A QuadraticSpline3d
is a quadratic Bézier curve
in 3D defined by a start point, control point and end point. This module
contains functionality for
Geometry.Types.QuadraticSpline3d
with : { startPoint : Point3d, controlPoint : Point3d, endPoint : Point3d } -> QuadraticSpline3d
Construct a spline from its start point, control point and end point:
exampleSpline =
QuadraticSpline3d.with
{ startPoint =
Point3d.fromCoordinates ( 1, 1, 1 )
, controlPoint =
Point3d.fromCoordinates ( 3, 2, 1 )
, endPoint =
Point3d.fromCoordinates ( 3, 3, 3 )
}
on : SketchPlane3d -> QuadraticSpline2d -> QuadraticSpline3d
Construct a 3D spline lying on a sketch plane by providing a 2D spline specified in XY coordinates within the sketch plane.
QuadraticSpline3d.on SketchPlane3d.xz <|
QuadraticSpline2d.with
{ startPoint =
Point2d.fromCoordinates ( 1, 1 )
, controlPoint =
Point2d.fromCoordinates ( 3, 4 )
, endPoint =
Point2d.fromCoordinates ( 5, 1 )
}
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 1, 0, 1 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 3, 0, 4 )
--> , endPoint =
--> Point3d.fromCoordinates ( 5, 0, 1 )
--> }
startPoint : QuadraticSpline3d -> Point3d
Get the start point of a spline.
QuadraticSpline3d.startPoint exampleSpline
--> Point3d.fromCoordinates ( 1, 1, 1 )
endPoint : QuadraticSpline3d -> Point3d
Get the end point of a spline. This is equal to the spline's last control point.
QuadraticSpline3d.endPoint exampleSpline
--> Point3d.fromCoordinates ( 3, 3, 3 )
controlPoint : QuadraticSpline3d -> Point3d
Get the control point of a spline.
QuadraticSpline3d.controlPoint exampleSpline
--> Point3d.fromCoordinates ( 3, 2, 1 )
startDerivative : QuadraticSpline3d -> Vector3d
Get the start derivative of a spline. This is equal to twice the vector from the spline's first control point to its second.
QuadraticSpline3d.startDerivative exampleSpline
--> Vector3d.fromComponents ( 4, 2, 0 )
endDerivative : QuadraticSpline3d -> Vector3d
Get the end derivative of a spline. This is equal to twice the vector from the spline's second control point to its third.
QuadraticSpline3d.endDerivative exampleSpline
--> Vector3d.fromComponents ( 0, 2, 4 )
boundingBox : QuadraticSpline3d -> BoundingBox3d
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 volume than the spline itself).
QuadraticSpline3d.boundingBox exampleSpline
--> BoundingBox3d.fromExtrema
--> { minX = 1
--> , maxX = 3
--> , minY = 1
--> , maxY = 3
--> , minZ = 1
--> , maxZ = 3
--> }
pointOn : QuadraticSpline3d -> Curve.ParameterValue.ParameterValue -> Point3d
Get the point along a spline at a given parameter value:
QuadraticSpline3d.pointOn exampleSpline 0
--> Point3d.fromCoordinates ( 1, 1, 1 )
QuadraticSpline3d.pointOn exampleSpline 0.5
--> Point3d.fromCoordinates ( 2.5, 2, 1.5 )
QuadraticSpline3d.pointOn exampleSpline 1
--> Point3d.fromCoordinates ( 3, 3, 3 )
pointsAt : List Curve.ParameterValue.ParameterValue -> QuadraticSpline3d -> List Point3d
Get points along a spline at a given set of parameter values:
exampleSpline
|> QuadraticSpline3d.pointsAt
(ParameterValue.steps 2)
--> [ Point3d.fromCoordinates ( 1, 1, 1 )
--> , Point3d.fromCoordinates ( 2.5, 2, 1.5 )
--> , Point3d.fromCoordinates ( 3, 3, 3 )
--> ]
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 QuadraticSpline3d.tangentDirection
and can be constructed using QuadraticSpline3d.nondegenerate
.
nondegenerate : QuadraticSpline3d -> Result Point3d Nondegenerate
Attempt to construct a nondegenerate spline from a general
QuadraticSpline3d
. If the spline is in fact degenerate (consists of a single
point), returns an Err
with that point.
