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Conversion functions for `Frame3d`

.

toMat4 : Frame3d -> Mat4

Convert a `Frame3d`

to a `Mat4`

. The resulting matrix can be thought of in
a couple of ways:

- It is the transformation matrix that takes the global XYZ frame and transforms it to the given frame
- It is the transformation matrix from local coordinates in the given frame to global coordinates

The first bullet implies that something like

```
Frame3d.xyz
|> Frame3d.translateBy displacement
|> Frame3d.rotateAround axis angle
|> Frame3d.mirrorAcross plane
|> Frame3d.toMat4
```

gives you a transformation matrix that is equivalent to applying the given displacement, then the given rotation, then the given mirror. The second bullet means that, for example,

```
Point3d.placeIn frame
```

is equivalent to

```
Point3d.transformBy (Frame3d.toMat4 frame)
```

and

```
Point3d.relativeTo frame
```

is equivalent to

```
Point3d.transformBy (Matrix4.inverseOrthonormal (Frame3d.toMat4 frame))
```

```
module OpenSolid.LinearAlgebra.Frame3d
exposing
( toMat4
)
{-| Conversion functions for `Frame3d`.
@docs toMat4
-}
import OpenSolid.Geometry.Types exposing (..)
import OpenSolid.Direction3d as Direction3d
import OpenSolid.Point3d as Point3d
import OpenSolid.Frame3d as Frame3d
import Math.Matrix4 exposing (Mat4)
{-| Convert a `Frame3d` to a `Mat4`. The resulting matrix can be thought of in
a couple of ways:
- It is the transformation matrix that takes the global XYZ frame and
transforms it to the given frame
- It is the transformation matrix from local coordinates in the given frame
to global coordinates
The first bullet implies that something like
Frame3d.xyz
|> Frame3d.translateBy displacement
|> Frame3d.rotateAround axis angle
|> Frame3d.mirrorAcross plane
|> Frame3d.toMat4
gives you a transformation matrix that is equivalent to applying the given
displacement, then the given rotation, then the given mirror. The second bullet
means that, for example,
Point3d.placeIn frame
is equivalent to
Point3d.transformBy (Frame3d.toMat4 frame)
and
Point3d.relativeTo frame
is equivalent to
Point3d.transformBy (Matrix4.inverseOrthonormal (Frame3d.toMat4 frame))
-}
toMat4 : Frame3d -> Mat4
toMat4 frame =
let
( m11, m21, m31 ) =
Direction3d.components (Frame3d.xDirection frame)
( m12, m22, m32 ) =
Direction3d.components (Frame3d.yDirection frame)
( m13, m23, m33 ) =
Direction3d.components (Frame3d.zDirection frame)
( m14, m24, m34 ) =
Point3d.coordinates (Frame3d.originPoint frame)
in
Math.Matrix4.fromRecord
{ m11 = m11
, m21 = m21
, m31 = m31
, m41 = 0
, m12 = m12
, m22 = m22
, m32 = m32
, m42 = 0
, m13 = m13
, m23 = m23
, m33 = m33
, m43 = 0
, m14 = m14
, m24 = m24
, m34 = m34
, m44 = 1
}
```