This is the original Wolfram Mathematica implementation of the Siggraph 2017 paper

https://zhipeiyan.github.io/projects/kCurves/kCurves.pdf

Put the .wl package in the same path of your notebook file and load the package

In[1]:= Get[NotebookDirectory[] <> "kCurves.wl"]

Compute an open curve interpolating the input points

In[2]:= kCurveOpen[{{-1,0},{0,1},{1,0},{2,1}}]
Out[2]:= {{{-1, 0}, {-0.08187, 1.71183}, {0.5, 0.5}}, {{0.5, 0.5}, {1.08187, -0.711831}, {2, 1}}}

Draw the curve

In[3]:= Graphics[BezierCurve[%]]
Out[3]:= 

Download the Eigen library.

Include the header file kCurves.h

The declaration of the two computing functions are

std::vector<std::vector<Eigen::Vector2d>> kCurveClosed(std::vector<Eigen::Vector2d>);
std::vector<std::vector<Eigen::Vector2d>> kCurveOpen(std::vector<Eigen::Vector2d>);

The input format std::vector<Eigen::Vector2d> is a list of your control points in 2D.

The output format is a list of triple points. Each triple represents the three control points of a quadratic Bezier curve segment according to your input points. The output vector has the same length as your input points for closed curves, and two shorter than your input points for open curves.

Manipulate the curves in real time.

In[4]:= Manipulate[Graphics[BezierCurve[kCurveClosed[pts]], PlotRange -> {{-5, 5}, {-5, 5}}], {{pts, {{-1, 0}, {0, 1}, {1, 0}, {2, 1}}}, Locator}]
Out[4]:= 

Check https://github.com/zhipeiyan/kappa-Curves/releases

No package files needed.

Plain Mathematica code format may be not reading friendly. Here's the screen shot of the code in the .nb format for some one doesnot have Wolfram Mathematica installed.

https://raw.githubusercontent.com/zhipeiyan/kappa-Curves/main/reading.svg