimentalEdward Kmett <ekmett@gmail.com> Safe-Inferred?An involutive ring @<Conjugate a value. This defaults to the trivial involution. conjugate (1 :+ 2) 1.0 :+ (-2.0) conjugate 11?@?@?@?@  non-portable experimentalEdward Kmett <ekmett@gmail.com>NoneA0A vector space that includes the basis elements F, G, B and C B   B :: Lens' (t a) a C   C :: Lens' (t a) a D   D :: Lens' (t a) (V3 a) E0A vector space that includes the basis elements F and G F   F :: Lens' (t a) a G   G :: Lens' (t a) a H Quaternions %quadrance of the imaginary component J norm of the imaginary component Kraise a H to a scalar power 1Helper for calculating with specific branch cuts 1Helper for calculating with specific branch cuts L with a specified branch cut. M with a specified branch cut. N with a specified branch cut. O with a specified branch cut. P with a specified branch cut. Q with a specified branch cut. R8Spherical linear interpolation between two quaternions. SApply a rotation to a vector. TT axis theta builds a H representing a  rotation of theta radians about axis. .ABCDEFGHIJKLMNOPQRSTABCDEFGHIJKLMNOPQRSTHIEFGABCDRLMNOPQJKST(ABCDEFGHIJKLMNOPQRST  non-portable experimentalEdward Kmett <ekmett@gmail.com>NoneU+A 4x3 matrix with row-major representation V+A 4x4 matrix with row-major representation W+A 3x3 matrix with row-major representation X+A 2x2 matrix with row-major representation YdMatrix product. This can compute mixed dense-dense, sparse-dense and sparse-sparse matrix products. :V2 (V3 1 2 3) (V3 4 5 6) !*! V3 (V2 1 2) (V2 3 4) (V2 4 5)V2 (V2 19 25) (V2 43 58)SV2 (fromList [(1,2)]) (fromList [(2,3)]) !*! fromList [(1,V3 0 0 1), (2, V3 0 0 5)]V2 (V3 0 0 2) (V3 0 0 15)ZMatrix * column vector $V2 (V3 1 2 3) (V3 4 5 6) !* V3 7 8 9 V2 50 122[Row vector * matrix "V2 1 2 *! V2 (V3 3 4 5) (V3 6 7 8) V3 15 18 21\Scalar-matrix product 5 *!! V2 (V2 1 2) (V2 3 4)V2 (V2 5 10) (V2 15 20)]Matrix-scalar product V2 (V2 1 2) (V2 3 4) !!* 5V2 (V2 5 10) (V2 15 20)^+Hermitian conjugate or conjugate transpose :adjoint (V2 (V2 (1 :+ 2) (3 :+ 4)) (V2 (5 :+ 6) (7 :+ 8)))LV2 (V2 (1.0 :+ (-2.0)) (5.0 :+ (-6.0))) (V2 (3.0 :+ (-4.0)) (7.0 :+ (-8.0)))_Compute the trace of a matrix trace (V2 (V2 a b) (V2 c d))a + d`$Build a rotation matrix from a unit H. a=Build a transformation matrix from a rotation expressed as a  H and a translation vector. bAConvert from a 4x3 matrix to a 4x4 matrix, extending it with the  [ 0 0 0 1 ] column vector c8Convert a 3x3 matrix to a 4x4 matrix extending it with 0's in the new row and column. d3x3 identity matrix. eye3#V3 (V3 1 0 0) (V3 0 1 0) (V3 0 0 1)e4x4 identity matrix. eye46V4 (V4 1 0 0 0) (V4 0 1 0 0) (V4 0 0 1 0) (V4 0 0 0 1)f@Extract the translation vector (first three entries of the last " column) from a 3x4 or 4x4 matrix g2x2 matrix determinant. det22 (V2 (V2 a b) (V2 c d)) a * d - b * ch3x3 matrix determinant. +det33 (V3 (V3 a b c) (V3 d e f) (V3 g h i))?a * (e * i - f * h) - d * (b * i - c * h) + g * (b * f - c * e)i2x2 matrix inverse. inv22 $ V2 (V2 1 2) (V2 3 4))Just (V2 (V2 (-2.0) 1.0) (V2 1.5 (-0.5)))j3x3 matrix inverse. +inv33 $ V3 (V3 1 2 4) (V3 4 2 2) (V3 1 1 1)JJust (V3 (V3 0.0 0.5 (-1.0)) (V3 (-0.5) (-0.75) 3.5) (V3 0.5 0.25 (-1.5)))UVWXYZ[\]^_`abcdefghijUVWXYZ[\]^_`abcdefghijYZ[]\^XWVUcbghijde_f`aUVWXYZ[\]^_`abcdefghij  non-portable experimentalEdward Kmett <ekmett@gmail.com>Nonek  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghij !"