Construct a spline from its four control points:

exampleSpline =
    CubicSpline3d.fromControlPoints
        ( Point3d.fromCoordinates ( 1, 1, 1 )
        , Point3d.fromCoordinates ( 3, 1, 1 )
        , Point3d.fromCoordinates ( 3, 3, 1 )
        , Point3d.fromCoordinates ( 3, 3, 3 )
        )

Construct a spline in Hermite form, from the position and derivative values at its start and end points, 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.

Construct a 3D spline lying on a sketch plane by providing a 2D spline specified in XY coordinates within the sketch plane.

CubicSpline3d.on SketchPlane3d.xz <|
    CubicSpline2d.fromControlPoints
        ( Point2d.fromCoordinates ( 1, 1 )
        , Point2d.fromCoordinates ( 3, 4 )
        , Point2d.fromCoordinates ( 5, 1 )
        , Point2d.fromCoordinates ( 7, 4 )
        )
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 1, 0, 1 )
-->     , Point3d.fromCoordinates ( 3, 0, 4 )
-->     , Point3d.fromCoordinates ( 5, 0, 1 )
-->     , Point3d.fromCoordinates ( 7, 0, 4 )
-->     )

Get the control points of a spline as a tuple.

( p1, p2, p3, p4 ) =
    CubicSpline3d.controlPoints exampleSpline




--> p1 = Point3d.fromCoordinates ( 1, 1, 1 )
--> p2 = Point3d.fromCoordinates ( 3, 1, 1 )
--> p3 = Point3d.fromCoordinates ( 3, 3, 1 )
--> p4 = Point3d.fromCoordinates ( 3, 3, 3 )

Get the start point of a spline. This is equal to the spline's first control point.

CubicSpline3d.startPoint exampleSpline
--> Point3d.fromCoordinates ( 1, 1, 1 )

Get the end point of a spline. This is equal to the spline's last control point.

CubicSpline3d.endPoint exampleSpline
--> Point3d.fromCoordinates ( 3, 3, 3 )

Get the start derivative of a spline. This is equal to three times the vector from the spline's first control point to its second.

CubicSpline3d.startDerivative exampleSpline
--> Vector3d.fromComponents ( 6, 0, 0 )

Get the end derivative of a spline. This is equal to three times the vector from the spline's third control point to its fourth.

CubicSpline3d.endDerivative exampleSpline
--> Vector3d.fromComponents ( 0, 0, 6 )

Get a point along a spline, based on a parameter that ranges from 0 to 1. A parameter value of 0 corresponds to the start point of the spline and a value of 1 corresponds to the end point.

CubicSpline3d.pointOn exampleSpline 0
--> Point3d.fromCoordinates ( 1, 1, 1 )


CubicSpline3d.pointOn exampleSpline 0.5
--> Point3d.fromCoordinates ( 2.75, 2, 1.25 )


CubicSpline3d.pointOn exampleSpline 1
--> Point3d.fromCoordinates ( 3, 3, 3 )

Get the deriative value at a point along a spline, based on a parameter that ranges from 0 to 1. A parameter value of 0 corresponds to the start derivative of the spline and a value of 1 corresponds to the end derivative.

CubicSpline3d.derivative exampleSpline 0
--> Vector3d.fromComponents ( 6, 0, 0 )


CubicSpline3d.derivative exampleSpline 0.5
--> Vector3d.fromComponents ( 1.5, 3, 1.5 )


CubicSpline3d.derivative exampleSpline 1
--> Vector3d.fromComponents ( 0, 0, 6 )

Note that the derivative interpolates linearly from end to end.

Evaluate a spline at a given parameter value, returning the point on the spline at that parameter value and the derivative with respect to that parameter value;

CubicSpline3d.evaluate spline t

is equivalent to

( CubicSpline3d.pointOn spline t
, CubicSpline3d.derivative spline t
)

but is more efficient.

Reverse a spline so that the start point becomes the end point, and vice versa.

CubicSpline3d.reverse exampleSpline
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 3, 3, 3 )
-->     , Point3d.fromCoordinates ( 3, 3, 1 )
-->     , Point3d.fromCoordinates ( 3, 1, 1 )
-->     , Point3d.fromCoordinates ( 1, 1, 1 )
-->     )

Scale a spline about the given center point by the given scale.

