Mastering Geometric Arc Construction in Matplotlib: A Comprehensive Guide

Advanced Geometric Construction: Drawing Arcs Between Two Points in Matplotlib

In the realm of scientific visualization and software engineering, the ability to represent relationships between data points is fundamental. While straight lines are the standard for most connections, there are numerous scenarios—ranging from geographic flight paths to hierarchical network diagrams—where a curved connection, or an arc, is aesthetically and functionally superior. However, Matplotlib, the cornerstone of Python’s plotting ecosystem, does not offer a single, direct "draw_arc_between(p1, p2)" function. Instead, it provides low-level primitives that require a solid grasp of coordinate geometry and trigonometry to manipulate correctly.

This technical guide explores the methodologies for constructing arcs between two arbitrary points. We will dissect the mathematical foundations, compare different implementation strategies such as the Arc patch and Bezier paths, and provide robust code samples for your production environments.

1. The Mathematical Foundation of Arc Construction

Before writing a single line of Python code, we must understand the geometry of a circular arc. A circular arc is defined as a segment of the circumference of a circle. To define this segment between two points, ##P_1(x_1, y_1)## and ##P_2(x_2, y_2)##, we need more information than just the endpoints. In Euclidean geometry, an infinite number of circles can pass through two specific points. To uniquely identify one arc, we usually require either a radius ##R## or a center point ##C(x_c, y_c)##.

The distance ##d## between the two points is calculated using the standard distance formula:

###d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}###

For a circle of radius ##R## to exist such that it passes through both points, the condition ##R \geq \frac{d}{2}## must be satisfied. If ##R = \frac{d}{2}##, the arc is a perfect semicircle, and the center ##C## is exactly at the midpoint ##M## of the segment ##P_1P_2##:

###M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)###

When ##R > \frac{d}{2}##, the center ##C## lies on the perpendicular bisector of the line segment joining the two points. The distance ##h## from the midpoint ##M## to the center ##C## can be found using the Pythagorean theorem applied to the triangle formed by ##C##, ##M##, and ##P_1##:

###h = \sqrt{R^2 - \left(\frac{d}{2}\right)^2}###

Finding the exact coordinates of ##C## involves calculating the unit vector perpendicular to the vector ##\vec{P_1P_2}##. If ##\vec{V} = (x_2 - x_1, y_2 - y_1)##, then a perpendicular vector ##\vec{V_{\perp}}## is ##(-(y_2 - y_1), x_2 - x_1)##. By normalizing this vector and scaling it by ##h##, we can locate the center of the arc.

2. Implementing the Matplotlib Arc Patch

Matplotlib provides the matplotlib.patches.Arc class. Unlike many drawing tools that define arcs by start and end coordinates, this class uses an ellipse-based definition. To use it, you must provide:

• xy: The center of the ellipse ##(x_c, y_c)##.
• width: The diameter of the horizontal axis.
• height: The diameter of the vertical axis.
• angle: The rotation of the ellipse in degrees.
• theta1: The starting angle of the arc in degrees.
• theta2: The ending angle of the arc in degrees.

The challenge lies in translating our two points ##P_1## and ##P_2## into these parameters. Specifically, the angles ##\theta_1## and ##\theta_2## must be calculated relative to the center ##C## using the atan2 function, which handles the signs of the coordinates to return the correct quadrant.

import matplotlib.pyplot as plt
import matplotlib.patches as patches
import numpy as np

def draw_circular_arc(p1, p2, radius, ax):
    # Calculate midpoint
    mid = (np.array(p1) + np.array(p2)) / 2
    
    # Distance between points
    d = np.linalg.norm(np.array(p2) - np.array(p1))
    
    if radius < d/2:
        raise ValueError("Radius is too small to connect the points.")
        
    # Distance from midpoint to center
    h = np.sqrt(radius**2 - (d/2)**2)
    
    # Vector p1 -> p2
    v = np.array(p2) - np.array(p1)
    # Unit perpendicular vector
    v_perp = np.array([-v[1], v[0]]) / d
    
    # Center of the circle (one of two possible centers)
    center = mid + v_perp * h
    
    # Calculate angles in degrees
    start_angle = np.degrees(np.arctan2(p1[1] - center[1], p1[0] - center[0]))
    end_angle = np.degrees(np.arctan2(p2[1] - center[1], p2[0] - center[0]))
    
    # Create the Arc patch
    # Note: width and height are diameters
    arc = patches.Arc(center, radius*2, radius*2, angle=0, 
                      theta1=start_angle, theta2=end_angle, 
                      color='blue', linewidth=2)
    ax.add_patch(arc)
    return center

One critical detail with patches.Arc is that it is often optimized for drawing and might not correctly display if the axes are not scaled equally. To ensure the arc looks circular, always use ax.set_aspect('equal').

3. Alternative Approach: Quadratic Bezier Curves

In many Software Engineering applications, a mathematically "perfect" circular arc is less important than a visually pleasing curve. In such cases, a Quadratic Bezier Curve is significantly easier to compute and more flexible. A Quadratic Bezier is defined by three points: the start ##P_1##, the end ##P_2##, and a control point ##P_{ctrl}##.

