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PositivelyOrientedPoints[{p1,p2,p3,…,pd+1}]

tests whether the sequence of points p1,p2,p3,…,pd+1 is positively oriented.

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Basic Examples  
Scope  
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Basic Applications  
Geometry  
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PositivelyOrientedPoints

PositivelyOrientedPoints[{p1,p2,p3,…,pd+1}]

tests whether the sequence of points p1,p2,p3,…,pd+1 is positively oriented.

Details

  • PositivelyOrientedPoints is also known as counterclockwise or anticlockwise in 2D and right‐hand rule in 3D.
  • Typically used to determine the orientation of a rotational motion with respect to a set of points.
  • A counterclockwise motion is one that proceeds in the opposite direction to the way in which the hands of a clock move around.
  • In two dimensions, the sequence of points p1, p2 and p3 is positively oriented if the orientation of the points is counterclockwise.
  • PositivelyOrientedPoints[{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 positively oriented points p1, p2 and p3, the determinant of the matrix {p2-p1,p3-p1} is positive.
  • In three dimensions, PositivelyOrientedPoints[{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 (p2-p1)(p3-p1).
  • For positively oriented points p1, p2, p3 and p4, the dot product of p4-p1 and (p2-p1)(p3-p1) is positive.
  • In d dimensions, d+1 points p1,p2,…,pd+1 are positively oriented if the determinant of the matrix {p2-p1,…,pd+1-p1} is positive.

Examples

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Basic Examples  (2)

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

Wolfram Language code: pts = {{0, 0}, {0.5, -1}, {1, 1}};
Wolfram Language code: PositivelyOrientedPoints[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 above a plane:

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

Scope  (3)

PositivelyOrientedPoints works with two-dimensional points:

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

Three-dimensional points:

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

-dimensional points:

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

PositivelyOrientedPoints works with numerical coordinates:

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

Symbolic coordinates:

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

PositivelyOrientedPoints over a set of coordinates:

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

List of points:

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

Multi-points:

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

Applications  (4)

Basic Applications  (2)

Graph positively oriented points:

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

Show the right-hand rule:

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

Geometry  (2)

PositivelyOrientedPoints over lines in 2D:

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

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

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

Show the robustness of PositivelyOrientedPoints:

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

Properties & Relations  (4)

PositivelyOrientedPoints returns False for collinear points:

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

Wolfram Language code: RegionMember[HalfPlane[{{0, 0}, {1, 0}}, {0, 1}], {1 / 3, 1 / 3}]
Wolfram Language code: PositivelyOrientedPoints[{{0, 0}, {1, 0}, {0, 1}}]

3D points that are not positively or negatively oriented are coplanar:

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), PositivelyOrientedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_positivelyorientedpoints, author="Wolfram Research", title="{PositivelyOrientedPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_positivelyorientedpoints, organization={Wolfram Research}, title={PositivelyOrientedPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html}, note=[Accessed: 13-June-2026]}