LaTeX TikZ Bézier curves of arbitrary degree

tikz_bezier.py

import string
from math import comb

def bernstein(n, i, var="t"):
    """Return the Bernstein polynomial term as a string."""
    return f"{comb(n, i)}*(1-{var})^({n-i})*{var}^({i})"

def generate_bezier_latex(
        nodes,
        samples=100,
        domain=(0, 1),
        width="10cm",
        height="10cm",
        anchors={},
        use_commands=True,
        var="t"
        ):
    r"""
    Generates LaTeX TikZ code for an arbitrary Bézier curve given an array of control points.


    TikZ natively supports Bézier curves until 3rd degree (start, end, 2 control points):
    ```
    \draw(p0)..controls (p1) and (p2)..(p3);
    ```
    Any higher order Bézier curve needs to be computed and drawn manually.
    $$B(t) = \sum_{i=0}^{n} B_i^n(t) P_i$$
    with
    $$B_i^n(t) = \binom{n}{i} (1 - t)^{n-i} t^i$$

    Parameters
    ----------
    nodes : np.ndarray or list
        - if `np.ndarray` array of shape (2, N), as used as input for python package `bezier`
        - if `list` list of coordinate tuples, i.e. `[(x0, y0), ..., (xN, yN)]`
    samples : int
        At how many points to evaluate the Bézier curve
    domain : tuple
        2-tuple indicating start and end value of curve parameter `t`
    width : str
        Width of plot
    height : str
        Height of plot
    anchors : dict, optional
        Dictionary mapping index of control point `i` to the anchor position for the point label
    use_commands : bool, optional
        - if `True`
            generate a macro for each control point coordinate and insert the macro at coordinate locations. Useful to manually change the coordinates without rerunning this function
        - if `False`
            insert the node coordinate values directly at coordinate locations
    var : str
        Variable name for Bézier parameter

    Returns
    -------
    str
        LaTeX TikZ code drawing an arbitrary Bézier curve
    """
    if isinstance(nodes, np.ndarray):
        coords = [(x_, y_) for x_, y_ in nodes.T]
    else:
        coords = nodes
    n = len(coords) - 1  # degree of the Bézier curve

    indent = "\t"

    def get_id(idx, alphabet=string.ascii_uppercase):
        """Get a unique coordinate ID by cycling `alphabet`."""
        result = ""
        idx += 1
        while idx > 0:
            idx, remainder = divmod(idx - 1, len(alphabet))
            result += alphabet[remainder]
        return result[::-1]

    def format_coordinate(i, x=None, y=None):
        """
        Format a coordinate either by inserting its value `x` or `y` or
        by inserting the coordinate's macro.
        """
        assert (x is None) ^ (y is None)
        if use_commands:
            coord = "X" if x is not None else "Y"
            return f"\\bezier{coord}{get_id(i)}"
        else:
            return x or y


    # define control point values and coordinates as macros
    if use_commands:
        s_def = f"{indent}% Define control point coordinates as commands\n"
        for i, (x, y) in enumerate(coords):
            namex = format_coordinate(i, x=x)
            namey = format_coordinate(i, y=y)
            s_def += f"{indent}\\def{namex}{{{x}}}\n"
            s_def += f"{indent}\\def{namey}{{{y}}}\n"
    else:
        s_def = ""

    # Build the expressions for the x(t) and y(t) components:
    terms_x = []
    terms_y = []
    for i, (x, y) in enumerate(coords):
        terms_x.append(f"({bernstein(n, i, var)})*({format_coordinate(i, x=x)})")
        terms_y.append(f"({bernstein(n, i, var)})*({format_coordinate(i, y=y)})")
    bezier_x = " + ".join(terms_x)
    bezier_y = " + ".join(terms_y)

    # Generate code for plotting the control points:
    control_points_code = "\n".join(
        f"{indent}{indent}\\addplot[only marks, mark=*, red] coordinates {{({format_coordinate(i, x=x)},{format_coordinate(i, y=y)})}};"
        for i, (x, y) in enumerate(coords)
    )

    # Generate code for labeling the control points:
    control_labels_code = "\n".join(
        f"{indent}{indent}\\node[anchor={anchors.get(i, 'north east')}] at ({format_coordinate(i, x=x)},{format_coordinate(i, y=y)}) {{\\texttt{{P{i}}} ({format_coordinate(i, x=x)},{format_coordinate(i, y=y)})}};"
        for i, (x, y) in enumerate(coords)
    )

    # Build the full LaTeX code using pgfplots
    latex_code = f"""\\documentclass{{article}}
\\usepackage{{pgfplots}}
\\pgfplotsset{{compat=1.17}}
\\begin{{document}}
\\begin{{tikzpicture}}
{s_def}
    \\begin{{axis}}[
        axis lines = middle,
        xlabel = {{\\(x\\)}},
        ylabel = {{\\(y\\)}},
        width={width},
        height={height},
        samples={samples},
        domain={domain[0]}:{domain[1]},
        variable={var}
    ]
        % Plot the Bézier curve
        \\addplot[
            thick, blue
        ] (
            {{ {bezier_x} }},
            {{ {bezier_y} }}
        );

