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# Geometry 3D

## Geometry 3D Angle Metrics

Icon Name Description / Applications Modules
AngleBetweenLines3D

Measures the smaller and the larger angle between two lines in 3D.

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AngleBetweenPlanes

Measures the smaller and the larger angle between two planes in 3D.

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AngleBetweenSegments3D

Measures the angle between two segments in 3D.

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AngleBetweenThreePoints3D

Measures the angle defined by three consecutive points in 3D.

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AngleBetweenVectors3D

Measures the angle between two vectors in 3D.

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## Geometry 3D Basics

Icon Name Description / Applications Modules
CreateBox3D

Creates a box in 3D.

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CreateSegment3D

Produces a segment in 3D of given parameters.

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Loads arrays of 3D points from text file.

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RandomPoint3D

Creates random 3D point inside given box.

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SkipEmptyBox3D

If the input box in 3D has all dimensions different from zero, then it is copied to the output; otherwise Nil is returned.

Secures against domain errors caused by empty boxes.

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TestBox3DEmpty

Tests whether the box in 3D is empty.

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TestBox3DInBox3D

Tests whether a box in 3D is contained in another one.

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TestBox3DIntersectsWith

Tests whether two boxes in 3D have non-empty intersection.

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TestBox3DNotEmpty

Tests whether the box in 3D is not empty.

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## Geometry 3D Constructions

Icon Name Description / Applications Modules
CircleThroughPoints3D

Computes a circle in 3D passing through three noncollinear 3D points.

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LineThroughPoint3D

Computes a line in 3D passing through a point with given direction.

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LineThroughPoints3D

Computes a line in 3D passing through two points.

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PlaneThroughLineAndPoint3D

Computes a plane passing through a line and a point in 3D.

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PlaneThroughPoints3D

Computes a plane passing through three distinct points in 3D.

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ProjectPointsOnLine3D

Projects points onto a line.

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VectorBetweenPoints3D

Computes a vector representing distance from one point in 3D to another.

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## Geometry 3D Distance Metrics

Icon Name Description / Applications Modules
CircleToPlaneDistance3D

Measures the distance between a circle in 3D and a plane.

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LineToLineDistance3D

Measures the distance between two lines in 3D.

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PointSequenceDistances3D

Measures the distances between consecutive points of a point sequence in 3D.

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PointToCircleDistance3D

Measures the distance between a 3D point and a circle in 3D.

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PointToLineDistance3D

Measures the distance between a 3D point and a line in 3D.

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PointToPlaneDistance3D

Measures the distance between a 3D point and a plane.

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PointToPointDistance3D

Measures the distance between two 3D points.

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PointToSegmentDistance3D

Measures the distance between a 3D point and a segment in 3D.

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SegmentToPlaneDistance3D

Measures the distance between a segment in 3D and a plane.

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SegmentToSegmentDistance3D

Measures the distance between two segments in 3D.

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## Geometry 3D Features

Icon Name Description / Applications Modules
Box3DCenter

Returns the center of the input box in 3D.

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Box3DCharacteristicPoint

Returns a characteristic point of a box in 3D.

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Box3DSurfaceArea

Returns the surface area of the input box in 3D.

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Box3DVolume

Returns the volume of the input box in 3D.

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Boxes3DBoundingBox3D_OrNil

Computes the bounding box 3D of given boxes in 3D; returns NIL if the array is empty.

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BoxesBoundingBox3D

Computes the bounding box 3D of given boxes in 3D.

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Circle3DArea

Computes the area of a circle in 3D.

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Circle3DBoundingBox

Computes the smallest box in 3D containing a circle in 3D.

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Circle3DPerimeterLength

Computes the length of a circle in 3D perimeter.

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Plane3DNormalVector

Computes the normal vector of a line.

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Plane3DOrientation

Computes the orientation of a plane as angles in the range from 0 to 180 between the plane and coordinate planes.

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Points3DMedian

Computes the geometric median of the input points.

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PointsBoundingBox3D

Computes the smallest box in 3D containing an array of points in 3D.

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PointsBoundingBox3D_OrNil

Computes the smallest box in 3D containing an array of points in 3D; returns NIL if the array is empty.

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PointsMassCenter3D

Computes the central point of the input points.

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Segment3DBisector

Computes a plane passing through the center of a segment in 3D at a right angle.

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Segment3DBoundingBox

Computes the smallest box in 3D containing a segment in 3D.

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Segment3DCenter

Computes the center point of a segment in 3D.

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Segment3DLength

Computes the length of a segment in 3D.

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Segment3DLine

Computes the line in 3D passing through a segment.

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Segment3DVector

Returns the vector [x2 - x1, y2 - y1, z2 - z1].

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Vector3DAzimuth

Computes the angle between a 3D vector's projection to the XY plane and the X axis measured toward Y axis, as an angle in the range from 0 to 360.

