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 righthand 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:

Plot the points:

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

Scope  (3)

PositivelyOrientedPoints works with two-dimensional points:

Three-dimensional points:

-dimensional points:

PositivelyOrientedPoints works with numerical coordinates:

Symbolic coordinates:

PositivelyOrientedPoints over a set of coordinates:

List of points:

Multi-points:

Applications  (5)

Basic Applications  (2)

Graph positively oriented points:

Show the right-hand rule:

Geometry  (3)

Sort points in counterclockwise order:

Sorting function:

PositivelyOrientedPoints over lines in 2D:

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

Show the robustness of PositivelyOrientedPoints:

Properties & Relations  (4)

PositivelyOrientedPoints returns False for collinear points:

NegativelyOrientedPoints returns False if positively oriented:

Use RegionMember to test whether points are positively oriented:

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

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.

BibTeX

@misc{reference.wolfram_2020_positivelyorientedpoints, author="Wolfram Research", title="{PositivelyOrientedPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html}", note=[Accessed: 16-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_positivelyorientedpoints, organization={Wolfram Research}, title={PositivelyOrientedPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html}, note=[Accessed: 16-January-2021 ]}

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