#$%&'())*+,-../012344567789:;<=>?@ABCDEFGH I J K L M N O P Q R R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~                 linear-0.7 Linear.VectorLinear.Epsilon Linear.Metric Linear.Core Linear.V2 Linear.V3 Linear.V4Linear.PluckerLinear.ConjugateLinear.Quaternion Linear.MatrixLinear.InstancesLinearAdditivezero^+^^-^lerpnegated*^^*^/basisbasisForEpsilonnearZeroMetricdot quadranceqddistancenormsignorm normalizeCorecoreR2_x_y_xyV2perpR3_z_xyzV3crosstripleR4_w_xyzwV4vectorpointPluckerpluckerp01p02p03p23p31p12p10p20p30p32p13p21 squaredError>< isotropic intersects Conjugate conjugate Hamiltonian_j_k_ijk Complicated_e_i Quaternionabsipowasinqacosqatanqasinhqacoshqatanhqslerprotate axisAngleM43M44M33M22!*!!**!*!!!!*adjointtracefromQuaternionmkTransformation m43_to_m44 m33_to_m44eye3eye4 translationdet22det33inv22inv33$fTraversable1Complex$fFoldable1Complex$fTraversableComplex$fFoldableComplex$fMonadComplex $fBindComplex$fApplicativeComplex$fApplyComplex$fFunctorComplex $fBindHashMap$fApplyHashMap setElement$fAdditiveComplex$fAdditive(->)$fAdditiveHashMap $fAdditiveMap$fAdditiveIntMap$fEpsilonDoublebaseGHC.Numabsghc-prim GHC.Classes<=$fEpsilonFloatGHC.BaseFunctor$fIxV2 $fStorableV2 $fEpsilonV2$fDistributiveV2$fCoreV2$fR2V2 $fMetricV2$fFractionalV2$fNumV2 $fMonadV2$fBindV2 $fAdditiveV2$fApplicativeV2 $fApplyV2$fTraversable1V2 $fFoldable1V2$fTraversableV2 $fFoldableV2 $fFunctorV2$fIxV3 $fEpsilonV3 $fStorableV3$fCoreV3$fR3V3$fR2V3$fDistributiveV3 $fMetricV3$fFractionalV3$fNumV3 $fMonadV3$fBindV3 $fAdditiveV3$fApplicativeV3 $fApplyV3$fTraversable1V3 $fFoldable1V3$fTraversableV3 $fFoldableV3 $fFunctorV3$fIxV4 $fEpsilonV4 $fStorableV4$fCoreV4$fR4V4$fR3V4$fR2V4$fDistributiveV4 $fMetricV4$fFractionalV4$fNumV4 $fMonadV4$fBindV4 $fAdditiveV4 $fApplyV4$fApplicativeV4$fTraversable1V4 $fFoldable1V4$fTraversableV4 $fFoldableV4 $fFunctorV4Num==anti$fEpsilonPlucker$fMetricPlucker$fStorablePlucker$fFractionalPlucker $fNumPlucker $fIxPlucker$fTraversable1Plucker$fFoldable1Plucker$fTraversablePlucker$fFoldablePlucker $fCorePlucker$fDistributivePlucker$fMonadPlucker $fBindPlucker$fAdditivePlucker$fApplicativePlucker$fApplyPlucker$fFunctorPlucker$fConjugateComplex$fConjugateFloat$fConjugateDouble$fConjugateWord8$fConjugateWord16$fConjugateWord32$fConjugateWord64$fConjugateWord$fConjugateInt8$fConjugateInt16$fConjugateInt32$fConjugateInt64$fConjugateInt$fConjugateIntegerqicutcutWith GHC.FloatasinacosatanasinhacoshatanhqNaN reimagine$fEpsilonQuaternion$fFloatingQuaternion$fConjugateQuaternion$fDistributiveQuaternion$fHamiltonianQuaternion$fComplicatedQuaternion$fComplicatedComplex$fMetricQuaternion$fFractionalQuaternion$fNumQuaternion$fStorableQuaternion$fTraversableQuaternion$fFoldableQuaternion$fCoreQuaternion$fIxQuaternion$fMonadQuaternion$fBindQuaternion$fAdditiveQuaternion$fApplicativeQuaternion$fApplyQuaternion$fFunctorQuaternionmkTransformationMat