QuadraticSpline3d.nondegenerate exampleSpline
--> Ok nondegenerateExampleSpline
fromNondegenerate : Nondegenerate -> QuadraticSpline3d
Convert a nondegenerate spline back to a general QuadraticSpline3d
.
QuadraticSpline3d.fromNondegenerate
nondegenerateExampleSpline
--> exampleSpline
tangentDirection : Nondegenerate -> Curve.ParameterValue.ParameterValue -> Direction3d
Get the tangent direction to a nondegenerate spline at a given parameter value:
QuadraticSpline3d.tangentDirection
nondegenerateExampleSpline
ParameterValue.zero
--> Direction3d.fromAzimuthAndElevation
--> (degrees 26.57)
--> (degrees 0)
QuadraticSpline3d.tangentDirection
nondegenerateExampleSpline
ParameterValue.half
--> Direction3d.fromAzimuthAndElevation
--> (degrees 45)
--> (degrees 35.26)
QuadraticSpline3d.tangentDirection
nondegenerateExampleSpline
ParameterValue.one
--> Direction3d.fromAzimuthAndElevation
--> (degrees 90)
--> (degrees 63.43)
tangentDirectionsAt : List Curve.ParameterValue.ParameterValue -> Nondegenerate -> List Direction3d
Get tangent directions to a nondegenerate spline at a given set of parameter values:
nondegenerateExampleSpline
|> QuadraticSpline3d.tangentDirectionsAt
(ParameterValue.steps 2)
--> [ Direction3d.fromAzimuthAndElevation
--> (degrees 26.57)
--> (degrees 0)
--> , Direction3d.fromAzimuthAndElevation
--> (degrees 45)
--> (degrees 35.26)
--> , Direction3d.fromAzimuthAndElevation
--> (degrees 90)
--> (degrees 63.43)
--> ]
sample : Nondegenerate -> Curve.ParameterValue.ParameterValue -> ( Point3d, Direction3d )
Get both the point and tangent direction of a nondegenerate spline at a given parameter value:
QuadraticSpline3d.sample nondegenerateExampleSpline
ParameterValue.half
--> ( Point3d.fromCoordinates ( 2.5, 2, 1.5 )
--> , Direction3d.fromAzimuthAndElevation
--> (degrees 45)
--> (degrees 35.26)
--> )
samplesAt : List Curve.ParameterValue.ParameterValue -> Nondegenerate -> List ( Point3d, Direction3d )
Get points and tangent directions of a nondegenerate spline at a given set of parameter values:
nondegenerateExampleSpline
|> QuadraticSpline3d.samplesAt
(ParameterValue.steps 2)
--> [ ( Point3d.fromCoordinates ( 1, 1, 1 )
--> , Direction3d.fromAzimuthAndElevation
--> (degrees 26.57)
--> (degrees 0)
--> )
--> , ( Point3d.fromCoordinates ( 2.5, 2, 1.5 )
--> , Direction3d.fromAzimuthAndElevation
--> (degrees 45)
--> (degrees 35.26)
--> )
--> , ( Point3d.fromCoordinates ( 3, 3, 3 )
--> , Direction3d.fromAzimuthAndElevation
--> (degrees 90)
--> (degrees 63.43)
--> )
--> ]
reverse : QuadraticSpline3d -> QuadraticSpline3d
Reverse a spline so that the start point becomes the end point, and vice versa.
QuadraticSpline3d.reverse exampleSpline
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 3, 3, 3 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 3, 2, 1 )
--> , endPoint =
--> Point3d.fromCoordinates ( 1, 1, 1 )
--> }
scaleAbout : Point3d -> Basics.Float -> QuadraticSpline3d -> QuadraticSpline3d
Scale a spline about the given center point by the given scale.
exampleSpline
|> QuadraticSpline3d.scaleAbout Point3d.origin 2
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 2, 2, 2 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 6, 4, 2 )
--> , endPoint =
--> Point3d.fromCoordinates ( 6, 6, 6 )
--> }
rotateAround : Axis3d -> Basics.Float -> QuadraticSpline3d -> QuadraticSpline3d
Rotate a spline counterclockwise around a given axis by a given angle (in radians).
