A CubicSpline2d
is a cubic Bézier curve
in 2D defined by a start point, end point and two control points. This module
contains functionality for
Geometry.Types.CubicSpline2d
with : { startPoint : Point2d, startControlPoint : Point2d, endControlPoint : Point2d, endPoint : Point2d } -> CubicSpline2d
Construct a spline from its endpoints and control points:
exampleSpline =
CubicSpline2d.with
{ startPoint =
Point2d.fromCoordinates ( 1, 1 )
, startControlPoint =
Point2d.fromCoordinates ( 3, 4 )
, endControlPoint =
Point2d.fromCoordinates ( 5, 1 )
, endPoint =
Point2d.fromCoordinates ( 7, 4 )
}
fromEndpoints : { startPoint : Point2d, startDerivative : Vector2d, endPoint : Point2d, endDerivative : Vector2d } -> CubicSpline2d
Construct a spline from a given start point with a given start derivative, to a given end point with a given end derivative, like so:
The spline is based on a parameter that ranges from 0 to 1; as a result, in most cases the length of each derivative vector should be roughly equal to the length of the resulting spline.
fromQuadraticSpline : QuadraticSpline2d -> CubicSpline2d
Convert a quadratic spline into the equivalent cubic spline (every quadratic spline can be represented exactly as a cubic spline).
quadraticSpline =
QuadraticSpline2d.with
{ startPoint =
Point2d.fromCoordinates ( 0, 0 )
, controlPoint =
Point2d.fromCoordinates ( 3, 0 )
, endPoint =
Point2d.fromCoordinates ( 3, 3 )
}
CubicSpline2d.fromQuadraticSpline quadraticSpline
--> CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 0, 0 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 2, 0 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 3, 1 )
--> , endPoint =
--> Point2d.fromCoordinates ( 3, 3 )
--> }
startPoint : CubicSpline2d -> Point2d
Get the start point of a spline.
CubicSpline2d.startPoint exampleSpline
--> Point2d.fromCoordinates ( 1, 1 )
endPoint : CubicSpline2d -> Point2d
Get the end point of a spline.
CubicSpline2d.endPoint exampleSpline
--> Point2d.fromCoordinates ( 7, 4 )
startControlPoint : CubicSpline2d -> Point2d
Get the start control point of a spline (the control point next to the start point).
CubicSpline2d.startControlPoint exampleSpline
--> Point2d.fromCoordinates ( 3, 4 )
endControlPoint : CubicSpline2d -> Point2d
Get the end control point of a spline (the control point next to the end point).
CubicSpline2d.endControlPoint exampleSpline
--> Point2d.fromCoordinates ( 5, 1 )
startDerivative : CubicSpline2d -> Vector2d
Get the start derivative of a spline. This is equal to three times the vector from the spline's start point to its start control point.
CubicSpline2d.startDerivative exampleSpline
--> Vector2d.fromComponents ( 6, 9 )
endDerivative : CubicSpline2d -> Vector2d
Get the end derivative of a spline. This is equal to three times the vector from the spline's end control point to its end point.
CubicSpline2d.endDerivative exampleSpline
--> Vector2d.fromComponents ( 6, 9 )
boundingBox : CubicSpline2d -> BoundingBox2d
Compute a bounding box for a given spline. It is not guaranteed that the result will be the smallest possible bounding box, since for efficiency the bounding box is computed from the spline's control points (which cover a larger area than the spline itself).
CubicSpline2d.boundingBox exampleSpline
--> BoundingBox2d.fromExtrema
--> { minX = 1
--> , maxX = 7
--> , minY = 1
--> , maxY = 4
--> }
pointOn : CubicSpline2d -> Curve.ParameterValue.ParameterValue -> Point2d
Get the point along a spline at a given parameter value:
CubicSpline2d.pointOn exampleSpline ParameterValue.zero
--> Point2d.fromCoordinates ( 1, 1 )
CubicSpline2d.pointOn exampleSpline ParameterValue.half
--> Point2d.fromCoordinates ( 4, 2.5 )
CubicSpline2d.pointOn exampleSpline ParameterValue.one
--> Point2d.fromCoordinates ( 7, 4 )
pointsAt : List Curve.ParameterValue.ParameterValue -> CubicSpline2d -> List Point2d
Get points along a spline at a given set of parameter values:
exampleSpline
|> CubicSpline2d.pointsAt
(ParameterValue.steps 2)
--> [ Point2d.fromCoordinates ( 1, 1 )
--> , Point2d.fromCoordinates ( 4, 2.5 )
--> , Point2d.fromCoordinates ( 7, 4 )
--> ]
If a curve has zero length (consists of just a single point), then we say that it is 'degenerate'. Some operations such as computing tangent directions are not defined on degenerate curves.
