Calculator Inputs
Formula Used
A constant angle spiral is a logarithmic spiral. Its radius is calculated with:
r = r₀ × scale × e^(b(θ - θ₀))
If the angle is measured against the radial line, then
b = cot(α). If the angle is measured against the circular tangent,
then b = tan(β).
Inward direction multiplies b by -1. The growth ratio is
r / r₀. The arc length estimate is
s = √(1 + b²) × |r - r₀| / |b|. When b = 0, the curve is circular and
s = r₀ × |θ - θ₀|.
How to Use This Calculator
- Enter the starting radius of the spiral.
- Enter a scale factor if your model uses normalized units.
- Add the starting angle and target angle.
- Choose degrees or radians.
- Enter the constant spiral angle.
- Select whether that angle is measured from the radial line or circular tangent.
- Choose outward or inward direction.
- Press the calculate button to view results, chart, CSV, and PDF options.
Example Data Table
| Start Radius | Start Angle | Target Angle | Spiral Angle | Direction | Approx. Radius |
|---|---|---|---|---|---|
| 2 | 0° | 180° | 80° radial | Outward | 3.477 |
| 5 | 0° | 360° | 85° radial | Outward | 8.665 |
| 10 | 0° | 90° | 8° circular | Outward | 12.469 |
| 4 | 0° | 270° | 70° radial | Outward | 22.223 |
| 12 | 0° | 180° | 5° circular | Inward | 9.117 |
Constant Angle Spiral Radius in Physics
What the Curve Means
A constant angle spiral is also called an equiangular spiral. It keeps the same angle between its tangent and the radius drawn from the origin. This simple rule creates a curve that grows by a fixed ratio for equal changes in angle.
How the Calculator Works
This calculator models that relation with a logarithmic spiral equation. You enter a starting radius, a starting angle, a target angle, and the constant spiral angle. The tool converts the angle setting, finds the growth coefficient, then estimates the final radius. It also reports radial change, growth ratio, and arc length along the curve.
Angle Reference Matters
The most important setting is the angle reference. Many physics and geometry texts define the constant angle against the radial line. Some design notes define it against the circular tangent. These two angles are complements. The calculator supports both styles, so you can match your source without rewriting the formula.
Direction and Scale
The outward option makes the radius grow as the angular distance increases. The inward option reverses the coefficient and makes the spiral shrink. This is useful for vortex paths, antenna curves, polar motion studies, and optical layouts where the same shape may be traced in either direction.
The scale multiplier helps when a model is drawn in normalized units but later needs real dimensions. A scale of one keeps the original radius. A larger scale expands the whole curve while preserving the same angular growth pattern.
Reading the Chart
The chart shows radius against polar angle. It helps reveal how quickly the curve expands. Small constant angles measured from the radial line can grow very fast. Angles near ninety degrees behave almost like circles because the growth coefficient approaches zero.
Exporting and Comparing
The data table supports comparison between cases. It shows how different target angles and spiral angles change radius. Export tools make the results easier to document. You can save a compact CSV file for spreadsheets. You can also create a simple PDF report for notes, lab records, or project handoffs.
Practical Use
Results should be interpreted as ideal geometric values. Real physics systems may include friction, turbulence, material limits, or numerical constraints. Use this calculator for theoretical radius estimates, design comparison, and checking logarithmic spiral geometry before deeper simulation.
FAQs
1. What is a constant angle spiral?
It is a spiral where the tangent meets the radial line at a constant angle. This creates logarithmic growth, so equal angular steps multiply the radius by a consistent factor.
2. Is this the same as a logarithmic spiral?
Yes. A constant angle spiral is commonly modeled as a logarithmic spiral. The radius changes exponentially with angular displacement from the starting angle.
3. Which angle reference should I choose?
Choose radial reference when your angle is measured from the radius line. Choose circular tangent reference when your angle is measured from the tangent of a circle at that point.
4. What does outward direction mean?
Outward direction means the radius increases as the angular travel increases. Inward direction reverses the growth coefficient and makes the spiral contract over the same angular travel.
5. Why does the radius grow so fast?
The radius follows an exponential formula. A small change in the growth coefficient can create a large change after many degrees or many revolutions.
6. What happens near ninety degrees?
When the radial reference angle approaches ninety degrees, the growth coefficient approaches zero. The spiral behaves more like a circle with nearly constant radius.
7. Can I use radians instead of degrees?
Yes. Select radians in the angle unit field. The calculator converts values internally and still reports angular travel in both radians and degrees.
8. Are the CSV and PDF exports exact?
They export the displayed calculated values and plotted samples. Small rounding differences can appear because the report uses readable formatted numbers.