exampleSpline
|> QuadraticSpline3d.rotateAround Axis3d.z
(degrees 90)
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( -1, 1, 1 )
--> , controlPoint =
--> Point3d.fromCoordinates ( -2, 3, 1 )
--> , endPoint =
--> Point3d.fromCoordinates ( -3, 3, 3 )
--> }
translateBy : Vector3d -> QuadraticSpline3d -> QuadraticSpline3d
Translate a spline by a given displacement.
displacement =
Vector3d.fromComponents ( 2, 3, 1 )
exampleSpline
|> QuadraticSpline3d.translateBy displacement
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 3, 4, 2 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 5, 5, 2 )
--> , endPoint =
--> Point3d.fromCoordinates ( 5, 6, 4 )
--> }
translateIn : Direction3d -> Basics.Float -> QuadraticSpline3d -> QuadraticSpline3d
Translate an arc in a given direction by a given distance;
QuadraticSpline3d.translateIn direction distance
is equivalent to
QuadraticSpline3d.translateBy
(Vector3d.withLength distance direction)
mirrorAcross : Plane3d -> QuadraticSpline3d -> QuadraticSpline3d
Mirror a spline across a plane.
QuadraticSpline3d.mirrorAcross Plane3d.xy exampleSpline
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 1, 1, -1 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 3, 2, -1 )
--> , endPoint =
--> Point3d.fromCoordinates ( 3, 3, -3 )
--> }
projectOnto : Plane3d -> QuadraticSpline3d -> QuadraticSpline3d
Find the orthographic projection of a spline onto a plane.
QuadraticSpline3d.projectOnto Plane3d.xy exampleSpline
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 1, 1, 0 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 3, 2, 0 )
--> , endPoint =
--> Point3d.fromCoordinates ( 3, 3, 0 )
--> }
relativeTo : Frame3d -> QuadraticSpline3d -> QuadraticSpline3d
Take a spline defined in global coordinates, and return it expressed in local coordinates relative to a given reference frame.
localFrame =
Frame3d.atPoint
(Point3d.fromCoordinates ( 1, 2, 3 ))
QuadraticSpline3d.relativeTo localFrame exampleSpline
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 0, -1, -2 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 2, 0, -2 )
--> , endPoint =
--> Point3d.fromCoordinates ( 2, 1, 0 )
--> }
placeIn : Frame3d -> QuadraticSpline3d -> QuadraticSpline3d
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 =
Frame3d.atPoint
(Point3d.fromCoordinates ( 1, 2, 3 ))
QuadraticSpline3d.placeIn localFrame exampleSpline
--> QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 2, 3, 4 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 4, 4, 4 )
--> , endPoint =
--> Point3d.fromCoordinates ( 4, 5, 6 )
--> }
projectInto : SketchPlane3d -> QuadraticSpline3d -> QuadraticSpline2d
Project a spline into a given sketch plane. Conceptually, finds the orthographic projection of the spline onto the plane and then expresses the projected spline in 2D sketch coordinates.
exampleSpline
|> QuadraticSpline3d.projectInto SketchPlane3d.yz
--> QuadraticSpline2d.with
--> { startPoint =
--> Point2d.fromCoordinates ( 1, 1 )
--> , controlPoint =
--> Point2d.fromCoordinates ( 2, 1 )
--> , endPoint =
--> Point2d.fromCoordinates ( 3, 3 )
--> }
bisect : QuadraticSpline3d -> ( QuadraticSpline3d, QuadraticSpline3d )
Split a spline into two roughly equal halves.
