The Lituus curve is studied by Roger Cotes in 1722 [Robert C Yates 1952].

The word lituus means a curved staff used by the augurs in quartering the heavens. (See: crosier), or A kind of trumpet of a somewhat curved form and shrill note.

Crosiere of arcbishop Heinrich of Finstingen
A crosier of Archbishop Heinrich of Finstingen, 1260 to 1286. 1223×2395 image source


Lituus is a spiral described by the polar equation r == 1/Sqrt[θ].


The curve is asymptotic to the positive x-axis, and the other end spiral in towards the pole. The above image is a plot from 0.1 to 20*π. As θ approachs infinity, the curve approaches the origin.


Polar equation: r == 1/Sqrt[θ].


It has the property that a circular sector produces the same area. That is, suppose P is a point on the curve, and X a point on the asymptote OP distance from the origin O. Suppose the area of the circular sector OPX is A. As P moves towards the center on the curve, the area remains the same.

inverse of parabolic spiral

The inverse of Lituus with respect to the center is the parabolic spiral.

lituus inv
Lituus (blue) and parabolic spiral (red).


The lituus spiral is a recurring shape in art called Volute.

violin scroll
The Scroll of a unfinished violin.

See Also

Archimedes' spiral, equiangular spiral, Mathematics of Seashell Shapes.

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.


The MacTutor History of Mathematics archive

Carving the scroll in making the violin:

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Plane Curves


  1. Conic Sections
  2. Parabola
  3. Hyperbola
  4. Ellipse
  5. Cissoid
  6. Conchoid
  7. Quadratrix
  8. Archimedean Spiral
  9. Equiangular Spiral
  10. Lituus
  11. Cornu Spiral


  1. Epitrochoid
  2. Hypotrochoid
  3. Epicycloid and Hypocycloid
  4. Rose Curve
  5. Astroid
  6. Deltoid
  7. Nephroid
  8. Cardioid
  9. Trochoid
  10. Cycloid

Calculus Era

  1. Cassinian Oval
  2. Cross Curve
  3. Folium of Descartes
  4. Piriform
  5. Semicubic Parabola
  6. Tractrix
  7. Trisectrix
  8. Trisectrix of Maclaurin
  9. Lemniscate of Bernoulli
  10. Lemniscate of Gerono
  11. Limacon Of Pascal
  12. Witch of Agnesi
  13. Sine Curve
  14. Catenary
  15. Bezier Curve


  1. Caustics
  2. Cissoid
  3. Conchoid
  4. Envelope
  5. Evolute
  6. Involute
  7. Geometric Inversion
  8. Orthoptic
  9. Parallel Curve
  10. Pedal Curve
  11. Radial Curve
  12. Roulette

Math of Curves

  1. Geometry: Coordinate Systems for Plane Curves
  2. Coordinate Transformation
  3. Vectors
  4. Naming and Classification of Curves
  1. Cusp
  2. Curvature