Wolfram Research

Function Repository Resource:

SplineCircle

Source Notebook

Generate a BSplineCurve primitive representing a circle

Contributed by: Carl Woll

ResourceFunction["SplineCircle"][{x,y}, r]

returns a BSplineCurve representation of a 2D circle with center {x,y} and radius r.

ResourceFunction["SplineCircle"][{x, y}, {r1,r2}]

returns a 2D ellipse with semimajor axis r1 and semiminor axis r2, and with the semimajor axis in the x direction.

ResourceFunction["SplineCircle"][{x, y}, {r1,r2}, u]

returns a 2D ellipse with the semimajor axis in the direction u.

ResourceFunction["SplineCircle"][{x, y}, {r1,r2}, u, {θ1, θ2}]

returns a 2D elliptical arc from θ1 to θ2 measured from the semimajor axis.

ResourceFunction["SplineCircle"][{x, y, z}, r]

returns a BSplineCurve representation of a 3D circle with center {x,y,z} and radius r, with the symmetry axis in the z direction.

ResourceFunction["SplineCircle"][{x, y, z},{r1,r2}]

returns a 3D ellipse with semimajor axis r1 and semiminor axis r2, and with the semimajor axis in the x direction.

ResourceFunction["SplineCircle"][{x, y, z},{r1,r2}, a]

returns a 3D ellipse with the symmetry axis in the direction a.

ResourceFunction["SplineCircle"][{x, y, z}, {r1,r2}, {u, v}]

returns a 3D ellipse with the semimajor axis in the direction u, and with the elliptical plane being spanned by u and v.

ResourceFunction["SplineCircle"][{x, y, z},{r1,r2}, p, {θ1, θ2}]

returns a 3D elliptical arc in the plane specified by p, where p can be the symmetry axis or the plane spanned by 3D vectors u and v.

Details and Options

Angles are measured counerclockwise from the semimajor axis direction.
In 2D, ResourceFunction["SplineCircle"][c,r,u] is equivalent to GeometricTransformation[Circle[c,r],RotationTransform[{{1,0},u},c]].
SplineCircle[c] is equivalent to SplineCircle[c, 1].
SplineCircle returns a BSplineCurve object.
SplineCircle can be used as a curve specification in primitives like Tube, Arrow, JoinedCurve and FilledCurve.

Examples

Basic Examples (7) 

A 2D circle:

In[1]:=
Graphics[ResourceFunction["SplineCircle"][{0, 0}, 1]]
Out[1]=

A 2D elliptical arc:

In[2]:=
Graphics[ResourceFunction[
  "SplineCircle"][{0, 0}, {2, 1}, {1, 0}, {Pi/2, 3 Pi/2}]]
Out[2]=

A 2D ellipse with semimajor axis in the direction {4, 1}:

In[3]:=
Graphics[{Arrow[{{0, 0}, {4, 1}}], ResourceFunction["SplineCircle"][{0, 0}, {2, 3}, {4, 1}]}]
Out[3]=

Differently styled ellipses:

In[4]:=
Table[Graphics[{d, ResourceFunction["SplineCircle"][{0, 0}, {1, 2}]}], {d, {Red, Dashed, Thick, Directive[Blue, Thickness[.1], Dashed]}}]
Out[4]=

A 3D circle:

In[5]:=
Graphics3D[
 ResourceFunction["SplineCircle"][{0, 0, 0}, 1],
 Boxed -> False,
 Axes -> True,
 AxesOrigin -> {0, 0, 0},
 AxesEdge -> {.5, .5, .5},
 AxesLabel -> {x, y, z}
 ]
Out[5]=

A 3D ellipse:

In[6]:=
Graphics3D[
 ResourceFunction[
  "SplineCircle"][{0, 0, 0}, {1, 3}, {{0, 1, 0}, {0, 0, 1}}],
 Boxed -> False,
 AxesOrigin -> {0, 0, 0},
 Axes -> True,
 AxesEdge -> {.5, .5, .5},
 AxesLabel -> {x, y, z}
 ]
Out[6]=

A 3D elliptical arc:

In[7]:=
Graphics3D[
 ResourceFunction[
  "SplineCircle"][{1, 1, 1}, {1, 3}, {{1, 0, 0}, {0, 0, 1}}, {0, Pi/2}],
 Axes -> True,
 AxesLabel -> {x, y, z}
 ]
Out[7]=

Scope (4) 

Use SplineCircle as a FilledCurve specification:

In[8]:=
Graphics[FilledCurve[{
   {ResourceFunction["SplineCircle"][{0, 0}, 2]},
   {ResourceFunction["SplineCircle"][{1, 1}, 1]}
   }]
 ]
Out[8]=

Use SplineCircle as a Tube curve specification:

In[9]:=
Graphics3D[Tube[ResourceFunction["SplineCircle"][{0, 0, 0}, {1, 2}]]]
Out[9]=

Use SplineCircle as an Arrow curve specification:

In[10]:=
SeedRandom[6]
points = RandomReal[1, {10, 3}];
axes = RandomReal[1, {10, 2, 3}];
angles = RandomReal[2 Pi, {10, 2}];
Graphics3D[
 MapThread[
  Arrow @* ResourceFunction["SplineCircle"], {points, ConstantArray[1, 10], axes, angles}]]
Out[14]=

Rendering of a fairly large number of circle tubes:

In[15]:=
Graphics3D[
 MapThread[
  Tube@*ResourceFunction["SplineCircle"], {RandomReal[3, {300, 3}], RandomReal[{2, 3}, 300], RandomReal[1, {300, 2, 3}]}]]
Out[15]=

Properties and Relations (2) 

A 2D SplineCircle with a non-default semimajor axis is equivalent to a geometric transformation of a Circle object:

In[16]:=
Graphics[{Arrow[{{0, 0}, {2, 1}}], ResourceFunction["SplineCircle"][{0, 0}, {1, 2}, {2, 1}]}, ImageSize -> 200]
Out[16]=

The Circle equivalent:

In[17]:=
Graphics[{Arrow[{{0, 0}, {2, 1}}], GeometricTransformation[Circle[{0, 0}, {1, 2}], RotationTransform[{{1, 0}, {2, 1}}, {0, 0}]]}, ImageSize -> 200]
Out[17]=

Possible Issues (3) 

When only a symmetry axis {x,y,z} is specified, the largest component of the symmetry axis is computed, and the third coordinate starting with the largest component in a RHS is determined. Then the semimajor axis is assumed to be perpendicular to both the third component and the symmetry axis. For example, suppose the symmetry axis is:

In[18]:=
axis = {1, 2, 3};

The largest component is in the z direction, and so the third coordinate would be the y coordinate. Then the semimajor axis is assumed to be in the direction:

In[19]:=
Normalize @ Cross[{0, 1, 0}, axis]
Out[19]=

Check:

In[20]:=
ResourceFunction["SplineCircle"][{0, 0, 0}, {1, 2}, {1, 2, 3}]
Out[20]=

Publisher

Carl Woll

Version History

  • 2.0.0 – 16 January 2020

Related Resources

Author Notes

I will add support for Scaled for both 2D and 3D graphics, and Offset and ImageScaled for 2D graphics.

License Information