|
Related Topics |
The Equal Arc Length projection option is used to project curves from an XC-YC plane to a single face, maintaining the arc lengths of the curves as closely as possible. This method can be used to transfer a tread pattern that is created on a plane, onto a 3D surface. It is the converse of world map making, where lines on the 3D earth’s surface are mapped to a planar representation.
Arc lengths are preserved by ‘mapping’ the XC-YC coordinates of points on the curves in the 2D representation, to parametric ‘u-v’ coordinates on the 3D face being projected to. There are several ‘Preserve Length’ options for mapping and they are discussed below.
Note:
The projected arclength will be truly equal (within modeling tolerance) only for special cases of the curve being projected, that is, for lines through the reference point and parallel to the XC and YC axes of the plane.
The projection can be made to only a single face. The 'face' you select must be a face, and not a plane or datum plane. If you select additional faces (and/or planes), they are ignored for the equal arclength projection.
Defining the XC-YC plane
After selecting a curve to project, and the face or faces to project to, choose the Equal Arc Length projection option (if it has not already been selected).
Define the XC-YC coordinate plane by making three selections.
Select the ‘reference point’. The reference point should be on or near the XC-YC plane of the curves. It becomes the origin of the XC-YC plane that you are defining.
Select the projection direction. It will appear as an arrow through the reference point. The ‘u’ parametric direction appears on the face being projected to. The projection direction becomes the normal (or ZC direction) for the XC-YC plane that you are defining. Note: The project operation will fail if the projection of the reference point does not fall on the face to project to. Keep this in mind when selecting the projection direction and the reference point.
Select an XC vector that corresponds to the ‘u’ parametric direction that is displayed on the face being projected to. The ‘v’ parametric direction appears on the face being projected to. The YC vector is displayed. It is perpendicular to the XC vector and the ZC vector, and in accordance with the right hand rule. This YC vector corresponds to the ‘v’ parametric direction on the face. The XC vector is adjusted if necessary so that it is perpendicular to the ZC vector.
The XC-YC plane that is the basis for the projection is now completely defined.
The (XC,YC,ZC) coordinate system is then used to define the parametric u-v coordinates on the face being projected to. The projected reference point (XC,YC,ZC) coordinate system origin, becomes the origin of the u-v coordinate system. The surface isolines that pass through the projected point become the u-v axes of the parametric coordinate system.
Note:
The reference point, the projection vector, and the X vector define the XC-YC plane (the Y vector is orthogonal to the projection vector and to the XC vector). The curves selected should be on (or parallel to) this plane for the arclength distances to be preserved by the projection.
The new 'u' and 'v' coordinates of the projected (XC,YC) points depend on the Keep Length option.
You can choose from one of the following options:
Both X and Y - 'u' is determined by measuring an arclength distance of XC along the u-isocurve, and 'v' is determined by measuring an arclength distance of YC along the v-isocurve. The projection of a line in the XC-YC plane, through the reference point and along the XC-vector or the YC-vector, will have an arclength equal to the length of the line.
First X, then Y - First 'u' is determined by measuring an arclength distance of XC along the u-isocurve, then an arclength distance of YC is measured along the v-isocurve to determine 'v'. The projection of a line in the XC-YC plane, through the reference point and along the XC-vector, will have an arclength equal to the length of the line. The projection of any line in the XC-YC plane parallel to the YC-vector will have an arclength equal to the length of the line.
First Y, then X - First 'v' is determined by measuring an arclength distance of YC along the v-isocurve, then an arclength distance of XC is measured along the u-isocurve to determine 'u'. The projection of the line in the XC-YC plane, through the reference point and along the YC-vector, will have an arclength equal to the length of the line. The projection of any line in the XC-YC plane parallel to the XC-vector will have an arclength equal to the length of the line.
X Only - 'u' is determined by measuring an arclength distance of XC along the u-isocurve, and 'v' is determined by measuring a distance of YC along the surface tangent vector in the v-isocurve direction. The projection of the line in the XC-YC plane, through the reference point and along the XC-vector, will have an arclength equal to the length of the line. For a line in the XC-YC plane parallel to the YC-vector, the distance along the surface tangent vector in the v-isocurve direction is preserved.
For a line parallel to the XC-vector, distance along the surface tangent vector in the u-isocurve direction is preserved.
Y Only - 'v' is determined by measuring an arclength distance of YC along the v-isocurve, and 'u' is determined by measuring a distance of XC along the surface tangent vector in the u-isocurve direction. The projection of the line in the XC-YC plane, through the reference point and along the YC-vector, will have an arc length equal to the length of the line.