MbTorusSurface Toroidal Surface

The MbTorusSurface class is declared in the surf_torus_surface.h file.

The MbTorusSurface toroidal surface belongs to the group of elementary surfaces MbElementarySurface. The toroidal surface is described by the major radius majorRadius and the tube radius minorRadius, defined in the local coordinate system MbPlacement3D position.

The first surface parameter is measured along the arc from the vector position.axisX in the direction of the vector position.axisY. The first surface parameter u takes values on the interval: umin ≤ u ≤ umax. The values u = 0 and u = 2π correspond to a point on the XZ plane. The surface can be periodic with respect to the first parameter. For a periodic surface, umax - umin = 2π; for a non-periodic surface, umax - umin < 2π.

The second surface parameter is measured along the arc from the XY plane of the local coordinate system of the surface in the direction of the vector position.axisZ. The second surface parameter v takes values on the interval: vmin ≤ v ≤ vmax. The values v = 0 and v = 2π correspond to a point on the XY plane of the local coordinate system of the surface. The surface can be periodic with respect to the second parameter when majorRadius > minorRadius. For a periodic surface, vmax - vmin = 2π; for a non-periodic surface, vmax - vmin < 2π.

In the PointOn( double u, double v, MbCartPoint3D & s ) method, the position vector of the surface s is described by the vector function

\begin{equation} \mathbf{s}(u, v) = \mathbf{position.origin} + \left( \mathit{majorRadius} + \mathit{minorRadius} \cdot \cos(v) \right) \left( \cos(u)\, \mathbf{position.axisX} + \sin(u)\, \mathbf{position.axisY} \right) + \\ \mathit{minorRadius} \cdot \sin(v)\, \mathbf{position.axisZ} \end{equation}

Figure shows the toroidal surface:

The tube radius must be greater than zero: minorRadius > 0. The major radius must be no less than the tube radius taken with the opposite sign: majorRadius > -minorRadius. If majorRadius < minorRadius, the surface has a pole at the parameter values v = arccos(majorRadius / minorRadius) and v = 2π - arccos(majorRadius / minorRadius). The following inequalities must hold for the boundary parameters of the surface: umin < umax, vmin < vmax.

The local coordinate system position can be either right-handed or left-handed. If the local coordinate system is right-handed, the normal is directed away from the tube of the surface; if the local coordinate system is left-handed, the normal is directed inward into the tube of the surface.