geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a cyclogon is the

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

traced by a vertex of a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

that

rolls Roll or Rolls may refer to: Movement about the longitudinal axis * Roll angle (or roll rotation), one of the 3 angular degrees of freedom of any stiff body (for example a vehicle), describing motion about the longitudinal axis ** Roll (aviation), ...

without slipping along a

straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...

. There are no restrictions on the nature of the polygon. It can be a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

like an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

or a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

. The polygon need not even be

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

: it could even be a

star-shaped polygon In geometry, a star-shaped polygon is a polygonal region in the plane that is a star domain, that is, a polygon that contains a point from which the entire polygon boundary is visible. Formally, a polygon is star-shaped if there exists a poi ...

. More generally, the curves traced by points other than vertices have also been considered. In such cases it would be assumed that the tracing point is rigidly attached to the polygon. If the tracing point is located outside the polygon, then the curve is called a ''prolate cyclogon'', and if it lies inside the polygon it is called a ''curtate cyclogon''. In the limit, as the number of sides increases to infinity, the cyclogon becomes a

cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cu ...

. The cyclogon has an interesting property regarding its area. Let denote the area of the region above the line and below one of the arches, let denote the area of the rolling polygon, and let denote the area of the disk that circumscribes the polygon. For every cyclogon generated by a regular polygon, :

A = P + 2C. \,

Examples

Cyclogons generated by an equilateral triangle and a square

Prolate cyclogon generated by an equilateral triangle

Curtate cyclogon generated by an equilateral triangle

Cyclogons generated by quadrilaterals

Cyclogon_generated_by_a_non-convex_quadrilateral

Cyclogon_generated_by_a_convex_quadrilateral

Cyclogon_generated_by_a_star-like_quadrilateral

Generalized cyclogons

A cyclogon is obtained when a polygon rolls over a straight line. Let it be assumed that the regular polygon rolls over the edge of another polygon. Let it also be assumed that the tracing point is not a point on the boundary of the polygon but possibly a point within the polygon or outside the polygon but lying in the plane of the polygon. In this more general situation, let a curve be traced by a point z on a regular polygonal disk with n sides rolling around another regular polygonal disk with m sides. The edges of the two regular polygons are assumed to have the same length. A point z attached rigidly to the n-gon traces out an arch consisting of n circular arcs before repeating the pattern periodically. This curve is called a trochogon — an ''epitrochogon'' if the n-gon rolls outside the m-gon, and a ''hypotrochogon'' if it rolls inside the m-gon. The trochogon is curtate if z is inside the n-gon, and prolate (with loops) if z is outside the n-gon. If z is at a vertex it traces an epicyclogon or a hypocyclogon.

References

{{Reflist Roulettes (curve) Piecewise-circular curves