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NegativelyOrientedPoints[{p1,p2,p3,…,pn}]

tests whether the sequence of points p1,p2,p3,…,pn is negatively oriented.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Applications  
Basic Applications  
Geometry  
Properties & Relations  
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NegativelyOrientedPoints

NegativelyOrientedPoints[{p1,p2,p3,…,pn}]

tests whether the sequence of points p1,p2,p3,…,pn is negatively oriented.

Details

  • NegativelyOrientedPoints is also known as clockwise in 2D and left‐hand rule in 3D.
  • Typically used to determine the orientation of a rotational motion with respect to a set of points.
  • In two dimensions, NegativelyOrientedPoints[{p1,p2,p3}] gives True if the point p3 is in the half-plane bounded by the line through p1 and p2 and extended in the direction of {1,0}.
  • For negatively oriented points p1, p2 and p3, the determinant of the matrix {p2-p1, p3-p1} is negative.
  • In three dimensions, NegativelyOrientedPoints[{p1,p2,p3,p4}] gives True if the point p4 is in the half-space bounded by the plane through the point p1 with normal direction (p3-p1)(p2-p1).
  • For negatively oriented points p1, p2, p3 and p4, the dot product of p4-p1 and (p3-p1)(p2-p1) is negative.
  • In d dimensions, d+1 points p1,p2,…,pd+1 are negatively oriented if the determinant of the matrix {p2-p1,…,pd+1-p1} is negative.

Examples

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Basic Examples  (2)

The points {0,0}, {1,1}, {.5,-1} are negatively oriented:

Wolfram Language code: pts = {{0, 0}, {1, 1}, {0.5, -1}};
Wolfram Language code: NegativelyOrientedPoints[pts]

Plot the points:

Wolfram Language code: Graphics[{Arrowheads[Small], Arrow /@ Partition[pts, 2, 1, 1], Red, Point[pts]}]

Find the condition for which a point is below a plane:

Wolfram Language code: NegativelyOrientedPoints[{{0, 0, 0}, {1, 1, -2}, {-1, 2, -1}, {x, y, z}}]

Scope  (3)

NegativelyOrientedPoints works with two-dimensional points:

Wolfram Language code: NegativelyOrientedPoints[{{1, 2}, {3, 4}, {5, 6}}]

Three-dimensional points:

Wolfram Language code: NegativelyOrientedPoints[{{1, 2, 1}, {3, 4, 1}, {5, 6, 1}, {7, 8, 1}}]

NegativelyOrientedPoints works with numerical coordinates:

Wolfram Language code: NegativelyOrientedPoints[{{1, 2}, {3, 4}, {5, 6}}]

Symbolic coordinates:

Wolfram Language code: NegativelyOrientedPoints[{{0, 0}, {a, b}, {c, d}}]

NegativelyOrientedPoints over a set of coordinates:

Wolfram Language code: NegativelyOrientedPoints[{{0, 0}, {a, b}, {c, d}}]

List of points:

Wolfram Language code: NegativelyOrientedPoints[{Point[{0, 0}], Point[{a, b}], Point[{c, d}]}]

Multi-points:

Wolfram Language code: NegativelyOrientedPoints[Point[{{0, 0}, {a, b}, {c, d}}]]

Generalizations & Extensions  (1)

Give assumptions to NegativelyOrientedPoints:

Wolfram Language code: pts = {{a, 2}, {b, 4}, {5, 6}};
Wolfram Language code: NegativelyOrientedPoints[pts]
Wolfram Language code: Assuming[a > 2b, NegativelyOrientedPoints[pts]]

Applications  (5)

Basic Applications  (2)

Graph negatively oriented points:

Wolfram Language code: pts = {{1, 1}, {1, 0}, {0, 0}};
Wolfram Language code: NegativelyOrientedPoints[pts]
Wolfram Language code: Graphics[{Arrowheads[.1], Arrow /@ Partition[pts, 2, 1]}]

Show the left-hand rule:

Wolfram Language code: pts = {{0, 0, 1}, {1, 0, 1}, {0, 1, 1}, {1, 0, 0}};
Wolfram Language code: NegativelyOrientedPoints[pts]
Wolfram Language code: Graphics3D[{Arrowheads[.1], Arrow /@ Partition[pts, 3, 1]}]

Geometry  (3)

Faces of a polyhedron are positively oriented:

Wolfram Language code: poly = OuterPolyhedron[Polyhedron[{{-1/2, -1/2, -1/2}, {-1/2, -1/2, 1/2}, {-1/2, 1/2, -1/2}, {-1/2, 1/2, 1/2}, {1/2, -1/2, -1/2}, {1/2, -1/2, 1/2}, {1/2, 1/2, -1/2}, {1/2, 1/2, 1/2}}, {{8, 4, 2, 6}, {8, 6, 5, 7}, {8, 7, 3, 4}, {4, 3, 1, 2}, {1, 3, 7, 5}, {2, 1, 5, 6}}]];
Wolfram Language code: faces = poly[[2]]
Wolfram Language code: NegativelyOrientedPoints[PolyhedronCoordinates[poly][[#]]]& /@ faces

NegativelyOrientedPoints over lines in 2D:

Wolfram Language code: NegativelyOrientedPoints[Line[{{0, 0}, {1, 0}, {1, 1}}]]

It is equivalent to the orientation of the consecutive vertices of the line:

Wolfram Language code: NegativelyOrientedPoints[{{0, 0}, {1, 0}, {1, 1}}]

Show the robustness of NegativelyOrientedPoints:

Wolfram Language code: pts = {{7.3000000000000194, 7.3000000000000167}, {24.000000000000068, 24.000000000000071}, {24.00000000000005, 24.000000000000053}};
Wolfram Language code: NegativelyOrientedPoints[pts]

Properties & Relations  (4)

NegativelyOrientedPoints returns False for collinear points:

Wolfram Language code: pts = {{0, 0}, {1, 2}, {2, 4}};
Wolfram Language code: CollinearPoints[pts]
Wolfram Language code: NegativelyOrientedPoints[pts]

NegativelyOrientedPoints returns False if positively oriented:

Wolfram Language code: pts = {{0, 0}, {1, 0}, {1, 1}};
Wolfram Language code: PositivelyOrientedPoints[pts]
Wolfram Language code: NegativelyOrientedPoints[pts]

Use RegionMember to test whether points are negatively oriented:

Wolfram Language code: RegionMember[HalfPlane[{{0, 0}, {1, 1}}, {0.5, -1}], {1 / 3, 1 / 3}]
Wolfram Language code: NegativelyOrientedPoints[{{0, 0}, {1, 1}, {0.5, -1}}]

3D points are coplanar if they are neither positively nor negatively oriented:

Wolfram Language code: pts = {{0, 0, 0}, {1, 1, -2}, {-1, 2, -1}, {3, -4, 1}};
Wolfram Language code: {PositivelyOrientedPoints[pts], NegativelyOrientedPoints[pts]}
Wolfram Language code: CoplanarPoints[pts]
Wolfram Research (2020), NegativelyOrientedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html.

Text

Wolfram Research (2020), NegativelyOrientedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html.

CMS

Wolfram Language. 2020. "NegativelyOrientedPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html.

APA

Wolfram Language. (2020). NegativelyOrientedPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html

BibTeX

@misc{reference.wolfram_2026_negativelyorientedpoints, author="Wolfram Research", title="{NegativelyOrientedPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_negativelyorientedpoints, organization={Wolfram Research}, title={NegativelyOrientedPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html}, note=[Accessed: 12-June-2026]}