ComputerAided_Design_Engineering_amp_Manufactur.pdf

Completeness of Dimensions
Completeness of dimensioning is determined by entities’ degrees of freedom, such as face, edge, vertex,
centerline of a cylindrical feature, and the center point of a spherical feature, where the degrees of freedom
are represented by a set of vectors that define the basis of a space.
A face has a single degree of freedom, with its directions aligned with face normal direction, which is
a basis of an E space. An edge can move along two joined faces, thus, it has two degrees of freedom on
an space. (The space is defined as a plane perpendicular to the edge; the vectors may not lie on
the joined faces.) The degrees of freedom of a vertex depend on the number of connected edges. The
largest number of vectors that form the basis of an space is three, so the degrees of freedom of a
vertex is equal to three. A centerline has two degrees of freedom, which form an basis describing the
position of this centerline. A centerpoint has three degrees of freedom to locate its position in an
space. (The basis vectors may not be along any one of the edges.) Thus, we have the degrees of freedom
of geometric entities listed as follows:
1
E 2
E 2
E 3
E 2
E 3
1. face � 1;
2. edge � 2;
3. vertex � 3;
4. centerline � 2; and
5. centerpoint � 3.
Each geometric entity is assigned the initial degrees of freedom as described above. In the case of the
dimension assigned between entities, the degrees of freedom are deducted. The deduction of degrees of
freedom according to the type of dimensioning scheme can be categorized as follows:
1. Size dimension between two faces: Faces are constrained in a relative position. The degree of
freedom of each face becomes zero. Each edge and vertex has one degree of freedom deducted
from its total degrees of freedom in the normal direction of the face.
2. Size dimension between a face and an edge (centerline): The degrees of freedom of all the edges,
vertices, and face are reduced by one as in 1 above.
3. Size dimension between two edges (centerlines): Each edge and vertex has one fewer degree of
freedom in the direction of a vector that is perpendicular to the edges and that is on a plane by
joining these two edges.
4. Size dimension between a face and a vertex (centerpoint): The degrees of freedom of all edges,
vertices, and face are reduced by one as in 1 above.
5. Size dimension between an edge (centerline) and a vertex (centerpoint): Each edge and vertex has
one fewer degree of freedom in the direction of a vector that is perpendicular to the edge and that
is on a plane obtained by joining the edge and vertex.
6. Angular dimension between two planes: The same as 1 above.
Over-dimensioning is detected when dimensioning on two geometric entities both have zero degree
of freedom. After all the dimensions have been put on a part, if there is any geometric entity with nonzero
degrees of freedom, then this is the under-dimensioning that acquires more dimensions on this
entity to reduce the degrees of freedom to zero. Figure 9.11 gives an example to deduct the degrees of
freedom according to the dimensions applied.
Once a feasible dimension scheme is obtained, the next step is to interpret these dimensions into
geometric constraints, which is described in the following section.
3-D Variational Geometry Constraints
Figure 9.12 illustrates types of constraints in dimensions of size and angle. The size dimensions are
point-to-plane, line-to-plane, plane-to-plane, and line-to-line; the angular dimensions are line-to-line,
line-to-plane, and plane-to-plane, where these constrained entities have to be valid and fully represented
in a geometric model. In practice, dimension and tolerance are drawn based on implicit or explicit
data. (A datam is not explicitly specified on drawing.) Size, angle, form, and correlative tolerances are
© 2001 by CRC Press LLC