Logarithmic Spiral Generator – Coordinates & Plot in 3 Easy Steps
Logarithmic Spiral Generator
Free logarithmic spiral generator – calculate Polar (r, θ) and Cartesian (X, Y) coordinates for an equiangular spiral (r = a·e^(bθ)).
Spiral Parameters
Angle & Steps
Use this free logarithmic spiral generator to plot an equiangular spiral and export its coordinates for design, CAD or coding.
Quick answer
A logarithmic spiral follows r = a·e^(bθ), where a is the starting radius and b controls how fast it grows. Each point converts to Cartesian form with x = r·cos θ and y = r·sin θ. Example: a = 1, b = 0.2 and θ from 0 to 10 radians gives r ≈ 7.389 at the end. This logarithmic spiral generator plots the curve and lists every (θ, r, x, y) point.
Table of Contents
The logarithmic spiral – also called the equiangular or growth spiral – appears in nautilus shells, hurricanes, galaxy arms and sunflower heads. Its special property is that it keeps the same shape as it grows: every turn is a scaled copy of the last.
Our free logarithmic spiral generator calculates as many points as you need, draws the spiral, and gives you polar and Cartesian coordinates you can paste into Excel, CAD, Python, JavaScript or a plotter.
Logarithmic Spiral Formulas
| Quantity | Formula |
|---|---|
| Polar equation | r = a·e^(bθ) |
| Cartesian x | x = a·e^(bθ)·cos θ |
| Cartesian y | y = a·e^(bθ)·sin θ |
| Constant angle | The curve crosses every radius at angle α where tan α = 1/b |
| Growth per turn | r grows by a factor e^(2πb) every full turn |
Effect of the growth factor b
- b > 0: spiral grows outward counter-clockwise.
- b < 0: spiral winds inward.
- Small b (e.g. 0.05): tightly wound spiral.
- b ≈ 0.3063: the golden spiral, which grows by the golden ratio φ every quarter turn.
How to Use the Logarithmic Spiral Generator
Generate your spiral in three steps.
Step 1: Set the Spiral Parameters
Enter the Initial Radius (a) and Growth Factor (b), then the Start Angle, End Angle, Angle Unit (radians or degrees) and number of Data Points. Click Generate Coordinates.
Step 2: View the Spiral Plot
The Spiral Visualization draws the curve so you can check its shape before using the numbers.
Step 3: Copy or Download the Coordinates
The Generated Coordinates table lists index, θ, r, X and Y for every point. Click Copy Results or Download CSV to use them in a spreadsheet or code.
Uses of the Logarithmic Spiral
- Graphic and logo design: golden-spiral layouts and organic shapes.
- Engineering: spiral gears, cams and antenna designs.
- Architecture: spiral staircases and decorative patterns.
- Generative art and games: particle paths and procedural patterns.
- Teaching: polar coordinates, exponentials and growth in nature.
Tips
- Use more data points for a smoother curve (100–500 for printing).
- Keep angles in radians for code; most programming languages use radians.
- For a golden spiral set b ≈ 0.3063 (ln φ ÷ (π/2)).
Model growth over time with the Exponential Growth & Decay Calculator, or work with points using the 3D Coordinate Distance Calculator.
Frequently Asked Questions
What is a logarithmic spiral?
A curve whose distance from the centre grows exponentially with the angle: r = a·e^(bθ). It keeps the same shape at every scale.
Is the golden spiral a logarithmic spiral?
Yes. The golden spiral is a logarithmic spiral with b ≈ 0.3063, growing by the golden ratio every 90°.
Can I use degrees instead of radians?
Yes. Choose Degrees in the Angle Unit menu; the tool converts them internally.
How do I make the spiral go inward?
Use a negative growth factor b.
Can I export the points?
Yes. Copy the results or download them as a CSV file.
Is the logarithmic spiral generator free?
Yes. It is free, runs in your browser and needs no sign-up.