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

arc =
    Arc3d.on SketchPlane3d.xz
        (Point2d.fromCoordinates ( 3, 1 )
            |> Arc2d.sweptAround
                (Point2d.fromCoordinates ( 1, 1 )
                (degrees 90)
        )


Arc3d.centerPoint arc
--> Point3d.fromCoordinates ( 1, 0, 1 )


Arc3d.radius arc
--> 2


Arc3d.startPoint arc
--> Point3d.fromCoordinates ( 3, 0, 1 )


Arc3d.endPoint arc
--> Point3d.fromCoordinates ( 1, 0, 3 )

Construct an arc by sweeping the given point around the given axis by the given angle:

exampleArc =
    Point3d.fromCoordinates ( 1, 1, 0 )
        |> Arc3d.sweptAround Axis3d.z (degrees 90)


Arc3d.centerPoint exampleArc
--> Point3d.origin


Arc3d.endPoint exampleArc
--> Point3d.fromCoordinates ( -1, 1, 0 )

Positive swept angles result in a counterclockwise (right-handed) rotation around the given axis and vice versa for negative swept angles. The center point of the returned arc will lie on the given axis.

Attempt to construct an arc that starts at the first given point, passes through the second given point and ends at the third given point. If the three points are collinear, returns Nothing.

p1 =
    Point3d.fromCoordinates ( 0, 0, 1 )


p2 =
    Point3d.origin


p3 =
    Point3d.fromCoordinates ( 0, 1, 0 )


Arc3d.throughPoints p1 p2 p3
--> Just
-->     (Arc3d.on SketchPlane3d.yz
-->         Point2d.fromCoordinates ( 0, 1 )
-->             |> Arc2d.sweptAround
-->                 (Point2d.fromCoordinates
-->                     ( 0.5, 0.5 )
-->                 )
-->                 (degrees 180)
-->     )

Get the axial direction of an arc.

Arc3d.axialDirection exampleArc
--> Direction3d.z

Get the central axis of an arc. The origin point of the axis will be equal to the center point of the arc.

Arc3d.axis exampleArc
--> Axis3d.z

Get the center point of an arc.

Arc3d.centerPoint exampleArc
--> Point3d.origin

Get the radius of an arc.

Arc3d.radius exampleArc
--> 1.4142

Get the start point of an arc.

Arc3d.startPoint exampleArc
--> Point3d.fromCoordinates ( 1, 1, 0 )

Get the end point of an arc.

Arc3d.endPoint exampleArc
--> Point3d.fromCoordinates ( -1, 1, 0 )

Get the swept angle of an arc in radians.

Arc3d.sweptAngle exampleArc
--> 1.5708

A positive swept angle means that the arc is formed by rotating the given start point counterclockwise around the central axis, and vice versa for a negative angle.

Get the point along an arc at a given parameter value:

Arc3d.pointOn exampleArc ParameterValue.half
--> Point3d.fromCoordinates ( 0, 1.4142, 0 )

Get points along an arc at a given set of parameter values.

exampleArc |> Arc3d.pointsAt (ParameterValue.steps 2)
--> [ Point3d ( 1, 1, 0 )
--> , Point3d ( 0, 1.4142, 0 )
--> , Point3d ( -1, 1, 0 )
--> ]

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 an arc that is definitely not degenerate. It is used as input to functions such as Arc3d.tangentDirection and can be constructed using Arc3d.nondegenerate.

Attempt to construct a nondegenerate arc from a general Arc3d. If the arc is in fact degenerate (consists of a single point), returns an Err with that point.

Arc3d.nondegenerate exampleArc
--> Ok nondegenerateExampleArc

Convert a nondegenerate arc back to a general Arc3d.

Arc3d.fromNondegenerate nondegenerateExampleArc
--> exampleArc

Get the tangent direction to a nondegenerate arc at a given parameter value:

Arc3d.tangentDirection nondegenerateExampleArc
    ParameterValue.zero
--> Direction3d.fromAzimuthAndElevation
-->     (degrees 135)
-->     (degrees 0)


Arc3d.tangentDirection nondegenerateExampleArc
    ParameterValue.half
--> Direction3d.negativeX


Arc3d.tangentDirection nondegenerateExampleArc
    ParameterValue.zero
--> Direction3d.fromAzimuthAndElevation
-->     (degrees 225)
-->     (degrees 0)

Get tangent directions to a nondegenerate arc at a given set of parameter values:

nondegenerateExampleArc
    |> Arc3d.tangentDirectionsAt
        (ParameterValue.steps 2)
--> [ Direction3d.fromAzimuthAndElevation
-->     (degrees 135)
-->     (degrees 0)
--> , Direction3d.negativeX
--> , Direction3d.fromAzimuthAndElevation
-->     (degrees 225)
-->     (degrees 0)
--> ]

Get both the point and tangent direction of a nondegenerate arc at a given parameter value:

Arc3d.sample nondegenerateExampleArc
    ParameterValue.zero
--> ( Point3d.fromCoordinates ( 1, 1, 0 )
--> , Direction3d.fromAzimuthAndElevation
-->     (degrees 135)
-->     (degrees 0)
--> )


Arc3d.sample nondegenerateExampleArc
    ParameterValue.half
--> ( Point3d.fromCoordinates ( 0, 1.4142, 0 )
--> , Direction3d.negativeX
--> )


Arc3d.sample nondegenerateExampleArc
    ParameterValue.one
--> ( Point3d.fromCoordinates ( -1, 1, 0 )
--> , Direction3d.fromAzimuthAndElevation
-->     (degrees 225)
-->     (degrees 0)
--> )

Get points and tangent directions of a nondegenerate arc at a given set of parameter values:

nondegenerateExampleArc
    |> Arc3d.samplesAt (ParameterValue.steps 2)
--> [ ( Point3d.fromCoordinates ( 1, 1, 0 )
-->   , Direction3d.fromAzimuthAndElevation
-->         (degrees 135)
-->         (degrees 0)
-->   )
--> , ( Point3d.fromCoordinates ( 0, 1.4142, 0 )
-->   , Direction3d.negativeX
-->   )
--> , ( Point3d.fromCoordinates ( -1, 1, 0 )
-->   , Direction3d.fromAzimuthAndElevation
-->         (degrees 225)
-->         (degrees 0)
-->   )
--> ]

If the arc is degenerate (start point and end point are equal), returns an empty list.