A Nondegenerate
value represents a spline that is definitely not degenerate.
It is used as input to functions such as CubicSpline2d.tangentDirection
and
can be constructed using CubicSpline2d.nondegenerate
.
nondegenerate : CubicSpline2d -> Result Point2d Nondegenerate
Attempt to construct a nondegenerate spline from a general CubicSpline2d
.
If the spline is in fact degenerate (consists of a single point), returns an
Err
with that point.
CubicSpline2d.nondegenerate exampleSpline
--> Ok nondegenerateExampleSpline
fromNondegenerate : Nondegenerate -> CubicSpline2d
Convert a nondegenerate spline back to a general CubicSpline2d
.
CubicSpline2d.fromNondegenerate
nondegenerateExampleSpline
--> exampleSpline
tangentDirection : Nondegenerate -> Curve.ParameterValue.ParameterValue -> Direction2d
Get the tangent direction to a nondegenerate spline at a given parameter value:
CubicSpline2d.tangentDirection
nondegenerateExampleSpline
ParameterValue.zero
--> Direction2d.fromAngle (degrees 56.31)
CubicSpline2d.tangentDirection
nondegenerateExampleSpline
ParameterValue.half
--> Direction2d.fromAngle (degrees 0)
CubicSpline2d.tangentDirection
nondegenerateExampleSpline
ParameterValue.one
--> Direction2d.fromAngle (degrees 56.31)
tangentDirectionsAt : List Curve.ParameterValue.ParameterValue -> Nondegenerate -> List Direction2d
Get tangent directions to a nondegenerate spline at a given set of parameter values:
nondegenerateExampleSpline
|> CubicSpline2d.tangentDirectionsAt
(ParameterValue.steps 2)
--> [ Direction2d.fromAngle (degrees 56.31)
--> , Direction2d.fromAngle (degrees 0)
--> , Direction2d.fromAngle (degrees 56.31)
--> ]
sample : Nondegenerate -> Curve.ParameterValue.ParameterValue -> ( Point2d, Direction2d )
Get both the point and tangent direction of a nondegenerate spline at a given parameter value:
CubicSpline2d.sample nondegenerateExampleSpline
ParameterValue.half
--> ( Point2d.fromCoordinates ( 4, 2.5 )
--> , Direction2d.fromAngle (degrees 0)
--> )
samplesAt : List Curve.ParameterValue.ParameterValue -> Nondegenerate -> List ( Point2d, Direction2d )
Get points and tangent directions of a nondegenerate spline at a given set of parameter values:
nondegenerateExampleSpline
|> CubicSpline2d.samplesAt
(ParameterValue.steps 2)
--> [ ( Point2d.fromCoordinates ( 1, 1 )
--> , Direction2d.fromAngle (degrees 56.31)
--> )
--> , ( Point2d.fromCoordinates ( 4, 2.5 )
--> , Direction2d.fromAngle (degrees 0)
--> )
--> , ( Point2d.fromCoordinates ( 7, 4 )
--> , Direction2d.fromAngle (degrees 56.31)
--> )
--> ]
reverse : CubicSpline2d -> CubicSpline2d
Reverse a spline so that the start point becomes the end point, and vice versa.
CubicSpline2d.reverse exampleSpline
--> CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 7, 4 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 5, 1 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 3, 4 )
--> , endPoint =
--> Point2d.fromCoordinates ( 1, 1 )
--> }
scaleAbout : Point2d -> Basics.Float -> CubicSpline2d -> CubicSpline2d
Scale a spline about the given center point by the given scale.
CubicSpline2d.scaleAbout Point2d.origin 2 exampleSpline
--> CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 2, 2 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 6, 8 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 10, 2 )
--> , endPoint =
--> Point2d.fromCoordinates ( 14, 8 )
--> }
rotateAround : Point2d -> Basics.Float -> CubicSpline2d -> CubicSpline2d
Rotate a spline counterclockwise around a given center point by a given angle (in radians).
exampleSpline
|> CubicSpline2d.rotateAround Point2d.origin
(degrees 90)
--> CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( -1, 1 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( -4, 3 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( -1, 5 )
--> , endPoint =
--> Point2d.fromCoordinates ( -4, 7 )
--> }
translateBy : Vector2d -> CubicSpline2d -> CubicSpline2d
Translate a spline by a given displacement.
displacement =
Vector2d.fromComponents ( 2, 3 )
CubicSpline2d.translateBy displacement exampleSpline
--> CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 3, 4 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 5, 7 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 7, 4 )
--> , endPoint =
--> Point2d.fromCoordinates ( 9, 7 )
--> }
translateIn : Direction2d -> Basics.Float -> CubicSpline2d -> CubicSpline2d
Translate a spline in a given direction by a given distance;
CubicSpline2d.translateIn direction distance
is equivalent to
CubicSpline2d.translateBy
(Vector2d.withLength distance direction)
mirrorAcross : Axis2d -> CubicSpline2d -> CubicSpline2d
Mirror a spline across an axis.
