Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 Biarc

Source Notebook

Generate a biarc through two points

Contributed by: Jan Mangaldan

ResourceFunction["Biarc"][p1,d1,p2,d2]

represents the biarc through the points p1 and p2, with corresponding tangent directions d1 and d2.

Details

A biarc is a smooth curve formed from two circular arcs, whose continuity is imposed by making the two arcs have the same tangent at the point where they meet.
ResourceFunction["Biarc"] produces a JoinedCurve object which can be used in Graphics (two-dimensional graphics).
ResourceFunction["Biarc"][p1,d1,p2,d2,"Circle"] gives a list of Circle objects that comprise the biarc.

Examples

Basic Examples (1) 

Show a biarc through two points, with given tangent directions:

In[1]:=
Graphics[ ResourceFunction["Biarc"][{0, 0}, AngleVector[\[Pi]/4], {1, 0}, AngleVector[\[Pi]/3]]]
Out[1]=

Scope (2) 

Show biarcs with varying tangent directions:

In[2]:=
Graphics[ Table[{Hue[1 - Rescale[\[Theta], {-\[Pi]/3, \[Pi]/3}]/2], ResourceFunction["Biarc"][{0, 0}, {2, 1}, {1, 0}, AngleVector[\[Theta]]]}, {\[Theta], -\[Pi]/2, \[Pi]/2, \[Pi]/12}], PlotRange -> All]
Out[2]=

Give the circle arcs comprising a biarc:

In[3]:=
ResourceFunction["Biarc"][{0, 0}, {2, 1}, {1, 0}, {-1, 2}, "Circle"]
Out[3]=

Show the circle arcs and their corresponding centers along with the biarc:

In[4]:=
Graphics[{{AbsoluteThickness[5], ColorData[97, 1], ResourceFunction[ "Biarc"][{0, 0}, {2, 1}, {1, 0}, {-1, 2}]}, {AbsoluteThickness[2], AbsolutePointSize[6], Transpose[{{ColorData[97, 4], ColorData[97, 3]}, %, Point[#1] & @@@ %}]}}]
Out[4]=

Applications (1) 

Interactively manipulate a biarc:

In[5]:=
Manipulate[ Graphics[{{ColorData[97, 1], AbsoluteThickness[2], ResourceFunction["Biarc"][pt[[1]], pt[[2]] - pt[[1]], pt[[3]], pt[[4]] - pt[[3]]]}, {Arrowheads[Medium], ColorData[97, 4], Arrow[{pt[[1]], pt[[2]]}], Arrow[{pt[[3]], pt[[4]]}]}}, PlotRange -> 4], {{pt, N[{{-2, 0}, {-2, 1}, {2, 0}, {2, 1}}]}, Locator}, SaveDefinitions -> True]
Out[5]=

Properties and Relations (1) 

Biarc gives a JoinedCurve object:

In[6]:=
ResourceFunction["Biarc"][{0, 0}, {2, 1}, {1, 0}, {-1, 2}] // Head
Out[6]=

Version History

  • 1.0.0 – 23 June 2022

Source Metadata

  • Citation:
    • Bertolazzi, E., Frego, M., "A note on robust biarc computation." Computer-Aided Design and Applications, vol. 16, no. 5, 822–835, 2019. DOI: 10.14733/cadaps.2019.822-835
    • Bolton, K. M., "Biarc curves." Computer-Aided Design, vol. 7, no. 2, 89–92, 1975. DOI: 10.1016/0010-4485(75)90086-X
    • Kurnosenko, A. I., "Biarcs and bilens." Computer Aided Geometric Design, vol. 30, no. 3, 310–330, 2013. DOI: 10.1016/j.cagd.2012.12.002

License Information