Tangents

A Python tool for visualizing tangent lines to mathematical functions using SymPy and Matplotlib.

Purpose

This tool graphs tangent lines for two types of functions:

1. Scalar functions: y = f(x) - standard single-variable functions
2. Parametric curves: r(t) = (x(t), y(t)) - curves defined by parametric equations

The visualization helps understand the behavior of derivatives and tangent lines at multiple points along a curve, making it useful for educational purposes or mathematical exploration.

Dependencies

This tool requires the following Python packages:

• numpy - for numerical computations
• sympy - for symbolic mathematics and automatic differentiation
• matplotlib - for plotting and visualization

Installation

1. Clone this repository:

git clone https://github.com/Lunaaaalj/tangents.git
cd tangents

2. Install the required dependencies:

pip install numpy sympy matplotlib

Usage

The tool supports two modes: scalar for scalar functions and param for parametric curves.

Scalar Functions

Plot tangent lines for a scalar function y = f(x):

python tangents.py scalar --f "sin(x) + x**2/8" --xmin -4 --xmax 4 --n 9 --segment 1.2

Arguments:

• --f: The function expression (required)
• --xmin: Minimum x value (required)
• --xmax: Maximum x value (required)
• --n: Number of tangent lines to draw (default: 7)
• --segment: Length of each tangent segment in x-units (default: 1.0)
• --samples: Number of points to sample for the curve (default: 800)
• --points: How to choose tangent points - even or random (default: even)

More Examples:

# Simple parabola
python tangents.py scalar --f "x**2" --xmin -3 --xmax 3 --n 5

# Trigonometric function
python tangents.py scalar --f "cos(x)" --xmin 0 --xmax 6.28318 --n 8 --segment 0.8

# Complex function with random tangent points
python tangents.py scalar --f "exp(-x**2/4)*sin(2*x)" --xmin -4 --xmax 4 --n 10 --points random

Parametric Curves

Plot tangent lines for a parametric curve r(t) = (x(t), y(t)):

python tangents.py param --x "cos(t)" --y "sin(t) + 0.25*sin(3*t)" --tmin 0 --tmax 6.28318 --n 12 --segment 0.6

Arguments:

• --x: The x(t) expression (required)
• --y: The y(t) expression (required)
• --tmin: Minimum t value (required)
• --tmax: Maximum t value (required)
• --n: Number of tangent lines to draw (default: 10)
• --segment: Length of each tangent segment in curve units (default: 0.8)
• --samples: Number of points to sample for the curve (default: 1200)
• --points: How to choose tangent points - even or random (default: even)

More Examples:

# Circle
python tangents.py param --x "cos(t)" --y "sin(t)" --tmin 0 --tmax 6.28318 --n 8

# Ellipse
python tangents.py param --x "2*cos(t)" --y "sin(t)" --tmin 0 --tmax 6.28318 --n 12

# Lissajous curve
python tangents.py param --x "sin(3*t)" --y "cos(4*t)" --tmin 0 --tmax 6.28318 --n 16 --segment 0.4

Supported Mathematical Functions

The tool supports the following mathematical functions and constants in expressions:

Constants:

• pi - π
• E - Euler's number (e)

Basic Functions:

• sqrt(x) - Square root
• exp(x) - Exponential (e^x)
• log(x) - Natural logarithm
• Abs(x) - Absolute value

Trigonometric Functions:

• sin(x), cos(x), tan(x) - Standard trig functions
• asin(x), acos(x), atan(x) - Inverse trig functions
• sinh(x), cosh(x), tanh(x) - Hyperbolic functions

Other Functions:

• Min(x, y), Max(x, y) - Minimum and maximum
• Piecewise(...) - Piecewise functions

How It Works

1. Expression Parsing: User-provided mathematical expressions are parsed using SymPy's symbolic mathematics engine with a restricted namespace for security
2. Automatic Differentiation: SymPy automatically computes the derivative of the function
3. Tangent Point Selection: Points are chosen either evenly spaced or randomly within the specified range
4. Tangent Line Calculation:
   • For scalar functions: Uses the derivative to compute slope m at each point, then draws the line y - y₀ = m(x - x₀)
   • For parametric curves: Uses the tangent vector (dx/dt, dy/dt) to determine the direction at each point
5. Visualization: Matplotlib renders the curve and tangent lines

Notes

• The tangent segments are normalized for parametric curves to ensure consistent visual length
• If a parametric curve has a degenerate point (where both derivatives are ~0), it's marked with an 'x' instead of a tangent line
• All expressions are evaluated in a safe, restricted namespace to prevent arbitrary code execution
• The aspect ratio for parametric plots is set to 'equal' when the data is finite to preserve the curve's shape

License

This project is provided as-is for educational and visualization purposes.