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Vector3DElevation

Computes the angle between a 3D vector's projection to the XY plane and itself measured toward Z axis, as an angle in the range from -90 to 90.

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Vector3DLength

Computes the length of a 3D vector.

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Vectors3DMedian

Computes the geometric median of the input vectors.

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## Geometry 3D Fitting

Icon Name Description / Applications Modules
AnalyzePoint3DGrid

Analyzes the steps in X and Y directions of a point 3D grid.

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FitCircle3DToHole

Fits a circle to a hole in a plane.

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FitCircleToPoints3D

Approximates points in 3D with a circle using selected outliers suppression method.

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FitLineToPoints3D

Approximates points in 3D with a line using the Least Squares method.

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FitLineToPoints3D_LTE

Approximates points in 3D with a line using Least Trimmed Error algorithm.

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FitLineToPoints3D_M

Approximates points in 3D with a line using selected M-estimator for outlier suppression.

Finding a locally optimal line in 3D. Good enough when the number of outliers is small.

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FitLineToPoints3D_RANSAC

Approximates points in 3D with a line using a RANSAC algorithm.

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FitPlaneToPoints

Approximates points with a plane using the Least Squares method.

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FitPlaneToPoints_M

Approximates points with a plane using selected M-estimator for outlier suppression.

Finding a locally optimal plane. Good enough when the number of outliers is small.

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FitSegmentToPoints3D

Approximates points in 3D with a segment using selected outliers suppression method.

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FitSegmentToPoints3D_LTE

Approximates points in 3D with a segment using a RANSAC algorithm.

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FitSegmentToPoints3D_RANSAC

Approximates points in 3D with a segment using a RANSAC algorithm.

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## Geometry 3D Interpolations

Icon Name Description / Applications Modules
LerpPoints3D

Linearly interpolates between two points in 3D.

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LerpSegments3D

Linearly interpolates between two segments in 3D.

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LerpVectors3D

Linearly interpolates between two 3D vectors.

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## Geometry 3D Intersections

Icon Name Description / Applications Modules
BoxIntersection3D

Computes the common part of two boxes in 3D.

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CirclePlaneIntersection3D

Computes common points of a circle in 3D and a plane.

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LineLineIntersection3D

Computes a common point of two lines in 3D.

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LinePlaneIntersection3D

Computes the intersection point of a line in 3D and a plane.

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PlanePlaneIntersection3D

Computes a common line in 3D of two planes.

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SegmentPlaneIntersection3D

Computes the intersection point of a segment in 3D and a plane.

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## Geometry 3D Normalizations

Icon Name Description / Applications Modules
RemoveInvalidPoints3D

Removes invalid 3D points from an array (i.e. points with NaN or INF coordinates).

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## Geometry 3D Relations

Icon Name Description / Applications Modules
TestLine3DThroughCircle3D

Tests whether a line in 3D goes through a circle in 3D.

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TestPoint3DInBox3D

Tests whether a point in 3D lies in a box in 3D.

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## Geometry 3D Spatial Transforms

Icon Name Description / Applications Modules
AlignPoint3DArrayToPlane

Rotates an array of 3D points with a rotation that transforms the input plane to be parallel to XY plane.

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Box2DToBox3D_AxesPlane

Projects a 2D box onto a plane defined by 3D coordinate axes.

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Box3DToBox2D_AxesPlane

Projects a 3D box onto a plane defined by coordinate axes.

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Circle2DToCircle3D_AxesPlane

Projects a 2D circle onto a plane defined by 3D coordinate axes.

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Circle3DToCircle2D_AxesPlane

Projects a 3D circle onto a plane defined by coordinate axes.

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Line2DToLine3D_AxesPlane

Projects a 2D line onto a plane defined by 3D coordinate axes.

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Line3DToLine2D_AxesPlane

Projects a 3D line onto a plane defined by coordinate axes.

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MirrorPoint3D

Mirrors a point according to the input plane.

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MirrorPoint3DArray

Mirrors each point from the array according to the input plane.

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Point2DToPoint3D_AxesPlane

Projects a 2D point onto a plane defined by 3D coordinate axes.

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Point3DToPoint2D_AxesPlane

Projects a 3D point onto a plane defined by coordinate axes.

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Points2DToPoints3D_AxesPlane

Projects an array of 2D points onto a plane defined by 3D coordinate axes.

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Points3DToPoints2D_AxesPlane

Projects an array of 3D points onto a plane defined by coordinate axes.

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ProjectLineOntoPlane3D

Projects a line 3D onto a plane.

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ProjectPoint2DOntoPlane3D

Projects a point on a plane Z=0 onto a given plane, translating it parallel to Z axis.

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ProjectPointOnLine3D

Projects a point 3D onto a line in 3D.

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ProjectPointOntoPlane3D

Projects a point 3D onto a plane.

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ProjectPointOrthogonal

Converts a point through a simple orthogonal projection, parallel to the Z axis.

