CirclePoints

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`CirclePoints`

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CirclePoints[n]

gives the positions of n points equally spaced around the unit circle.

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CirclePoints[r,n]

gives the positions of n points equally spaced around a circle of radius r.

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CirclePoints[{r,θ₁},n]

starts at angle θ₁ with respect to the axis.

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CirclePoints[{x,y},rspec,n]

centers the circle at {x,y}.

Details

For positive integer n, CirclePoints[n] generates a list of vertices for a regular n-sided polygon, oriented so its base is horizontal.
In CirclePoints[n], n does not have to be an exact integer. The angles between successive vectors are always .
Unless explicitly given as a Quantity object, the angle θ₁ is assumed to be in radians, counterclockwise starting from the axis. (Multiply by Degree to convert from degrees.)
If the angle θ₁ is not given, it is assumed to be π/n-π/2, so that for integer n the vectors correspond to a regular polygon with its base horizontal.
All arguments of CirclePoints except n can be symbolic. They can also be Quantity objects.

Examples

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Basic Examples (3)Summary of the most common use cases

Corners of an equilateral triangle:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-nupkob

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Draw a pentagon:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-0wablo

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Draw unit vectors to the corners of a regular heptagon:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-sxpxtv

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Scope (6)Survey of the scope of standard use cases

Corners of a centered square with horizontal and vertical sides:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-9h1zsa

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Corners of a square of size :

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-z2pf8e

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Four unitary vectors aligned with the axes:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-7srgdn

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Displace them to a different point:

In[2]:=2

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https://wolfram.com/xid/0rswwyniu-x2otro

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The four corners of an arbitrary square, at any point and any size or orientation:

In[3]:=3

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https://wolfram.com/xid/0rswwyniu-jyo4kf

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Use a Quantity radius:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-hwdwdw

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Specify the initial angle as a Quantity object:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-ojbhta

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Draw polygons, with their lowest side oriented horizontally by default:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-h658hl

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Draw them all together, with opacity:

In[2]:=2

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https://wolfram.com/xid/0rswwyniu-48azw2

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Properties & Relations (2)Properties of the function, and connections to other functions

CirclePoints returns pairs that can be reinterpreted as real and imaginary parts of the roots of unity:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-5hvjyg

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rswwyniu-em154p

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0rswwyniu-1afeq5

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CirclePoints is equivalent to a collection of AngleVector calls:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-tong5y

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In[2]:=2

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https://wolfram.com/xid/0rswwyniu-k3qzwj

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Interactive Examples (1)Examples with interactive outputs

Construct polygons of different sizes and number of sides:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-kcuc25

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Neat Examples (2)Surprising or curious use cases

Combine Fold and CirclePoints:

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https://wolfram.com/xid/0rswwyniu-s9fmxe

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Combine CirclePoints with BezierCurve:

In[1]:=1

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https://wolfram.com/xid/0rswwyniu-m3ze2e

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In[2]:=2

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https://wolfram.com/xid/0rswwyniu-d63t6

Out[2]=2

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