Lemniscate of Gerono

figure 8 curve
Lemniscate of Gerono

History

Studied by Camille-Christophe Gerono (1799 – 1891).

Description

Lemniscate of Gerono is also known as the figure eight curve. It can be constructed as follows: Let there be a circle of radius 1 centered on the Origin. Let P be a point on the circle. Let M be the intersection of x==1 and a horizontal line passing P. Let Q be the intersection of line OM and a vertical line passing P. The trace of Q as P moves on the circle is the Figure Eight curve.

figure 8 curve
Figure 8 Curve

Formula

The parametric formula is directly derived. Starting with P:={Cos[t],Sin[t]}. The vertical line passing P would be x==Cos[t]. The point M would be {1,Sin[t]}. Solving for the insersection of line OM and x==Cos[t] results the formula.

For the Cartesian formula, eliminate the trig terms in x==Cos[t], y==Cos[t]*Sin[t] by squaring both side and add the two equations and use the identity Cos[t]^2+Sin[t]^2==1.

Properties

Second Construction

figure 8 curve

Let there be a circle centered on {1/2,0} of radius 1/2. Let N be the intersection of the circle and line OP. The intersection Q of vertical line thru P and horizontal line thru N traces the curve.

Figure 8 Curve Const

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

The Construction property is from http://www.mathcurve.com/courbes2d/gerono/gerono.shtml by Robert Ferreol, Jacques Mandonnet.

Lemniscate of Gerono.

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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