In the plane , a rose (or rhodonea) is acurve , given by the polar equation , where is a positive rational number in lowest terms. This is a directing curve of a special rose surface presented in this Demonstration.
is the system of circles with diameters that are perpendicular to the plane, where is the origin and is a point of the rose .
This Demonstration shows the parts of special rose surfaces , where is the simplest form of the rational number that you choose using the setter. For every choice of and there are three possibilities for surface presentation: red colored surface (Snapshot 1), petals of surface colored by different colors (Snapshot 2), and the illustration of surface construction as the system of circles (Snapshot 3).
is an algebraic surface with the following properties:
• If is odd and , the order of is , is an -fold point, is an -fold line, and there are petals.
• If is even and , the order of is , is -fold point, is a -fold line, and and there are petals.
• If is odd and , the order of is , is a -fold point, and there are petals.
• If is even and , the order of is , is a -fold point, and there are petals.