QuadraticSpline3d.bisect exampleSpline
--> ( QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 1, 1, 1 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 2, 2.5 )
--> , endPoint =
--> Point3d.fromCoordinates ( 3, 2.5 )
--> }
--> , QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 3, 2.5 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 4, 2.5 )
--> , endPoint =
--> Point3d.fromCoordinates ( 3, 3, 3 )
--> }
--> )
Equivalent to QuadraticSpline3d.splitAt ParameterValue.half
.
splitAt : Curve.ParameterValue.ParameterValue -> QuadraticSpline3d -> ( QuadraticSpline3d, QuadraticSpline3d )
Split a spline at a particular parameter value, resulting in two smaller splines.
parameterValue =
ParameterValue.clamped 0.75
QuadraticSpline3d.splitAt parameterValue exampleSpline
--> ( QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 1, 1, 1 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 2, 1.5, 1 )
--> , endPoint =
--> Point3d.fromCoordinates ( 2.5, 2, 1.5 )
--> }
--> , QuadraticSpline3d.with
--> { startPoint =
--> Point3d.fromCoordinates ( 2.5, 2, 1.5 )
--> , controlPoint =
--> Point3d.fromCoordinates ( 3, 2.5, 2 )
--> , endPoint =
--> Point3d.fromCoordinates ( 3, 3, 3 )
--> }
--> )
A spline that has been parameterized by arc length.
arcLengthParameterized : { maxError : Basics.Float } -> QuadraticSpline3d -> 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
|> QuadraticSpline3d.arcLengthParameterized
{ maxError = 1.0e-4 }
arcLength : ArcLengthParameterized -> Basics.Float
Find the total arc length of a spline.
QuadraticSpline3d.arcLength parameterizedSpline
--> 3.8175
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 Point3d
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
:
QuadraticSpline3d.pointAlong parameterizedSpline
(arcLength / 4)
--> Just <|
--> Point3d.fromCoordinates
--> ( 1.8227, 1.4655, 1.1083 )
Note that this is not the same as evaulating at a parameter value of 1/4:
QuadraticSpline3d.pointOn exampleSpline 0.25
--> Point3d.fromCoordinates ( 1.875, 1.5, 1.125 )
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 Direction3d
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
:
QuadraticSpline3d.tangentDirectionAlong
parameterizedSpline
(arcLength / 4)
--> Just <|
--> Direction3d.fromAzimuthAndElevation
--> (degrees 33.09)
--> (degrees 14.26)
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 ( Point3d, Direction3d )
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
:
QuadraticSpline3d.sampleAlong parameterizedSpline
(0.25 * arcLength)
--> Just
--> ( Point3d.fromCoordinates
--> ( 1.8227, 1.4655, 1.1083 )
--> , Direction3d.fromAzimuthAndElevation
--> (degrees 33.09)
--> (degrees 14.26)
--> )
If the given arc length is less than zero or greater than the arc length of the
spline (or if the spline is degenerate), Nothing
is returned.
An ArcLengthParameterized
value is a combination of an
ArcLengthParameterization
and an
underlying QuadraticSpline3d
. If you need to do something fancy, you can
extract these two values separately.
arcLengthParameterization : ArcLengthParameterized -> Curve.ArcLengthParameterization.ArcLengthParameterization
fromArcLengthParameterized : ArcLengthParameterized -> QuadraticSpline3d
Get the original QuadraticSpline3d
from which an ArcLengthParameterized
value was constructed.
You are unlikely to need to use these functions directly, but they are useful if you are writing low-level geometric algorithms.
firstDerivative : QuadraticSpline3d -> Curve.ParameterValue.ParameterValue -> Vector3d
Get the first derivative of a spline at a given parameter value:
QuadraticSpline3d.derivative exampleSpline
ParameterValue.zero
--> Vector3d.fromComponents ( 4, 2, 0 )
QuadraticSpline3d.derivative exampleSpline
ParameterValue.half
--> Vector3d.fromComponents ( 2, 2, 2 )
QuadraticSpline3d.derivative exampleSpline
ParameterValue.one
--> Vector3d.fromComponents ( 0, 2, 4 )
Note that the derivative interpolates linearly from end to end.
firstDerivativesAt : List Curve.ParameterValue.ParameterValue -> QuadraticSpline3d -> List Vector3d
Evaluate the first derivative of a spline at a range of parameter values:
exampleSpline
|> QuadraticSpline3d.firstDerivativesAt
(ParameterValue.steps 2)
--> [ Vector3d.fromComponents ( 4, 2, 0 )
--> , Vector3d.fromComponents ( 2, 2, 2 )
--> , Vector3d.fromComponents ( 0, 2, 4 )
--> ]
secondDerivative : QuadraticSpline3d -> Vector3d
Get the second derivative of a spline (for a quadratic spline, this is a constant).