Semicubic Parabola

semicubic parabola
The V-shaped boundary is the semicubic parabola. The droplet-shaped boundaries are parallels of a pedal of the semicubic parabola {t^3,t^2}, with respect to the point {0,-20}.

Mathematica Notebook for This Page.


From Robert Yates:

a*y^2==x^3 was the first algebraic curve rectified (Neil 1659). Leibnitz in 1687 proposed the problem of finding the curve down which a particle may descend under the force of gravity, falling equal vertical distances in equal time intervals with initial velocity different from zero. Huygens announced the solution as a semi-cubic parabola with a vertical cusp tangent.


Semicubic parabola is defined as the evolute of parabola. In the figure, normals (light blue) of a parabola (purple) are drawn. The envelope of these lines is the semicubic parabola.

semicubic parabola


{t^3, t^2} is the evolute of the parabola {(-4 t)/9, -8/27 + t^2/3}.



The parabola {(-4 t)/9, -8/27 +t^2/3} is the parabola {t,1/4 t^2} scaled by a factor of 4/27 and translated by {0, -8/27}.

Semicubic parabola inherets the property from parabola that when streched horizontally or vertically, the curve remain unchanged. That is, the curve {a t^3, b t^2} is equivalent to {t^3, t^2}*(b^3/a^2). This is why sometimes you'll see different equations like {2 t^3, 3 t^2} for it.

The pedal of a semicubic parabola with respect to the focus of a parabola the semicubic parabola is derived from is another parabola. In the figure below, the pink curve is the original parabola, blue dot is its focus. The evolute of it is the semicubic parabola (bluish-white). Its pedal is rendered as red dots, which is another parabola.

semicubic parabola

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

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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