Trisectrix of Maclaurin

trisectrix of Maclaurin
Trisectrix of Maclaurin

Mathematica Notebook for This Page.

History

This is a curve Colin Maclaurin (1698 – 1746) used to trisect a angle.

Description

Formula

The following formula has asymtote at x=1 and loop point at origin:

Polar: r==Sec[t] - 4 Cos[t], -π/2 < t < π/2.

Parametric: (1-4 Cos[t]^2)*{1, Tan[t]}, -π/2 < t < π/2.

Parametric: {(-3 + t^2)/(1 + t^2), (t*(-3 + t^2))/(1 + t^2)}, -∞ < t < ∞.

Cartesian: y^2 (1-x) == x^2 (x + 3)

Polar: r==Sec[t/3], -3/2 π < t < 3/2 π. (vertex at {1,0}, double point at {-2,0})

Properties

Relation to Parabola

It is the pedal of parabola with repect to its focus reflect by the directrix.

trisectrix of Maclaurin

Trisecting a Angle

Suppose we have a trisectrix with node at the origin and vertex at point {-3,0}, and let P be any point on the loop of the curve. angle[{-3,0},{-2,0},P] == 3*angle[{-2,0},{0,0},P].

trisectrix of Maclaurin triOfMaclaurin
Moving Point P

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

The MacTutor History of Mathematics archive

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Plane Curves

Ancient

  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

Cyclodal

  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

Methods

  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