Hypotrochoid
   HOME

TheInfoList



Click Here for Items Related To - Hypotrochoid
OR:
Hypotrochoid on:   [Wikipedia]   [Google]   [Amazon]

In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a hypotrochoid is a
roulette Roulette (named after the French language, French word meaning "little wheel") is a casino game which was likely developed from the Italy, Italian game Biribi. In the game, a player may choose to place a bet on a single number, various grouping ...
traced by a point attached to a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
of
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
rolling around the inside of a fixed circle of radius , where the point is a
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
from the center of the interior circle. The
parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...
s for a hypotrochoid are: :\begin & x (\theta) = (R - r)\cos\theta + d\cos\left(\theta\right) \\ & y (\theta) = (R - r)\sin\theta - d\sin\left(\theta\right) \end where is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because is not the polar angle). When measured in radian, takes values from 0 to 2 \pi \times \tfrac (where is
least common multiple In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...
). Special cases include the
hypocycloid In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid creat ...
with and the
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
with and . The eccentricity of the ellipse is :e=\frac becoming 1 when d=r (see
Tusi couple The Tusi couple (also known as Tusi's mechanism) is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller ...
). The classic
Spirograph Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well-known toy version was developed by British engineer Denys Fisher and first sold in ...
toy traces out hypotrochoid and
epitrochoid In geometry, an epitrochoid ( or ) is a roulette traced by a point attached to a circle of radius rolling around the outside of a fixed circle of radius , where the point is at a distance from the center of the exterior circle. The parametric ...
curves. Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.


See also

*
Cycloid In geometry, a cycloid is the curve traced by a point on a circle as it Rolling, rolls along a Line (geometry), straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette (curve), roulette, a curve g ...
*
Cyclogon In geometry, a cyclogon is the curve traced by a vertex of a regular polygon that Rolling, rolls without slipping along a straight line. In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid. The cyclogon h ...
* Epicycloid *
Rosetta (orbit) A Rosetta orbit is a complex type of orbit. In astronomy, a Rosetta orbit occurs when there is a periastron shift during each orbital cycle. A retrograde Newtonian shift can occur when the central mass is extended rather than a point gravitation ...
*
Apsidal precession In celestial mechanics, apsidal precession (or apsidal advance) is the precession (gradual rotation) of the line connecting the apsis, apsides (line of apsides) of an orbiting body, astronomical body's orbit. The apsides are the orbital poi ...
*
Spirograph Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well-known toy version was developed by British engineer Denys Fisher and first sold in ...


References


External links

*
Flash Animation of Hypocycloid
from Visual Dictionary of Special Plane Curves, Xah Lee
Interactive hypotrochoide animation
*{{MacTutor, class=Curves, id=Hypotrochoid, title=Hypotrochoid Roulettes (curve) de:Zykloide#Epi- und Hypozykloide ja:トロコイド#内トロコイド