        % Plot control points
{control_points_code}

        % Label control points
{control_labels_code}

    \\end{{axis}}
\\end{{tikzpicture}}
\\end{{document}}
"""
    return latex_code

# Example usage:
control_points = np.round(np.random.rand(2, 10), 2)
latex_code = generate_bezier_latex(control_points)

Output:

summary

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\begin{document}
\begin{tikzpicture}
	% Define control point coordinates as commands
	\def\bezierXA{0.94}
	\def\bezierYA{0.96}
	\def\bezierXB{0.27}
	\def\bezierYB{0.89}
	\def\bezierXC{0.75}
	\def\bezierYC{0.53}
	\def\bezierXD{0.73}
	\def\bezierYD{0.03}
	\def\bezierXE{0.46}
	\def\bezierYE{0.09}
	\def\bezierXF{0.82}
	\def\bezierYF{0.04}
	\def\bezierXG{0.38}
	\def\bezierYG{0.98}
	\def\bezierXH{0.53}
	\def\bezierYH{0.34}
	\def\bezierXI{0.76}
	\def\bezierYI{0.88}
	\def\bezierXJ{0.3}
	\def\bezierYJ{0.26}

    \begin{axis}[
        axis lines = middle,
        xlabel = {\(x\)},
        ylabel = {\(y\)},
        xmin=0, xmax=1.,
        ymin=0, ymax=1.,
        width=10cm,
        height=10cm,
        samples=100,
        domain=0:1,
        variable=t
    ]
        % Plot the Bézier curve
        \addplot[
            thick, blue
        ] (
            { (1*(1-t)^(9)*t^(0))*(\bezierXA) + (9*(1-t)^(8)*t^(1))*(\bezierXB) + (36*(1-t)^(7)*t^(2))*(\bezierXC) + (84*(1-t)^(6)*t^(3))*(\bezierXD) + (126*(1-t)^(5)*t^(4))*(\bezierXE) + (126*(1-t)^(4)*t^(5))*(\bezierXF) + (84*(1-t)^(3)*t^(6))*(\bezierXG) + (36*(1-t)^(2)*t^(7))*(\bezierXH) + (9*(1-t)^(1)*t^(8))*(\bezierXI) + (1*(1-t)^(0)*t^(9))*(\bezierXJ) },
            { (1*(1-t)^(9)*t^(0))*(\bezierYA) + (9*(1-t)^(8)*t^(1))*(\bezierYB) + (36*(1-t)^(7)*t^(2))*(\bezierYC) + (84*(1-t)^(6)*t^(3))*(\bezierYD) + (126*(1-t)^(5)*t^(4))*(\bezierYE) + (126*(1-t)^(4)*t^(5))*(\bezierYF) + (84*(1-t)^(3)*t^(6))*(\bezierYG) + (36*(1-t)^(2)*t^(7))*(\bezierYH) + (9*(1-t)^(1)*t^(8))*(\bezierYI) + (1*(1-t)^(0)*t^(9))*(\bezierYJ) }
        );

        % Plot control points
		\addplot[only marks, mark=*, red] coordinates {(\bezierXA,\bezierYA)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXB,\bezierYB)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXC,\bezierYC)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXD,\bezierYD)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXE,\bezierYE)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXF,\bezierYF)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXG,\bezierYG)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXH,\bezierYH)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXI,\bezierYI)};
		\addplot[only marks, mark=*, red] coordinates {(\bezierXJ,\bezierYJ)};

        % Label control points
		\node[anchor=north east] at (\bezierXA,\bezierYA) {\texttt{P0} (\bezierXA,\bezierYA)};
		\node[anchor=north east] at (\bezierXB,\bezierYB) {\texttt{P1} (\bezierXB,\bezierYB)};
		\node[anchor=north east] at (\bezierXC,\bezierYC) {\texttt{P2} (\bezierXC,\bezierYC)};
		\node[anchor=north east] at (\bezierXD,\bezierYD) {\texttt{P3} (\bezierXD,\bezierYD)};
		\node[anchor=north east] at (\bezierXE,\bezierYE) {\texttt{P4} (\bezierXE,\bezierYE)};
		\node[anchor=north east] at (\bezierXF,\bezierYF) {\texttt{P5} (\bezierXF,\bezierYF)};
		\node[anchor=north east] at (\bezierXG,\bezierYG) {\texttt{P6} (\bezierXG,\bezierYG)};
		\node[anchor=north east] at (\bezierXH,\bezierYH) {\texttt{P7} (\bezierXH,\bezierYH)};
		\node[anchor=north east] at (\bezierXI,\bezierYI) {\texttt{P8} (\bezierXI,\bezierYI)};
		\node[anchor=north east] at (\bezierXJ,\bezierYJ) {\texttt{P9} (\bezierXJ,\bezierYJ)};

    \end{axis}
\end{tikzpicture}
\end{document}

For random points, generates: