curtate cycloid

A slight 2 dimensional extension of the cycloid. See also cyclohedron for an extension of the cycloid into three dimensions.

The curtate cycloid is defined to be a cycloid measured at a point outside the generating circle:

                 ,.onOK@@@@@HQme.,
            ,.szF'`             `'Tux.,
          ,z'`                       `'c,
        ,x'`                           `'w,
      .u'`                               `'n.
     dy                                     qb
    /7                                       VA
   4y                                         \D
  ,I'                                          `U,
  dp                                            qb
 ,j'                                            `t,
 AV                                              VA
 69                                              96--a--* (x,y)  <-- example point of measure,
 VA                                              AV  distance "a" from the edge of the circle.
 `t,                                            ,j'
  qb                                            dp
  `I,                                          ,U'
   \D                                         4y
    VA                                       /7
     qb                                     dy
      `'n.                               .u'`
        `'w,                           ,x'`
          `'c,                       ,z'`
            `'Tux.,             ,.szF'`
                 `'TTOK@@@@@HQTT'`

Let (x,y) be a point of the curtate cycloid, where the generating circle has radius r, and the shortest distance from (x,y) to the edge of the circle is a at all angles t. Then,

x=r*(t-sin(t))-a*sin(t)

and

y=r*(1-cos(t))-a*cos(t)