CubicSpline2d.mirrorAcross Axis2d.x exampleSpline
--> CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 1, -1 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 3, -4 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 5, -1 )
--> , endPoint =
--> Point2d.fromCoordinates ( 7, -4 )
--> }
relativeTo : Frame2d -> CubicSpline2d -> CubicSpline2d
Take a spline defined in global coordinates, and return it expressed in local coordinates relative to a given reference frame.
localFrame =
Frame2d.atPoint (Point2d.fromCoordinates ( 1, 2 ))
CubicSpline2d.relativeTo localFrame exampleSpline
--> CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 0, -1 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 2, 2 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 4, -1 )
--> , endPoint =
--> Point2d.fromCoordinates ( 6, 2 )
--> }
placeIn : Frame2d -> CubicSpline2d -> CubicSpline2d
Take a spline considered to be defined in local coordinates relative to a given reference frame, and return that spline expressed in global coordinates.
localFrame =
Frame2d.atPoint (Point2d.fromCoordinates ( 1, 2 ))
CubicSpline2d.placeIn localFrame exampleSpline
--> CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 2, 3 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 4, 6 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 6, 3 )
--> , endPoint =
--> Point2d.fromCoordinates ( 8, 6 )
--> }
bisect : CubicSpline2d -> ( CubicSpline2d, CubicSpline2d )
Split a spline into two roughly equal halves.
CubicSpline2d.bisect exampleSpline
--> ( CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 1, 1 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 2, 2.5 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 3, 2.5 )
--> , endPoint =
--> Point2d.fromCoordinates ( 4, 2.5 )
--> }
--> , CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 4, 2.5 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 5, 2.5 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 6, 2.5 )
--> , endPoint =
--> Point2d.fromCoordinates ( 7, 4 )
--> }
--> )
Equivalent to CubicSpline2d.splitAt ParameterValue.half
.
splitAt : Curve.ParameterValue.ParameterValue -> CubicSpline2d -> ( CubicSpline2d, CubicSpline2d )
Split a spline at a particular parameter value, resulting in two smaller splines.
parameterValue =
ParameterValue.clamped 0.75
CubicSpline2d.splitAt parameterValue exampleSpline
--> ( CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 1, 1 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 2.5, 3.25 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 4, 2.125 )
--> , endPoint =
--> Point2d.fromCoordinates ( 5.5, 2.6875 )
--> }
--> , CubicSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 5.5, 2.6875 )
--> , startControlPoint =
--> Point2d.fromCoordinates ( 6, 2.875 )
--> , endControlPoint =
--> Point2d.fromCoordinates ( 6.5, 3.25 )
--> , endPoint =
--> Point2d.fromCoordinates ( 7, 4 )
--> }
--> )
A spline that has been parameterized by arc length.
arcLengthParameterized : { maxError : Basics.Float } -> CubicSpline2d -> ArcLengthParameterized
Build an arc length parameterization of the given spline, with a given
accuracy. Generally speaking, all operations on the resulting
ArcLengthParameterized
value will be accurate to within the specified maximum
error.
parameterizedSpline =
exampleSpline
|> CubicSpline2d.arcLengthParameterized
{ maxError = 1.0e-4 }
The accuracy of the parameterization affects the accuracy of results returned
from functions such as arcLength
and pointAlong
.
arcLength : ArcLengthParameterized -> Basics.Float
Find the total arc length of a spline:
arcLength =
CubicSpline2d.arcLength parameterizedSpline
arcLength
--> 7.0952
In this example, the result will be accurate to within 1.0e-4
since that was
the tolerance used when constructing parameterizedSpline
.
pointAlong : ArcLengthParameterized -> Basics.Float -> Maybe Point2d
Try to get the point along a spline at a given arc length. For example, to
get the point a quarter of the way along exampleSpline
, using arcLength
as
computed above:
CubicSpline2d.pointAlong parameterizedSpline
(0.25 * arcLength)
--> Just (Point2d.fromCoordinates ( 2.2681, 2.2114 ))
Note that this is not the same as evaulating at a parameter value of 0.25:
CubicSpline2d.pointOn exampleSpline
(ParameterValue.clamped 0.25)
--> Point2d.fromCoordinates ( 2.5, 2.3125 )
If the given arc length is less than zero or greater than the arc length of the
spline, returns Nothing
.
