The point (0, 0, 0).

Point3d.origin
--> Point3d ( 0, 0, 0 )

Construct a point halfway between two other points.

p1 =
    Point3d ( 1, 1, 1 )


p2 =
    Point3d ( 3, 7, 9 )


Point3d.midpoint p1 p2
--> Point3d ( 2, 4, 5 )

Construct a point by interpolating between two other points based on a parameter that ranges from zero to one.

startPoint =
    Point3d ( 1, 2, 4 )


endPoint =
    Point3d ( 1, 2, 8 )


Point3d.interpolate startPoint endPoint 0.25
--> Point3d ( 1, 2, 5 )

Partial application may be useful:

interpolatedPoint : Float -> Point3d
interpolatedPoint =
    Point3d.interpolate startPoint endPoint


List.map interpolatedPoint [ 0, 0.5, 1 ]
--> [ Point3d ( 1, 2, 4 )
--> , Point3d ( 1, 2, 6 )
--> , Point3d ( 1, 2, 8 )
--> ]

You can pass values less than zero or greater than one to extrapolate:

interpolatedPoint -0.5
--> Point3d ( 1, 2, 2 )


interpolatedPoint 1.25
--> Point3d ( 1, 2, 9 )

Construct a point along an axis at a particular distance from the axis' origin point.

Point3d.along Axis3d.z 2
--> Point3d ( 0, 0, 2 )

Positive and negative distances are interpreted relative to the direction of the axis:

horizontalAxis =
    Axis3d
        { originPoint = Point3d ( 1, 1, 1 )
        , direction = Direction3d ( -1, 0, 0 )
        }


Point3d.along horizontalAxis 3
--> Point3d ( -2, 1, 1 )


Point3d.along horizontalAxis -3
--> Point3d ( 4, 1, 1 )

Get the coordinates of a point as a tuple.

( x, y, z ) =
    Point3d.coordinates point

Get the X coordinate of a point.

Point3d.xCoordinate (Point3d ( 2, 1, 3 ))
--> 2

Get the Y coordinate of a point.

Point3d.yCoordinate (Point3d ( 2, 1, 3 ))
--> 1

Get the Z coordinate of a point.

Point3d.zCoordinate (Point3d ( 2, 1, 3 ))
--> 3

Compare two points within a tolerance. Returns true if the distance between the two given points is less than the given tolerance.

firstPoint =
    Point3d ( 2, 1, 3 )


secondPoint =
    Point3d ( 2.0002, 0.9999, 3.0001 )


Point3d.equalWithin 1e-3 firstPoint secondPoint
--> True


Point3d.equalWithin 1e-6 firstPoint secondPoint
--> False

Find the vector from one point to another.

startPoint =
    Point3d ( 1, 1, 1 )


endPoint =
    Point3d ( 4, 5, 6 )


Point3d.vectorFrom startPoint endPoint
--> Vector3d ( 3, 4, 5 )

Find the distance between two points.

p1 =
    Point3d ( 1, 1, 2 )


p2 =
    Point3d ( 2, 3, 4 )


Point3d.distanceFrom p1 p2
--> 3

Partial application can be useful:

points =
    [ Point3d ( 3, 4, 5 )
    , Point3d ( 10, 10, 10 )
    , Point3d ( -1, 2, -3 )
    ]


distanceFromOrigin : Point3d -> Float
distanceFromOrigin =
    Point3d.distanceFrom Point3d.origin


List.sortBy distanceFromOrigin points
--> [ Point3d ( -1, 2, -3 )
--> , Point3d ( 3, 4, 5 )
--> , Point3d ( 10, 10, 10 )
--> ]

Find the square of the distance from one point to another. squaredDistanceFrom is slightly faster than distanceFrom, so for example

Point3d.squaredDistanceFrom p1 p2 > tolerance * tolerance

is equivalent to but slightly more efficient than

Point3d.distanceFrom p1 p2 > tolerance

since the latter requires a square root under the hood. In many cases, however, the speed difference will be negligible and using distanceFrom is much more readable!

Determine how far along an axis a particular point lies. Conceptually, the point is projected perpendicularly onto the axis, and then the distance of this projected point from the axis' origin point is measured. The result will be positive if the projected point is ahead the axis' origin point and negative if it is behind, with 'ahead' and 'behind' defined by the direction of the axis.

axis =
    Axis3d
        { originPoint = Point3d ( 1, 0, 0 )
        , direction = Direction3d.x
        }


point =
    Point3d ( 3, 3, 3 )


Point3d.distanceAlong axis point
--> 2


Point3d.distanceAlong axis Point3d.origin
--> -1

Find the perpendicular (nearest) distance of a point from an axis.

point =
    Point3d ( -3, 4, 0 )


Point3d.radialDistanceFrom Axis3d.x point
--> 4


Point3d.radialDistanceFrom Axis3d.y point
--> 3


Point3d.radialDistanceFrom Axis3d.z point
--> 5

Note that unlike in 2D, the result is always positive (unsigned) since there is no such thing as the left or right side of an axis in 3D.

