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# OpenSolid.WebGLMath.Vector2d

Transformation functions for `Vector2d`.

transformBy : Float3x3 -> Vector2d -> Vector2d

Transform a `Vector2d` by a `Float3x3`; note that

``````Vector2d.transformBy matrix vector
``````

is similar to but not in general equivalent to

``````Vector2d (Matrix3.transform matrix (Vector2d.components vector))
``````

since `Matrix3.transform` implicitly assumes that the given argument represents a point, not a vector, and therefore applies translation to it. Transforming a vector by a 3x3 matrix should in fact ignore any translation component of the matrix, which this function does. For example:

``````vector =
Vector2d ( 2, 1 )

-- 90 degree rotation, followed by a translation by (5, 5)
matrix =
( ( 0, -1, 5 )
, ( 1, 0, 5 )
, ( 0, 0, 1 )
)

Vector2d.transformBy matrix vector
--> Vector2d ( -1, 2 )
``````
``````module OpenSolid.WebGLMath.Vector2d
exposing
( transformBy
)

{-| Transformation functions for `Vector2d`.

@docs transformBy

-}

import OpenSolid.Geometry.Types exposing (..)
import OpenSolid.Vector2d as Vector2d
import Matrix3 exposing (Float3x3)

{-| Transform a `Vector2d` by a `Float3x3`; note that

Vector2d.transformBy matrix vector

is similar to but *not* in general equivalent to

Vector2d (Matrix3.transform matrix (Vector2d.components vector))

since `Matrix3.transform` implicitly assumes that the given argument represents
a point, not a vector, and therefore applies translation to it. Transforming a
vector by a 3x3 matrix should in fact ignore any translation component of the
matrix, which this function does. For example:

vector =
Vector2d ( 2, 1 )

-- 90 degree rotation, followed by a translation by (5, 5)
matrix =
( ( 0, -1, 5 )
, ( 1, 0, 5 )
, ( 0, 0, 1 )
)

Vector2d.transformBy matrix vector
--> Vector2d ( -1, 2 )

-}
transformBy : Float3x3 -> Vector2d -> Vector2d
transformBy matrix vector =
let
( ( m11, m12, m13 ), ( m21, m22, m23 ), ( m31, m32, m33 ) ) =
matrix

( x, y ) =
Vector2d.components vector
in
Vector2d
( m11 * x + m12 * y
, m21 * x + m22 * y
)
```
```