Cross Curve

cross curve contour plot
The Cross Curve function ploted as surface contours. crossCurve.nb.zip

Description

Construction of the cross curve:

Let there be a circle centered on the origin. Let P be a point on the circle and O be the origin. Let the intersections of the tangent at P with the x and y axes be M and N. Let a line passing M and perpendicular to x-axis be m. Similarly a line n for N and y-axis. The trace of the intersection of m and n is the Cross Curve, as P moves on the circle.

cross curve construction
cross curve construction.

History

Formula

Start with a circle centered on the origin with radius one {Cos[t],Sin[t]}. Take its tangent vector, then find its tangent equation, then find their intersections with the x and y axes, we easily derive the parametric formula {Sec[t],Csc[t]}. To get the Cartesian equation, we try the art of eliminating a variable. With luck, we use formula Sin[t]^2+Cos[t]^2==1 and fairly easily derive the Cartesian equation x^2+y^2==x^2*y^2.

Note that in deriving the Cartesian equation from the parametric, we have squared both sides of x==Sec[t] and y=Csc[t]. That means, we may obtain extraneous solution when x==0 or y==0, which is the case here. The Cartesian equation x^2+y^2-x^2*y^2==0 has a isolated point {0,0}.

Properties

cross curve surface
The cross curve x^2+y^2-x^2*y^2 plotted as a surface.
cross curve surface cross curve surface as density
The surface plotted (not to scale), and as density.
2006-05

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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