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PositivelyOrientedPoints
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PositivelyOrientedPoints

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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

open allclose all

Basic Examples  (2)Summary of the most common use cases

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

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-7f2o6y
In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-7kqtw5
Out[2]=2

Plot the points:

In[3]:=3
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https://wolfram.com/xid/0b7e7i326acexymu-ofnazw
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-5nyoxu
Out[1]=1

Scope  (3)Survey of the scope of standard use cases

PositivelyOrientedPoints works with two-dimensional points:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-o520tf
Out[1]=1

Three-dimensional points:

In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-tzeutr
Out[2]=2

-dimensional points:

In[3]:=3
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https://wolfram.com/xid/0b7e7i326acexymu-3kckar
Out[3]=3

PositivelyOrientedPoints works with numerical coordinates:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-vaa7iv
Out[1]=1

Symbolic coordinates:

In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-loksgh
Out[2]=2

PositivelyOrientedPoints over a set of coordinates:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-iosq0l
Out[1]=1

List of points:

In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-pbm87t
Out[2]=2

Multi-points:

In[3]:=3
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https://wolfram.com/xid/0b7e7i326acexymu-odxneo
Out[3]=3

Applications  (4)Sample problems that can be solved with this function

Basic Applications  (2)

Graph positively oriented points:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-0e61c
In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-x5hix
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b7e7i326acexymu-p801bp
Out[3]=3

Show the right-hand rule:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-bb3lo1
In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-c5qez8
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b7e7i326acexymu-qklxhp
Out[3]=3

Geometry  (2)

PositivelyOrientedPoints over lines in 2D:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-9jvrr0
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-566q3k
Out[2]=2

Show the robustness of PositivelyOrientedPoints:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-s8g3om
In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-o3ft7h
Out[2]=2

Properties & Relations  (4)Properties of the function, and connections to other functions

PositivelyOrientedPoints returns False for collinear points:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-2afa08
In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-4449mp
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b7e7i326acexymu-36y6ii
Out[3]=3

NegativelyOrientedPoints returns False if positively oriented:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-8gvtvm
In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-isc17j
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b7e7i326acexymu-wxx8sf
Out[3]=3

Use RegionMember to test whether points are positively oriented:

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-j0931q
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-m9ot9p
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0b7e7i326acexymu-xr9x2n
In[2]:=2
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https://wolfram.com/xid/0b7e7i326acexymu-cq6slx
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b7e7i326acexymu-2gfk11
Out[3]=3
Wolfram Research (2020), PositivelyOrientedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html.
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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.

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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.

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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

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

BibTeX

@misc{reference.wolfram_2025_positivelyorientedpoints, author="Wolfram Research", title="{PositivelyOrientedPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html}", note=[Accessed: 30-May-2025 ]}

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@misc{reference.wolfram_2025_positivelyorientedpoints, author="Wolfram Research", title="{PositivelyOrientedPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html}", note=[Accessed: 30-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_positivelyorientedpoints, organization={Wolfram Research}, title={PositivelyOrientedPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html}, note=[Accessed: 30-May-2025 ]}

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@online{reference.wolfram_2025_positivelyorientedpoints, organization={Wolfram Research}, title={PositivelyOrientedPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/PositivelyOrientedPoints.html}, note=[Accessed: 30-May-2025 ]}