Find the square of the perpendicular distance of a point from an axis. As with distanceFrom/squaredDistanceFrom this is slightly more efficient than radialDistanceFrom since it avoids a square root.

Find the perpendicular distance of a point from a plane. The result will be positive if the point is 'above' the plane and negative if it is 'below', with 'up' defined by the normal direction of the plane.

plane =
    Plane3d
        { originPoint = Point3d ( 1, 2, 3 )
        , normalDirection = Direction3d.y
        }


point =
    Point3d ( 3, 3, 3 )


Point3d.signedDistanceFrom plane point
--> 1


Point3d.signedDistanceFrom plane Point3d.origin
--> -2

This means that flipping a plane (reversing its normal direction) will also flip the sign of the result of this function:

flippedPlane =
    Plane3d.flip plane


Point3d.signedDistanceFrom flippedPlane point
--> -1

Perform a uniform scaling about the given center point. The center point is given first and the point to transform is given last. Points will contract or expand about the center point by the given scale. Scaling by a factor of 1 is a no-op, and scaling by a factor of 0 collapses all points to the center point.

centerPoint =
    Point3d ( 1, 1, 1 )


point =
    Point3d ( 1, 2, 3 )


Point3d.scaleAbout centerPoint 3 point
--> Point3d ( 1, 4, 7 )


Point3d.scaleAbout centerPoint 0.5 point
--> Point3d ( 1, 1.5, 2 )

Avoid scaling by a negative scaling factor - while this may sometimes do what you want it is confusing and error prone. Try a combination of mirror and/or rotation operations instead.

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

axis =
    Axis3d.x


angle =
    degrees 45


point =
    Point3d ( 3, 1, 0 )


Point3d.rotateAround axis angle point
--> Point3d ( 3, 0.7071, 0.7071 )

Rotation direction is given by the right-hand rule, counterclockwise around the direction of the axis.

Translate a point by a given displacement.

point =
    Point3d ( 3, 4, 5 )


displacement =
    Vector3d ( 1, 2, 3 )


Point3d.translateBy displacement point
--> Point3d ( 4, 6, 8 )

Mirror a point across a plane. The result will be the same distance from the plane but on the opposite side.

point =
    Point3d ( 1, 2, 3 )


-- Plane3d.xy is the plane Z=0
Point3d.mirrorAcross Plane3d.xy point
--> Point3d ( 1, 2, -3 )


-- Plane3d.yz is the plane X=0
Point3d.mirrorAcross Plane3d.yz point
--> Point3d ( -1, 2, 3 )

The plane does not have to pass through the origin:

-- offsetPlane is the plane Z=1
offsetPlane =
    Plane3d.offsetBy 1 Plane3d.xy


-- The origin point is 1 unit below the offset
-- plane, so its mirrored copy is one unit above
Point3d.mirrorAcross offsetPlane Point3d.origin
--> Point3d ( 0, 0, 2 )

Project a point perpendicularly onto a plane.

point =
    Point3d ( 1, 2, 3 )


Point3d.projectOnto Plane3d.xy point
--> Point3d ( 1, 2, 0 )


Point3d.projectOnto Plane3d.yz point
--> Point3d ( 0, 2, 3 )

The plane does not have to pass through the origin:

offsetPlane =
    Plane3d.offsetBy 1 Plane3d.xy


Point3d.projectOnto offsetPlane point
--> Point3d ( 1, 2, 1 )

Take a point defined in global coordinates, and return it expressed in local coordinates relative to a given reference frame.

localFrame =
    Frame3d.at (Point3d ( 1, 2, 3 ))


Point3d.relativeTo localFrame (Point3d ( 4, 5, 6 ))
--> Point3d ( 3, 3, 3 )


Point3d.relativeTo localFrame (Point3d ( 1, 1, 1 ))
--> Point3d ( 0, -1, -2 )

Take a point defined in local coordinates relative to a given reference frame, and return that point expressed in global coordinates.

localFrame =
    Frame3d.at (Point3d ( 1, 2, 3 ))


Point3d.placeIn localFrame (Point3d ( 3, 3, 3 ))
--> Point3d ( 4, 5, 6 )


Point3d.placeIn localFrame (Point3d ( 0, -1, -2 ))
--> Point3d ( 1, 1, 1 )

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

point =
    Point3d ( 2, 1, 3 )


Point3d.projectInto SketchPlane3d.xy point
--> Point2d ( 2, 1 )


Point3d.projectInto SketchPlane3d.yz point
--> Point2d ( 1, 3 )


Point3d.projectInto SketchPlane3d.zx point
--> Point2d ( 3, 2 )

Construct a bounding box containing both of the given points.

point1 =
    Point3d ( 2, 1, 3 )


point2 =
    Point3d ( -1, 5, -2 )


Point3d.hull point1 point2
--> BoundingBox3d
-->     { minX = -1
-->     , maxX = 2
-->     , minY = 1
-->     , maxY = 5
-->     , minZ = -2
-->     , maxZ = 3
-->     }