Hypotrochoid

hypotrochoid spiralgraph

Mathematica Notebook for This Page

History

See Epicycloid and Hypocycloid.

Description

Hypotrochoid describes a family of curves. Hypotrochoid and epitrochoid are roulette of two circles. If one circle is inside another, the curve traced are called hypotrochoids. Otherwise (one rolls outside another), the curves are called epitrochoids. If the tracing point is on the (circumference) rolling circle, the curves are called hypocycloid or epicycloids. Hypotrochoids are also known as SpiroGraph — a tradmark of a toy that trace limited cases of hypotrochoids. Many famous curves are special cases of hypotrochoid. (See: Curve Family Index).

hypotrochoid hypotrochoidGen1
{1,2/3,h}
hypotrochoid hypotrochoidGen2
{1,2/5,1}

Formula

See Epicycloid and Hypocycloid.

Properties

Special Cases

h == 0, Circle.

b == h, hypocycloid with q cusps, where q is the denominator of b written as a irreducible fraction.

a/2 == b == h, Line of length 2 a.

hypotrochoidLineGen
{1,1/2,1/2}

a/2 == b AND h ≠ b, ellipse with semimajor axis b+h and semiminor axis Abs[b-h].

hypotrochoidEllipse1
{1,1/2,1/4}
hypotrochoidEllipse2
{1,1/2,1}

a/3 == b == h or 2/3 a == b == h, deltoid

hypotrochoidDeltoidGen1
{1,1/3,1/3}
hypotrochoidDeltoidGen2
{1,2/3,2/3}

a/4 == b == h or 3/4 a == b == h, astroid

hypotrochoidAstroidGen1
{1,1/4,1/4}
hypotrochoidAstroidGen2
{1,3/4,3/4}

a = n/(n+1), b = (n-1)/(2 (n+1)), h = 1/2, n is a integer greater than 2, rose with n petals if n is odd, otherwise 2*n petals. One can multiply all parameters by (n+1)/n to get {1, (-1 + n)/(2 n), (1 + n)/(2 n)}. find the formula for double generation for rose

hypotrochoidRose2Gen
Quadrifolium {1,1/4,3/4}
hypotrochoidRose3Gen
Trifolium {1,1/3,2/3}
hypotrochoidRose4Gen
8 leafed rose {1,3/8,5/8}

Two fun movies: spiralgrap_switch.mov spiralgraphs.mov Mathematica spiral-graph generation template

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

If you have a question, put $5 at patreon and message me.

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