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ProjectPointPerspective

Converts a point through a perspective projection, parallel to the Z axis.

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RescaleBox3D

Changes the distances of box in 3D to a reference point.

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RescaleCircle3D

Changes radius of a circle in 3D and translates its center in relation to a reference point.

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RescaleLine3D

Changes the input line in 3D distance to a reference point.

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RescalePlane

Changes the input plane distance to a reference point.

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RescalePoint3D

Changes the distance of a point to a reference point.

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RescalePoint3DArray

Changes the distances of array points to a reference point.

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RescaleSegment3D

Lengthens or shortens a segment in 3D relatively.

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RescaleVector3D

Lengthens or shortens a 3D vector relatively preserving its direction.

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ResizeBox3D

Changes the dimensions of a box in 3D.

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ResizeBox3D_Delta

Changes the dimensions of a box in 3D by adding some values.

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ResizeBox3D_Relative

Resizes a box in 3D to relatively defined dimensions.

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ResizeCircle3D

Changes radius of a circle in 3D.

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ResizeCircle3D_Delta

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ResizeSegment3D

Lengthens or shortens a segment in 3D to a new length preserving its orientation and center point.

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ResizeSegment3D_Delta

Changes length of a segment in 3D by adding a value preserving its orientation and center point.

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ResizeVector3D

Lengthens or shortens a 3D vector preserving its direction.

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ResizeVector3D_Delta

Extends length of a 3D vector by adding a value preserving its direction.

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ReverseSegment3D

Swaps the two endpoints of a segment in 3D.

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RotateCircle3D

Rotates a circle in 3D around an axis in 3D.

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RotateCircle3D_AroundSegment

Rotates a circle in 3D around a segment (the axis) in 3D.

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RotateLine3D

Rotates a line in 3D around an axis in 3D.

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RotateLine3D_AroundSegment

Rotates a line in 3D around a segment (the axis) in 3D.

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RotatePlane

Rotates a plane in 3D around an axis in 3D.

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RotatePlane_AroundSegment

Rotates a plane in 3D around a segment (the axis) in 3D.

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RotatePoint3D

Rotates a 3D point around an axis in 3D.

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RotatePoint3DArray

Rotates an array of 3D points around an axis in 3D.

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RotatePoint3DArray_AroundSegment

Rotates an array of 3D points around a segment (the axis).

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RotatePoint3D_AroundSegment

Rotates a 3D point around a segment (the axis).

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RotateSegment3D

Rotates a segment in 3D around an axis in 3D.

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RotateSegment3D_AroundSegment

Rotates a segment in 3D around a segment (the axis) in 3D.

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RotateVector3D

Rotates a 3D vector around another 3D vector (the axis).

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Segment2DToSegment3D_AxesPlane

Projects a 2D segment onto a plane defined by 3D coordinate axes.

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Segment3DToSegment2D_AxesPlane

Projects a 3D segment onto a plane defined by coordinate axes.

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Segments2DToSegments3D_AxesPlane

Projects an array of 2D segments onto a plane defined by 3D coordinate axes.

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Segments3DToSegments2D_AxesPlane

Projects an array of 3D segments onto a plane defined by coordinate axes.

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SplitPointsByPlane

Separates the points being on one side of the input plane from the others.

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TransformCircle3D

Applies a general transformation expressed by a matrix to a 3D circle.

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TransformLine3D

Applies a general transformation expressed by a matrix to a 3D line.

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TransformPlane

Applies a general transformation expressed by a matrix to a plane.

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TransformPoint3D

Applies a general transformation expressed by a matrix to a 3D point.

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TransformPoint3DArray

Applies a general transformation expressed by a matrix to an array of 3D points.

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TransformSegment3D

Applies a general transformation expressed by a matrix to a 3D segment.

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TransformVector3D

Applies a general transformation expressed by a matrix to a 3D vector.

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TranslateBox3D

Shifts a box in 3D by a vector.

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TranslateCircle3D

Shifts a circle in 3D by a vector.

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TranslateLine3D

Shifts a line in 3D by a vector.

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TranslatePlane

Shifts a plane in 3D by a vector.

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TranslatePoint3D

Translates a point by a vector.

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TranslatePoint3DArray

Translates an array of points by a vector.

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TranslatePoint3D_Toward

Translates a point in 3D towards another point in 3D by a specified distance.

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TranslateSegment3D

Shifts a segment in 3D by a vector.

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TrimLine3D

Creates a segment contained in a box in 3D from a line in 3D.

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TrimPoint3DArray

Removes points not contained in a box in 3D from an array.

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TrimSegment3D

Creates a segment contained in a box in 3D from another segment in 3D.

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Vector2DToVector3D_AxesPlane

Projects a 2D vector onto a plane defined by 3D coordinate axes.

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Vector3DToVector2D_AxesPlane

Projects a 3D vector onto a plane defined by coordinate axes.

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