Space Geometry Intersections of Lines and Planes

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Space Geometry Intersections of Lines and Planes
It is important to study space geometry to
improve your spatial comprehension. Line
Piercing A Plane Edge View Method SOLUTION
PRINCIPLE The intersection of a line and a
plane is found in a view showing the plane as an
edge. The exact location of the point of
intersection is found on the line in a view
adjacent to this edge view.
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• Given oblique plane and piercing line.

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• Procedure Layout Horizontal line in F.

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• Procedure Project into H.

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• Procedure Layout TL in the Horizontal Plane.

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• Procedure Create an auxiliary hinge line
  perpendicular to the TL line in the horizontal
  plane.

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• Procedure Project the plane and the line into
  the auxiliary plane.

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• Procedure Offset the points related to the
  distance in the Frontal plane.

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• Procedure Layout the Edge View of ABC plane and
  line GH.

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Procedure Project the intersection of line GH
and plane ABC, in auxiliary plane 1, back into H
where it intersects the line (GH).
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Procedure Project the intersection point on
line GH (in H) to line GH in the Frontal plane.
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Procedure Place a Point on the intersection
points in H and F.
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Procedure These are the locations where line GH
pierces the plane ABC.
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Procedure Solve for Visibility. Project a line
from the crossover of a plane edge and the line.
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Procedure Interrogate only line GH and plane
edge AB.
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Procedure Notice that the projection line
encounters line GH in the Frontal plane before
plane edge AB. This means that line GH is above
(visible) in the Horizontal plane.
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Procedure Extend line GH in the Horizontal
plane over plane edge AB to the piercing point.
Trim line GH after the piercing point to the
opposite plane edge.
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Procedure Solve for visibility in the Frontal
plane.
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Procedure Project the crossover of line GH in
the Frontal view and an edge on plane ABC.
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Procedure The projection line strikes plane
edge BC first. This makes edge BC visible in the
Frontal plane.
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Procedure Trim line GH in the Frontal plane so
that it passes behind plane edge BC.
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Procedure This solution shows the piercing
point of the line and the plane and also the
proper visibility.
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Space Geometry Intersections of Lines and Planes
It is important to study space geometry to
improve your spatial comprehension. Line
Piercing A Plane Cutting Plane Method SOLUTION
PRINCIPLE If two given oblique planes are cut
by an auxiliary plane, the two lines of
intersection, or traces, it forms with the given
planes will intersect at a point common to both
planes, thus defining the line of intersection of
the given planes.
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• Given oblique plane and piercing line.

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• Use the given line as a cutting plane (on edge)
  which passes through the given plane.

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• Project the points of intersection from the edges
  of the plane into the adjacent view. Be sure to
  project to the correct edges.

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Connect the two intersecting points to form the
trace.
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• Notice that the trace crosses Line GFHF and
  locates the intersection of the line and the
  plane.

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• Project the point into the Horizontal plane.

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• Solve for visibility in the Frontal Plane.

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• Solve for visibility in the Horizontal Plane.

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Space Geometry Intersections of Lines and Planes
It is important to study space geometry to
improve your spatial comprehension. Line of
Intersection of Two Planes Edge View
Method SOLUTION PRINCIPLE If two intersecting
planes are viewed so that the edge view of one
plane cuts across the other plane, this line of
cut is the line of intersection of the two
planes. Note Two planes will intersect in a
straight line.
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Given Two intersecting oblique planes.
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Procedure Layout a TL line in the Horizontal
plane.
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Procedure Create a hinge line perpendicular to
the TL line.
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Procedure Project the Edge View of ABC and the
oblique view of EHG.
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Procedure The line of intersection is where the
Edge View of ABC cuts through EHG.
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Procedure Project the intersecting points back
into the adjacent planes.
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Procedure Project the intersection of plane ABC
and edge GH into the Horizontal view. Be sure to
project to the correct edge!
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Procedure Do the same for ABC and edge GE. Be
sure to project to the correct edge!
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Procedure Layout the line of intersection.
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Procedure Project the intersecting points into
the Frontal view.
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Procedure Layout the line of intersection in the
Front view.
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Procedure Solve for visibility.
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Procedure Project from edge crossovers.
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Procedure The projection line from crossover HE
and AC strikes AC in the Frontal view first.
Therefore, AC is visible in the Horizontal view.
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Procedure Draw the proper visibility of the two
planes in the Horizontal view.
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Procedure Project from a crossover in the
Frontal view to the first edge in the Horizontal
view. (BC and GE)
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Procedure BC is first so it is visible in the
Frontal view.
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Procedure Solve for visibility.
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Procedure Proper solution of line of
intersection and visibility.
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Space Geometry Intersections of Lines and Planes
It is important to study space geometry to
improve your spatial comprehension. Dihedral
Formed by Two Planes Edge View Method SOLUTION
PRINCIPLE The true angle between two planes is
found in a view showing the point view of their
line of intersection and both planes as edges
(EV).
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Given two oblique planes. PROBLEM Find the true
angle between ABC and EFG planes.
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• Graphically solve the line of intersection. Find
  one plane in EV and project back the line of cut
  to the other plane.

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• Since the solution is the PV of the line of
  intersection you will need to get the TL of the
  line of intersection first.

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• Layout a hinge line parallel to the line of
  intersection.

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• Project both planes into the auxiliary elevation
  plane.

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• Solve for visibility.

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• Orient a hinge line perpendicular to the TL line
  of intersection.

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• Project the planes into auxiliary plane 2. They
  will both be in Edge View since you have found
  the PV on the line of intersection. (The line of
  intersection lies on both planes).

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• Measure the acute angle.