Lituus
r²θ = a²: a shepherd’s-crook spiral named by Roger Cotes, whose sectors have equal area whenever their angles have equal ratio.
Adjust parameters
Export
Formulas and key values
- Polar equation
- Asymptote
- Area · sector between θ₁ and θ₂
a sets the scale (a = 1 in the graph); θ > 0.
Explainer video
Make a video of the Lituus
Opens a video template with the parameters on screen. Every number in the video is computed before it renders; choose landscape or vertical.
Start the video01 · Definition
What is the Lituus?
Roger Cotes named this spiral in 1722 after the lituus, the curved staff of a Roman augur, which it resembles: a long straight handle that curls into a coil. In polar form r = a/√θ.
Its sectors have a neat property. The area swept between angles θ₁ and θ₂ is (a²/2)·ln(θ₂/θ₁), so it depends only on their ratio. As θ → 0 the curve straightens out along the x-axis, its asymptote; as θ grows it coils ever more tightly around the pole.
02 · Properties
Key properties
- Polar equation r²θ = a², i.e. r = a/√θ.
- Sector area between θ₁ and θ₂: (a²/2)·ln(θ₂/θ₁).
- The x-axis is an asymptote as θ → 0.
- Winds into the pole as θ → ∞.
- Named by Roger Cotes in 1722.
03 · Questions
Lituus: common questions
- What is the equation of the Lituus?
- Polar equation: r^2θ = a^2, r = a/√θ. a sets the scale (a = 1 in the graph); θ > 0.
- What is the area of the Lituus?
- Area (sector between θ₁ and θ₂): A = ((a^2)/2)ln (θ₂)/(θ₁). a sets the scale (a = 1 in the graph); θ > 0.
04 · Keep exploring