Reviewing a few concepts in plane and solid geometry will help immensely in
implementing your Geomet system. It is important to develop a feeling for 3D spatial
elements such as lines, planes, cylinders, etc., and for their spatial relationships.
Understanding the concept of the coordinate system and its elements is absolutely
essential.
Named after the French mathematician Rene Descartes, the Cartesian coordinate system is
a three dimensional grid consisting of
orthogonal
X, Y, and Z axes with distances marked out along them. A common road map is a special
case: It is a two dimensional Cartesian coordinate system which shows where landscape
features, cities and towns lie relative to each other. The coordinate system we will be
dealing with determine where the part features lie relative to part datum.
A three dimensional Cartesian coordinate system represented by an X, a Y, and a Z axis
is shown in figure 1. The figure also contains a number of geometric elements that make up
the GeoWidget: intersecting planes, bores, cylinders, counter bores, slots, cones and
spheres. In the following paragraphs we will discuss these basic and other geometric
elements.
Points are dimensionless locations is 3D space. Points have no size. The location of a
point is specified by its X, Y, and Z coordinate in a particular coordinate system. There
are many points that Geomet support. These include 1D, 2D and 3D points.
A line is the trail left by a point which moves in a constant direction. Lines are
infinitely long and have no width. A line is characterized by a point somewhere along its
length and a direction. In Geomet a line is characterized by a PCS base plane pierce point
and two or three base plane projection angles.
A plane is the trail left by a line which moves in a constant direction not parallel to
itself. Planes are infinitely wide and long but have no thickness. A plane is
characterized by a point and the direction of the plane's unit
normal vector. In Geomet the plane is
characterized by a PCS axis pierce point and three base plane projection angles.
A circle is the trail left by a point traveling inside a plane and at a constant
distance from a point in that plane called its center. A circle is also the intersection
of a plane whose axis is normal (90°) to the plane. A circle is characterized by a two
dimensional center location and a diameter or radius.
A cylinder is the trail left by a line which revolves around and at a constant distance
away from a parallel line called its axis. A cylinder is characterized by a point on its
axis, the direction of its axis and by its diameter or radius.
A sphere is the trail left by a point traveling in every conceivable direction at a
constant distance from another point called its center. A sphere is also the surface
generated by a rotating circle about a coplanar axis through its center. The sphere is
characterized by its diameter and a three dimensional center location.
A cone is the trail left by a line which rotates about another non-parallel and
non-perpendicular, intersecting line called its axis. The intersection of two lines is
called the apex. The cone is characterized by a point on its axis (PCS base plane pierce
point), the direction of its axis, its apex, its included angle and its diameter at the
pierce point.
Intersections
An intersection is the geometric figure created by the meeting of two other geometric
figures. Common intersections are planes with planes (creates lines), lines with planes
(creates points) and lines with lines (creates points).
Projections
A projection is the image of one geometric figure cast onto another geometric figure.
The projection of a line onto a plane is also a line, The projection of a sphere onto a
plane is a circle.
