| | Describes a torus.
A torus is defined by its major and minor radii, and
positioned in space with a coordinate system (a
gp_Ax3 object) as follows:
- The origin is the center of the torus.
- The surface is obtained by rotating a circle around
the "main Direction". This circle has a radius equal
to the minor radius, and is located in the plane
defined by the origin, "X Direction" and "main <br>
Direction". It is centered on the "X Axis", on its
positive side, and positioned at a distance from the
origin equal to the major radius. This circle is the
"reference circle" of the torus.
- The plane defined by the origin, the "X Direction"
and the "Y Direction" is called the "reference plane" of the torus.
This coordinate system is the "local coordinate <br>
system" of the torus. The following apply:
- Rotation around its "main Axis", in the trigonometric
sense given by "X Direction" and "Y Direction",
defines the u parametric direction.
- The "X Axis" gives the origin for the u parameter.
- Rotation around an axis parallel to the "Y Axis" and
passing through the center of the "reference circle"
gives the v parameter on the "reference circle".
- The "X Axis" gives the origin of the v parameter on
the "reference circle".
- The v parametric direction is oriented by the
inverse of the "main Direction", i.e. near 0, as v
increases, the Z coordinate decreases. (This
implies that the "Y Direction" orients the reference
circle only when the local coordinate system is direct.)
- The u isoparametric curve is a circle obtained by
rotating the "reference circle" of the torus through
an angle u about the "main Axis".
The parametric equation of the torus is :
P(u, v) = O + (R + r*cos(v)) * (cos(u)*XDir +
sin(u)*YDir ) + r*sin(v)*ZDir, where:
- O, XDir, YDir and ZDir are respectively the
origin, the "X Direction", the "Y Direction" and the "Z <br>
Direction" of the local coordinate system,
- r and R are, respectively, the minor and major radius.
The parametric range of the two parameters is:
- [ 0, 2.*Pi ] for u
- [ 0, 2.*Pi ] for v
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1.6.3