Cusps

cusp cusp
Three curves {t^2, t^3} (blue), {t^2, t^5} (red), {t^2 + t^3, t^4} (green). All have a cusp at t==0. The left image shows a plot, the right is a close up.

Their curvature functions are:

 (6*t^2)/(t^2*(4 + 9*t^2))^(3/2) 
 (30*t^4)/(t^2*(4 + 25*t^6))^(3/2)
 (-4*t^3*(2 + 6*t) + 12*t^2*(2*t + 3*t^2))/(16*t^6 + t^2*(2 + 3*t)^2)^(3/2)

Curvature at t=0 is undefined, but we can compute the limit at t=0. The following are plots of each curve and their Evolute. From these we learn that cusps can have a well-defined curvature, and may be different.

cusp cusp
{t^2, t^3} and its evolute. As one can see, as t approches the cusp, the radius of osculating circles (shown as yellow lines) decreases, and becomes 0 as a limit at the cusp. In other words, as t approaches the cusp, the curve gets more and more curved, and is infinitly curved like circle of radius 0 at the cusp.
cusp cusp
{t^2, t^5} and its evolute. The curve gets flatter and flatter near the cusp. The curvature at origin is 0. (as flat as a line).
cusp cusp
{t^2 + t^3, t^4} and its evolute. The curvature at origin is 2 (like the arc of a circle with radius 1/2).

cusps.nb.zip

Example on this page are from C G Gibson's Elementary Geometry of Differential Curves Buy at amazon.

Related Web Sites

Cusp (singularity).

2006-11

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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