tangentDirectionAlong : ArcLengthParameterized -> Basics.Float -> Maybe Direction2d
Try to get the tangent direction along a spline at a given arc length. To
get the tangent direction a quarter of the way along exampleSpline
:
CubicSpline2d.tangentDirectionAlong parameterizedSpline
(0.25 * arcLength)
--> Just (Direction2d.fromAngle (degrees 26.5611))
If the given arc length is less than zero or greater than the arc length of the
spline (or if the spline is degenerate), returns Nothing
.
sampleAlong : ArcLengthParameterized -> Basics.Float -> Maybe ( Point2d, Direction2d )
Try to get the point and tangent direction along a spline at a given arc
length. To get the point and tangent direction a quarter of the way along
exampleSpline
:
CubicSpline2d.sampleAlong parameterizedSpline
(0.25 * arcLength)
--> Just
--> ( Point2d.fromCoordinates ( 2.2681, 2.2114 )
--> , Direction2d.fromAngle (degrees 26.5611)
--> )
If the given arc length is less than zero or greater than the arc length of the
spline (or if the spline is degenerate), returns Nothing
.
An ArcLengthParameterized
value is a combination of an
ArcLengthParameterization
and an
underlying CubicSpline2d
. If you need to do something fancy, you can extract
these two values separately.
arcLengthParameterization : ArcLengthParameterized -> Curve.ArcLengthParameterization.ArcLengthParameterization
fromArcLengthParameterized : ArcLengthParameterized -> CubicSpline2d
You are unlikely to need to use these functions directly, but they are useful if you are writing low-level geometric algorithms.
firstDerivative : CubicSpline2d -> Curve.ParameterValue.ParameterValue -> Vector2d
Get the first derivative of a spline at a given parameter value:
CubicSpline2d.firstDerivative exampleSpline
ParameterValue.zero
--> Vector2d.fromComponents ( 6, 9 )
CubicSpline2d.firstDerivative exampleSpline
ParameterValue.half
--> Vector2d.fromComponents ( 6, 0 )
CubicSpline2d.firstDerivative exampleSpline
ParameterValue.one
--> Vector2d.fromComponents ( 6, 9 )
firstDerivativesAt : List Curve.ParameterValue.ParameterValue -> CubicSpline2d -> List Vector2d
Evaluate the first derivative of a spline at a given set of parameter values:
exampleSpline
|> CubicSpline2d.firstDerivativesAt
(ParameterValue.steps 2)
--> [ Vector2d.fromComponents ( 6, 9 )
--> , Vector2d.fromComponents ( 6, 0 )
--> , Vector2d.fromComponents ( 6, 9 )
--> ]
secondDerivative : CubicSpline2d -> Curve.ParameterValue.ParameterValue -> Vector2d
Evaluate the second derivative of a spline at a given parameter value:
CubicSpline2d.secondDerivativeAt 0 exampleSpline
--> Just (Vector2d.fromComponents ( 0, -36 ))
CubicSpline2d.secondDerivativeAt 0.5 exampleSpline
--> Just (Vector2d.fromComponents ( 0, 0 ))
CubicSpline2d.secondDerivativeAt 1 exampleSpline
--> Just (Vector2d.fromComponents ( 0, 36 ))
secondDerivativesAt : List Curve.ParameterValue.ParameterValue -> CubicSpline2d -> List Vector2d
Evaluate the second derivative of a spline at a given set of parameter values:
exampleSpline
|> CubicSpline2d.secondDerivativesAt
(ParameterValue.steps 2)
--> [ Vector2d.fromComponents ( 0, -36 )
--> , Vector2d.fromComponents ( 0, 0 )
--> , Vector2d.fromComponents ( 0, 36 )
--> ]
thirdDerivative : CubicSpline2d -> Vector2d
Get the third derivative of a spline (for a cubic spline, this is a constant):
CubicSpline2d.thirdDerivative exampleSpline
--> Vector2d.fromComponents ( 0, 72 )
maxSecondDerivativeMagnitude : CubicSpline2d -> Basics.Float
Find a conservative upper bound on the magnitude of the second derivative of a spline. This can be useful when determining error bounds for various kinds of linear approximations.
exampleSpline
|> CubicSpline2d.maxSecondDerivativeMagnitude
--> 36