A Direction2d
represents a direction like 'up' or 'north' or 'forwards'.
They are represented using X and Y components, and can be converted to vectors
if necessary, but should be thought of as conceptually different. Directions
have several uses, such as:
Geometry.Types.Direction2d coordinates
x : Direction2d coordinates
Synonym for Direction2d.positiveX
.
y : Direction2d coordinates
Synonym for Direction2d.positiveY
.
positiveX : Direction2d coordinates
The positive X direction.
negativeX : Direction2d coordinates
The negative X direction.
positiveY : Direction2d coordinates
The positive Y direction.
negativeY : Direction2d coordinates
The negative Y direction.
degrees : Basics.Float -> Direction2d coordinates
Construct a direction from a number of degrees, given counterclockwise from the positive X axis:
Direction2d.degrees 0
--> Direction2d.positiveX
Direction2d.degrees 90
--> Direction2d.positiveY
This is a convenient shorthand for using Direction2d.fromAngle
and
Angle.degrees
if you want to construct a direction at a fixed angle in
degrees.
radians : Basics.Float -> Direction2d coordinates
Construct a direction from a number of radians, given counterclockwise from the positive X axis:
Direction2d.radians pi
--> Direction2d.negativeX
Direction2d.radians (-pi / 2)
--> Direction2d.negativeY
from : Geometry.Types.Point2d units coordinates -> Geometry.Types.Point2d units coordinates -> Maybe (Direction2d coordinates)
Attempt to construct the direction from the first given point to the second.
If the two points are coincident, returns Nothing
.
point =
Point2d.meters 1 1
Direction2d.from Point2d.origin point
--> Just (Direction2d.degrees 45)
Direction2d.from point Point2d.origin
--> Just (Direction2d.degrees -135)
Direction2d.from point point
--> Nothing
perpendicularTo : Direction2d coordinates -> Direction2d coordinates
Construct a direction perpendicular to the given direction, by rotating the
given direction 90 degrees counterclockwise. Synonym for
rotateCounterclockwise
.
Direction2d.perpendicularTo Direction2d.x
--> Direction2d.y
Direction2d.perpendicularTo Direction2d.y
--> Direction2d.negativeX
orthonormalize : Vector2d units coordinates -> Vector2d units coordinates -> Maybe ( Direction2d coordinates, Direction2d coordinates )
Attempt to form a pair of perpendicular directions from the two given vectors by performing Gram-Schmidt normalization:
If either of the given vectors are zero, or if the two vectors are parallel,
returns Nothing
.
Direction2d.orthonormalize
(Vector2d.meters 3 3)
(Vector2d.meters 0 -2)
--> Just
--> ( Direction2d.degrees 45
--> , Direction2d.degrees -45
--> )
Direction2d.orthonormalize
(Vector2d.meters 3 3)
(Vector2d.meters -2 -2)
--> Nothing
orthogonalize : Direction2d coordinates -> Direction2d coordinates -> Maybe ( Direction2d coordinates, Direction2d coordinates )
Attempt to form a pair of perpendicular directions from the two given directions;
Direction2d.orthogonalize xDirection yDirection
is equivalent to
Direction2d.orthonormalize
(Direction2d.toVector xDirection)
(Direction2d.toVector yDirection)
fromAngle : Angle -> Direction2d coordinates
Construct a direction from an Angle given counterclockwise from the positive X direction.
Direction2d.fromAngle (Angle.degrees 90)
--> Direction2d.y
toAngle : Direction2d coordinates -> Angle
Convert a direction to a polar angle (the counterclockwise angle from the positive X direction). The result will be in the range -180 to 180 degrees.
Direction2d.toAngle Direction2d.negativeY
--> Angle.degrees -90
toVector : Direction2d coordinates -> Vector2d Quantity.Unitless coordinates
Convert a direction to a unitless vector of length 1.
Direction2d.toVector Direction2d.x
--> Vector2d.unitless 1 0
components : Direction2d coordinates -> ( Basics.Float, Basics.Float )
Get the X and Y components of a direction as a tuple.
Direction2d.components (Direction2d.degrees 135)
--> ( -0.7071, 0.7071 )
xComponent : Direction2d coordinates -> Basics.Float
Get the X component of a direction.
Direction2d.xComponent Direction2d.x
--> 1
Direction2d.xComponent Direction2d.y
--> 0
yComponent : Direction2d coordinates -> Basics.Float
Get the Y component of a direction.
