Worksurfaces are defined parametrically in the special coordinate systems:

•cylindrical;

•spherical;

•toroidal (ring-shaped).

The parametric coordinates are always counted with respect to an orthogonal (Cartesian) coordinate system. (The meaning and the range of the coordinates depend on the surface type.) The particular Cartesian system can be either the world coordinate system, or a specifically selected local coordinate system (LCS).

The position of a point in the cylindrical coordinates is defined by three parameters: the radius of the cylinder, the shift along the cylinder axis (Z-axis) and the angle (or the arc length) between X-axis and the radius-vector of the point projection onto the X-Y plane.

The position of a point in the spherical coordinates is defined by three parameters: the radius of the sphere, the angle between X-axis and the radius-vector of the point projection onto the X-Y plane, and the angle between the point radius vector and Z-axis.

The position of a point in the toroidal coordinates is defined by four parameters: the two defining radii of the torus, the angle between X-axis and the point projection on the X-Y plane and the angle between the point radius vector and Z-axis direction.

The parameters defining the radius of the cylinder, that of the sphere or the two torus radii are called fixed, as those are the same for all points belonging to one surface. This is a constant characteristic of the given surface. The cylinder and the sphere have one fixed parameter – the radius, while the torus – two (the two torus radii).

Therefore, the position of a point belonging to a surface of any type is actually defined in a special coordinate system by two orthogonal parameters – the coordinates (U and V). Due to this fact, a mapping can be established between any of the special coordinate systems and a rectangular region on a plane (a parametric 2D region). This 2D region represents an "unfolding" of the surface onto the plane (according to the rules of mapping the UV-coordinates into the Cartesian coordinates for the selected type of surface), playing the same role as the image of the workplane on a page of a 2D drawing. Besides, the 2D region defines the range of the UV-coordinates.

Introducing a parametric 2D region allows working with a surface as if with a common workplane.

Spherical coordinates

Graphic Illustration |
Parametric 2D region |
Profile in 3D |

U – angle from X-axis (0-2p) V – angle from Z-axis (0-p) R – radius of the sphere |

Cylindrical coordinates

Graphic Illustration |
Parametric 2D region |
Profile in 3D |

U – angle from X-axis (0-2π) or arc length counted from X-axis (0-2πR) V - Z-coordinate (-,+) R – radius of the cylinder |

Toroidal coordinates

Graphic Illustration |
Parametric 2D region |
Profile in 3D |

U – angle from X-axis (0-2p) V – angle from the vector R in the plane ZOR (-p,+p) R – torus major radius, r – torus minor radius |

Data required for creating a worksurface

When creating a worksurface of any type, you need to define:

•The parametric 2D region (a rectangular region in the 2D drawing) – the unfolding onto a plane of the surface being created;

•The fixed parameter value for this surface (the radius of the sphere or cylinder, or the two torus radii);

•The original Cartesian coordinate system (the world coordinate system or an arbitrary LCS), with respect to which the parametric surface coordinates are defined.

Worksurface Representation in 3D Window

The ability of a worksurface to be displayed in 3D window depends on the status of the Show in 3D Window checkbox in worksurface parameters dialog.