SGAC-3 INTERSECTIONS
Pierce Point Of A Line And A Plane
CONCEPT:
Assuming that the straight line is neither parallel to nor in the plane, it will intersect the plane at a point common to both the line and the plane.
Piercing Point – Two-View Method – Descriptive Geometry by Paré, Loving, Hill, & Paré, Macmillan, 1982.
The piercing point of line EG with the plane ABC may be found using only the given views as follows.
(a) Any convenient cutting plane containing line EG is introduced, Figure 6.3(b). A cutting plane perpendicular to one of the principal planes is convenient because it appears in edge view in a principal view. This simplifies the following step.
(b) The line of intersection 1-2 between this cutting plane and plane ABC is determined.
(c) Since lines EG and 1-2 both lie in the cutting plane, they intersect, locating point P.
(d) Since line 1-2 also lies in plane ABC, point P is the required piercing point of line EG with the plane ABC.
PROBLEM 1 – Load Drawing Template H191-3D
A plane (yellow) is intersected by a control cable (red). The coordinates of the plane and the cable are given below. Locate the pierce point of the control cable with the plane. Show the pierce point in all views with a point (+).
Plane (yellow)
(0, 0, 0) (3, 0, -0.75) (3, -2, -2) (0, -2, -1.25) [CLOSE]
Control Cable (red)
(1.8, -2.6, 0.1) (1.2, 0.5, -2.1)
SOLUTION
1) Set up the following Layers with colors and line styles:
Plane Yellow Continuous
Control Cable Red Continuous
Solution Cyan Continuous
2) Draw the line strings on the appropriate layers (enter coordinates).
3) Zoom Extents in each viewport so all the lines can be seen.
4) Change to the current layer to Solution.
5) Set the Top View (upper left viewport) as the active viewport.
6) Start the Line command.
7) [Shift + Right Click] to invoke the Snap Menu – Select Apparent Intersection.
8) Select upper yellow line, then select red line.
9) Repeat Step 7.
10) Select lower yellow line, then select red line. (notice line appears in all viewports)
11) Set the Front View (lower left viewport) as the active viewport.
12) Draw point at the intersection of the Control Cable and the Cyan Line.
13) Use the List Command [LI ¿] to determine coordinates of point (to the thousandth).
Prove your point is the pierce point by getting an edge view of the plane. The point will be on both the edge view and the line in this view. If it is not, then you have made an error and need to redo the problem.
STEPS
1) Open the UCS Menu by right-clicking on one of the command menus and selecting ACAD, then UCS
2) Activate the Right Viewport and make sure OSNAP is on.
3) Click on the ‘Z Axis Vector UCS’ command in the UCS Menu.
4) Hit [Enter] to keep the same origin point
(one of the corners of the plane is at the origin)
5) Select one of the adjacent end-points as a point on the positive portion of the Z-axis.
NOTE: This changes the local coordinate system to one that has the z-axis aligned with an edge of the plane. Looking down the z-axis (plan view) will show the plane as an edge
6) Change to the Plan View [PLAN¿¿]
The plane should now appear as an edge with the control cable intersecting it. If you select the point you created at the intersection (use a enclosing window selection), it should appear at the intersection of the edge view of the plane and the control cable.
You will need to repeat the above steps when creating a layout for printing. Label the view as the “Edge View of the Plane”.
Make a layout using the front view, top view, and the auxiliary view showing the plane as an edge and the coordinates of the pierce point.
Intersection of Planes
CONCEPT:
Two planes intersect in space to form a line. The cutting plane method can be executed twice to establish tow pierce points that will define the line of intersection between the planes.
PROBLEM 2 – Load Drawing Template H191-3D
A yellow plane and a red plane are defined by the coordinates below. Determine the line of intersection between the two planes. Draw the line of intersection in cyan.
Yellow Plane
(0, 0, 0) (1.75, 0.50, 1.75) (0.75, 1.50, 1.25) [CLOSE]
Red Plane
(-0.25, 1, 1) (1.75, 1, 1.25) (1, -0.25, 0.25) [CLOSE]
SOLUTION
1) Set up the following Layers with colors and line styles:
Yellow Plane Yellow Continuous
Red Plane Red Continuous
Solution Cyan Continuous
2) Draw the line strings on the appropriate layers (enter coordinates).
3) Zoom Extents in each viewport so all the lines can be seen.
4) Change to the current layer to Solution.
5) Find the Pierce point of one of the lines of either plane with the other plane in the same manner as described in the instructions for finding the pierce point of a line and plane.
6) Repeat the process for another line.
7) Connect the two pierce points – this line defines the line of intersection of the planes.
8) Use the extend command to extend the line to the edges of the each plane.
Prove that the line of intersection determined above is the true intersection between the planes by creating a new view that yields both planes as an edge (use the Z Axis Vector UCS command – the same technique used on the previous page). If the line of intersection is true, both planes will appear as an edge at this point.
Make a layout using the front and top views of the planes and the line of intersection. Also include the view that shows a point view of the line of intersection and the dihedral angle. Dimension the dihedral angle in your layout, in paper-space, using the angle command in the Dimension Menu (click on command, select edges of planes).
rev. 10/02 Page 3 of 3