CubicSpline3d.scaleAbout Point3d.origin 2 exampleSpline
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 2, 2, 2 )
-->     , Point3d.fromCoordinates ( 6, 2, 2 )
-->     , Point3d.fromCoordinates ( 6, 6, 2 )
-->     , Point3d.fromCoordinates ( 6, 6, 6 )
-->     )

Rotate a spline counterclockwise around a given axis by a given angle (in radians).

exampleSpline
    |> CubicSpline3d.rotateAround Axis3d.z (degrees 90)
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( -1, 1, 1 )
-->     , Point3d.fromCoordinates ( -1, 3, 1 )
-->     , Point3d.fromCoordinates ( -3, 3, 1 )
-->     , Point3d.fromCoordinates ( -3, 3, 3 )
-->     )

Translate a spline by a given displacement.

displacement =
    Vector3d.fromComponents ( 2, 3, 1 )


CubicSpline3d.translateBy displacement exampleSpline
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 3, 4, 2 )
-->     , Point3d.fromCoordinates ( 5, 4, 2 )
-->     , Point3d.fromCoordinates ( 5, 6, 2 )
-->     , Point3d.fromCoordinates ( 5, 6, 4 )
-->     )

Mirror a spline across a plane.

CubicSpline3d.mirrorAcross Plane3d.xy exampleSpline
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 1, 1, -1 )
-->     , Point3d.fromCoordinates ( 3, 1, -1 )
-->     , Point3d.fromCoordinates ( 3, 3, -1 )
-->     , Point3d.fromCoordinates ( 3, 3, -3 )
-->     )

Project a spline onto a plane.

CubicSpline3d.projectOnto Plane3d.xy exampleSpline
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 1, 1, 0 )
-->     , Point3d.fromCoordinates ( 3, 1, 0 )
-->     , Point3d.fromCoordinates ( 3, 3, 0 )
-->     , Point3d.fromCoordinates ( 3, 3, 0 )
-->     )

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 ))


CubicSpline3d.relativeTo localFrame exampleSpline
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 0, -1, -2 )
-->     , Point3d.fromCoordinates ( 2, -1, -2 )
-->     , Point3d.fromCoordinates ( 2, 1, -2 )
-->     , Point3d.fromCoordinates ( 2, 1, 0 )
-->     )

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 ))


CubicSpline3d.placeIn localFrame exampleSpline
--> CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 2, 3, 4 )
-->     , Point3d.fromCoordinates ( 4, 3, 4 )
-->     , Point3d.fromCoordinates ( 4, 5, 4 )
-->     , Point3d.fromCoordinates ( 4, 5, 6 )
-->     )

Project a spline into a given sketch plane. Conceptually, this projects the spline onto the plane and then expresses the projected spline in 2D sketch coordinates.

exampleSpline
    |> CubicSpline3d.projectInto SketchPlane3d.yz
--> CubicSpline2d.fromControlPoints
-->     ( Point2d.fromCoordinates ( 1, 1 )
-->     , Point2d.fromCoordinates ( 1, 1 )
-->     , Point2d.fromCoordinates ( 3, 1 )
-->     , Point2d.fromCoordinates ( 3, 3 )
-->     )

Split a spline into two roughly equal halves. Equivalent to splitAt 0.5.

CubicSpline3d.bisect exampleSpline
--> ( CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 1, 1, 1 )
-->     , Point3d.fromCoordinates ( 2, 1, 1 )
-->     , Point3d.fromCoordinates ( 2.5, 1.5, 1 )
-->     , Point3d.fromCoordinates ( 2.75, 2, 1.25 )
-->     )
--> , CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 2.75, 2, 1.25 )
-->     , Point3d.fromCoordinates ( 3, 2.5, 1.5 )
-->     , Point3d.fromCoordinates ( 3, 3, 2 )
-->     , Point3d.fromCoordinates ( 3, 3, 3 )
-->     )
--> )

Split a spline at a particular parameter value (in the range 0 to 1), resulting in two smaller splines.

CubicSpline3d.splitAt 0.75 exampleSpline
--> ( CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 1, 1, 1 )
-->     , Point3d.fromCoordinates ( 2.5, 1, 1 )
-->     , Point3d.fromCoordinates ( 2.88, 2.13, 1 )
-->     , Point3d.fromCoordinates ( 2.97, 2.69, 1.84 )
-->     )
--> , CubicSpline3d.fromControlPoints
-->     ( Point3d.fromCoordinates ( 2.97, 2.69, 1.84 )
-->     , Point3d.fromCoordinates ( 3, 2.88, 2.13 )
-->     , Point3d.fromCoordinates ( 3, 3, 2.5 )
-->     , Point3d.fromCoordinates ( 3, 3, 3 )
-->     )
--> )