Approximate an arc as a polyline, within a given tolerance:

exampleArc |> Arc3d.toPolyline { maxError = 0.1 }
--> Polyline3d.fromVertices
-->     [ Point3d.fromCoordinates ( 1, 1, 0 )
-->     , Point3d.fromCoordinates ( 0.366, 1.366, 0 )
-->     , Point3d.fromCoordinates ( -0.366, 1.366, 0 )
-->     , Point3d.fromCoordinates ( -1, 1, 0 )
-->     ]

In this example, every point on the returned polyline will be within 0.1 units of the original arc.

Reverse the direction of an arc, so that the start point becomes the end point and vice versa. The resulting arc will have the same axis as the original but a swept angle with the opposite sign.

Arc3d.reverse exampleArc
--> Arc3d.sweptAround Axis3d.z
-->     (degrees -90)
-->     (Point3d.fromCoordinates ( -1, 1, 0 ))

Scale an arc about the given center point by the given scale.

point =
    Point3d.fromCoordinates ( 0, -1, 0 )


Arc3d.scaleAbout point 2 exampleArc
--> Arc3d.sweptAround
-->     (Axis3d.withDirection Direction3d.z
-->         (Point3d.fromCoordinates ( 0, 1, 0 ))
-->     )
-->     (degrees 90)
-->     (Point3d.fromCoordinates ( 2, 3, 0 ))

Rotate an arc around a given axis by a given angle (in radians).

Arc3d.rotateAround Axis3d.x (degrees 90) exampleArc
--> Arc3d.sweptAround (Axis3d.reverse Axis3d.y)
-->     (degrees 90)
-->     (Point3d.fromCoordinates ( 1, 0, 1 ))

Translate an arc by a given displacement.

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


Arc3d.translateBy displacement exampleArc
--> Arc3d.sweptAround
-->     (Axis3d.withDirection Direction3d.z
-->         (Point3d ( 2, 1, 3 ))
-->     )
-->     (degrees 90)
-->     (Point3d.fromCoordinates ( 3, 2, 3 ))

Translate an arc in a given direction by a given distance;

Arc3d.translateIn direction distance

is equivalent to

Arc3d.translateBy
    (Vector3d.withLength distance direction)

Mirror an arc across a given plane.

Arc3d.mirrorAcross Plane3d.xy exampleArc
--> Arc3d.sweptAround (Axis3d.reverse Axis3d.z)
-->     (degrees -90)
-->     (Point3d.fromCoordinates ( 1, 1, 0 ))

Note that this flips the sign of the arc's swept angle.

Project an arc into a sketch plane.

axis : Axis3d
axis =
    Axis3d.through
        (Point3d.fromCoordinates ( 1, 2, 3 ))
        (Direction3d.fromAzimuthAndElevation
            (degrees 0)
            (degrees 45)
        )


arc : Arc3d
arc =
    Arc3d.sweptAround axis
        (degrees 45)
        (Point3d.fromCoordinates ( 1, 4, 3 ))


Arc3d.projectInto SketchPlane3d.xy arc
--> EllipticalArc2d.with
-->     { centerPoint =
-->         Point2d.fromCoordinates ( 1, 2 )
-->     , xDirection = Direction2d.y
-->     , xRadius = 2
-->     , yRadius = 1.4142
-->     , startAngle = degrees 0
-->     , sweptAngle = degrees 45
-->     }

Take an arc 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 ))


Arc3d.relativeTo localFrame exampleArc
--> Arc3d.sweptAround
-->     (Axis3d.withDirection Direction3d.z
-->         (Point3d ( -1, -2, -3 ))
-->     )
-->     (degrees 90)
-->     (Point3d.fromCoordinates ( 0, -1, -3 ))

Take an arc considered to be defined in local coordinates relative to a given reference frame, and return that arc expressed in global coordinates.

localFrame =
    Frame3d.atPoint
        (Point3d.fromCoordinates ( 1, 2, 3 ))


Arc3d.placeIn localFrame exampleArc
--> Arc3d.sweptAround
-->     (Axis3d.withDirection Direction3d.z
-->         (Point3d.fromCoordinates ( 1, 2, 3 ))
-->     )
-->     (degrees 90)
-->     (Point3d.fromCoordinates ( 2, 3, 3 ))

Get the first derivative of an arc at a given parameter value.

Arc3d.firstDerivative exampleArc ParameterValue.zero
--> Vector3d.fromComponents ( -1.5708, 1.5708, 0 )


Arc3d.firstDerivative exampleArc ParameterValue.one
--> Vector3d.fromComponents ( -1.5708, -1.5708, 0 )

Evaluate the first derivative of an arc at a range of parameter values.

exampleArc
    |> Arc3d.firstDerivativesAt
        (ParameterValue.steps 2)
--> [ Vector3d ( -1.5708, 1.5708, 0 )
--> , Vector3d ( -2.2214, 0, 0 )
--> , Vector3d ( -1.5708, -1.5708, 0 )
--> ]