The curve ##B(t)## is defined by the formula:

###B(t) = (1-t)^2 P_1 + 2(1-t)t P_{ctrl} + t^2 P_2, \quad t \in [0, 1]###

To create an arc-like curve, we place the control point ##P_{ctrl}## above the midpoint of ##P_1P_2##. The "height" of this control point determines the curvature. This method avoids trigonometric calculations and is highly performant for rendering large sets of edges in graph visualizations.

from matplotlib.path import Path

def draw_bezier_arc(p1, p2, bend=0.2, ax=None):
    p1 = np.array(p1)
    p2 = np.array(p2)
    
    # Calculate midpoint
    mid = (p1 + p2) / 2
    
    # Vector p1 to p2
    v = p2 - p1
    # Perpendicular vector
    v_perp = np.array([-v[1], v[0]])
    
    # Define control point by shifting the midpoint perpendicular to the line
    ctrl = mid + v_perp * bend
    
    # Define the Path
    path_data = [
        (Path.MOVETO, p1),
        (Path.CURVE3, ctrl),
        (Path.CURVE3, p2),
    ]
    codes, verts = zip(*path_data)
    path = Path(verts, codes)
    
    patch = patches.PathPatch(path, facecolor='none', edgecolor='red', lw=2)
    if ax:
        ax.add_patch(patch)
    return patch

The bend parameter allows for dynamic adjustment of the arc's depth. A positive value bends the arc in one direction, while a negative value bends it in the opposite direction. This is particularly useful for Data Science projects where you need to distinguish between bidirectional flows between nodes.

4. Handling Coordinate Transformations

One common pitfall when drawing arcs in Matplotlib is the difference between Data Coordinates and Display Coordinates. If your axes have different scales (e.g., the X-axis ranges from 0 to 1000 while the Y-axis ranges from 0 to 1), a circular arc defined in data space will appear as a highly distorted ellipse on the screen.

To maintain a visual "circle" regardless of the data scale, you may need to apply a transform. Matplotlib's transforms module allows you to define patches in terms of pixels or "axes fraction" instead of data units. However, for most scientific plots, it is preferable to keep the data consistent and adjust the aspect ratio of the plot using:

plt.gca().set_aspect('equal', adjustable='box')

If you are working with geographic data, you should consider using the Cartopy library, which handles the complex projections required to draw "Great Circle" arcs on a spherical earth representation. For standard 2D Cartesian plots, the geometric derivations provided in Section 1 remain the standard.

5. Customizing Appearance: Styling and Arrows

An arc often represents a direction or a flow. Adding an arrow to an arc in Matplotlib requires using FancyArrowPatch. This class is powerful because it can take a Path object (like our Bezier curve) and automatically place an arrowhead at the end, correctly aligned with the curve's tangent.

def draw_arrow_arc(p1, p2, bend=0.3, ax=None):
    # FancyArrowPatch supports 'connectionstyle'
    # 'arc3' is essentially a quadratic Bezier
    arrow = patches.FancyArrowPatch(
        p1, p2,
        connectionstyle=f"arc3,rad={bend}",
        arrowstyle='->',
        mutation_scale=20, # Size of the arrow head
        color='green',
        lw=2
    )
    if ax:
        ax.add_patch(arrow)

In this snippet, the rad parameter in connectionstyle defines the curvature. This is often the most efficient way to draw arcs between points when precise geometric properties (like a specific radius) are less important than visual clarity and directional indication.

For more detailed information on path styling, the official Matplotlib Path Documentation offers an exhaustive list of codes and vertex manipulations available for complex shapes.

6. Advanced Implementation: The Sagitta Method

In architecture and civil engineering, arcs are often defined by their endpoints and the sagitta. The sagitta is the vertical distance from the center of the arc's chord to the arc itself. If we denote the length of the chord as ##L## and the sagitta as ##s##, the radius ##R## can be derived as:

###R = \frac{s}{2} + \frac{L^2}{8s}###

This formulation is particularly useful when you want to control the "bulge" of the arc directly without calculating radii. Implementing this in a Python function involves calculating the chord length ##L## between ##P_1## and ##P_2##, then finding ##R##, and finally proceeding with the Arc patch logic described earlier.

When visualizing large-scale networks, consider the following optimization tips:

1. Collection Usage: If drawing thousands of arcs, use PatchCollection to improve rendering performance.
2. Clipping: Ensure that clip_on=True is set (it is by default) to prevent arcs from drawing outside the axes boundaries.
3. Z-Order: Use the zorder parameter to ensure arcs sit behind markers but in front of grid lines.

By mastering these various methods—from the rigid precision of trigonometric circles to the fluid flexibility of Bezier curves—you can enhance your data visualizations with sophisticated, clear, and professional-looking curved connections. Whether you are building a custom UI or a complex scientific report, these geometric techniques provide the control necessary to handle any 2D spatial requirement in Matplotlib.

For further reading on coordinate systems and advanced rendering, explore the NumPy Linear Algebra documentation to sharpen the vector math underlying these visualizations.