Direction2d.yComponent Direction2d.x
--> 0
Direction2d.yComponent Direction2d.y
--> 1
componentIn : Direction2d coordinates -> Direction2d coordinates -> Basics.Float
Find the component of one direction in another direction. This is equal to the cosine of the angle between the directions, or equivalently the dot product of the two directions converted to unit vectors.
direction =
Direction2d.degrees 60
Direction2d.componentIn Direction2d.x direction
--> 0.5
Direction2d.componentIn direction direction
--> 1
Direction2d.componentIn Direction2d.x Direction2d.y
--> 0
This is more general and flexible than using xComponent
or yComponent
, both
of which can be expressed in terms of componentIn
; for example,
Direction2d.xComponent direction
is equivalent to
Direction2d.componentIn Direction2d.x direction
angleFrom : Direction2d coordinates -> Direction2d coordinates -> Angle
Find the counterclockwise angle from the first direction to the second. The result will be in the range -180 to 180 degrees.
referenceDirection =
Direction2d.degrees 30
Direction2d.angleFrom referenceDirection Direction2d.y
--> Angle.degrees 60
Direction2d.angleFrom referenceDirection Direction2d.x
--> Angle.degrees -30
equalWithin : Angle -> Direction2d coordinates -> Direction2d coordinates -> Basics.Bool
Compare two directions within an angular tolerance. Returns true if the absolute value of the angle between the two given directions is less than the given tolerance.
firstDirection =
Direction2d.degrees 45
secondDirection =
Direction2d.degrees 47
Direction2d.equalWithin (Angle.degrees 5)
firstDirection
secondDirection
--> True
Direction2d.equalWithin (Angle.degrees 1)
firstDirection
secondDirection
--> False
reverse : Direction2d coordinates -> Direction2d coordinates
Reverse a direction.
rotateClockwise : Direction2d coordinates -> Direction2d coordinates
Rotate a direction by 90 degrees clockwise.
Direction2d.rotateClockwise Direction2d.x
--> Direction2d.negativeY
rotateCounterclockwise : Direction2d coordinates -> Direction2d coordinates
Rotate a direction by 90 degrees counterclockwise.
Direction2d.rotateClockwise Direction2d.x
--> Direction2d.y
rotateBy : Angle -> Direction2d coordinates -> Direction2d coordinates
Rotate a direction counterclockwise by a given angle.
Direction2d.rotateBy (Angle.degrees 180) Direction2d.x
--> Direction2d.negativeX
mirrorAcross : Geometry.Types.Axis2d units coordinates -> Direction2d coordinates -> Direction2d coordinates
Mirror a direction across a particular axis. Note that only the direction of the axis affects the result, since directions are position-independent.
slopedAxis =
Axis2d.through
(Point2d.meters 100 200)
(Direction2d.degrees 45)
Direction2d.mirrorAcross slopedAxis Direction2d.x
--> Direction2d.y
Direction2d.mirrorAcross slopedAxis Direction2d.y
--> Direction2d.x
Like other transformations, coordinate transformations of directions depend only on the orientations of the relevant frames, not the positions of their origin points.
For the examples, assume the following frames have been defined:
upsideDownFrame =
Frame2d.atOrigin |> Frame2d.reverseY
rotatedFrame =
Frame2d.atOrigin |> Frame2d.rotateBy (Angle.degrees 30)
relativeTo : Geometry.Types.Frame2d units globalCoordinates { defines : localCoordinates } -> Direction2d globalCoordinates -> Direction2d localCoordinates
Take a direction defined in global coordinates, and return it expressed in local coordinates relative to a given reference frame.
Direction2d.relativeTo upsideDownFrame Direction2d.y
--> Direction2d.negativeY
Direction2d.relativeTo rotatedFrame Direction2d.x
--> Direction2d.degrees -30
Direction2d.relativeTo rotatedFrame Direction2d.y
--> Direction2d.degrees 60
placeIn : Geometry.Types.Frame2d units globalCoordinates { defines : localCoordinates } -> Direction2d localCoordinates -> Direction2d globalCoordinates
Take a direction defined in local coordinates relative to a given reference frame, and return that direction expressed in global coordinates.
Direction2d.placeIn upsideDownFrame Direction2d.y
--> Direction2d.negativeY
Direction2d.placeIn rotatedFrame Direction2d.x
--> Direction2d.degrees 30
Direction2d.placeIn rotatedFrame Direction2d.y
--> Direction2d.degrees 120
random : Random.Generator (Direction2d coordinates)
A random generator for 2D directions.
unsafe : { x : Basics.Float, y : Basics.Float } -> Direction2d coordinates
Construct a direction directly from its X and Y components. Note that you must ensure that the sum of the squares of the given components is exactly one:
Direction2d.unsafe { x = 1, y = 0 }
Direction2d.unsafe { x = 0, y = -1 }
Direction2d.unsafe { x = 0.6, y = 0.8 }
are all valid but
Direction2d.unsafe { x = 2, y = 0 }
Direction2d.unsafe { x = 1, y = 1 }
are not. Instead of using Direction2d.unsafe
, it may be easier to use
constructors like degrees
or fromAngle
(which will
always result in a valid direction) or start with existing directions and
transform them as necessary.
unwrap : Direction2d coordinates -> { x : Basics.Float, y : Basics.Float }
Extract the X and Y components of a direction as a record.