Hello, everyone
We are going to learn the relative relation between straight lines and planes
especially intersection relation between
the general-position line and the general-position plane
Objectives and requirements
Master the method of solving intersection point when the general-position line intersects with the general-position plane
Master the method of determining the visibility using the points which have overlapping projections
For the intersection of the general-position line and the general-position plane
since both are in general-position, the line does not have accumulative projection, nor does the plane
We learned from the last lesson about the special-position situation that
the accumulative projection can help us directly obtain
one of projection of the intersection point or the intersection line
Herein there is no accumulative projection. How to solve it
We will introduce an auxiliary plane
Make a plane R including DE, and the intersection line of the plane R and ABC is MN
The intersection point of MN and DE is exactly the intersection point of DE and ABC
This is why we introduce the auxiliary plane R.
Turn the problem of the intersection of line and plane into the problem of the intersection between lines, which is easier to be solved
However, the introduction of the plane R requires solving the intersection line between planes
Is the intersection line easy to be solved
It depends on the plane we introduce
If the auxiliary plane including the line is perpendicular to the projection plane, it is relatively easy to solve the intersection line
In the last lesson, we have learned that the intersection line of the special-position plane and general-position plane can be obtained directly
So the auxiliary plane we made must be in special-position
Since the special-position plane including the general-position line can only be the plane perpendicular to the projection plane
the plane R including DE is perpendicular to the projection plane
Step two, We get the intersection line MN
Step three, the intersection point K of the line and the plane is the intersection point K of MN and DE, which has become the intersection point of two lines
Step four, determine the visibility
In the followings we will follow these four steps to finish the drawing
Solve the intersection point of the general-position line FE and the general-position plane ABC
Step one, make a plane perpendicular to the projection plane and including FE
What kind of the plane? Both planes perpendicular to H-plane and V-plane are acceptable
We make the H-perpendicular plane
Using the representation of trace line, it is represented by its horizontal trace line, QH
Step one is done
Step two, solve the intersection line I II of the H-perpendicular plane Q and the plane ABC
For the intersection line of the special-position plane and the general-position plane
its projection in H-plane 12 can be obtained directly
and its projection in V-plane, which is the common line of the two planes, can be obtained in ABC
The point I is on AB, and the point II is on AC
According to the principle of lengthwise alignment, the projections of I and II in V-plane are obtained
We have completed the solution of the intersection line, that is, step two
Step three, solve the intersection point of the intersection line and the known straight line EF
which is the intersection point of the line and the plane, K
In step three, we can figure out the intersection point
Step four, determine the visibility,
by using the points which have overlapping projections
Herein the plane does not have accumulative projection, nor does the line
How to use the points with overlapping projections to determine the visibility
In both V-plane and H-plane, we have found two points with overlapping projections
Any one of them can be used to determine the visibility in the corresponding projection plane
We pick out the two points I and II
These two points are considered to be points on FE and BC, which have overlapping projections
The overlapping projections of the points on FE and BC, will be used to judge their front-back relation
According to the principle of lengthwise alignment, we find that the point on BC is in front, and the point on FE is at the back
The point 2’ is in a bracket
BC is in front, and FE is at the back,
so the overlapping part of FK is drawn as a dashed line
With the intersection point as boundary, this part of KE is drawn as a solid line
Similarly, for the projections in H-plane,
we pick one of the two points with overlapping projections
We pick out the two points III and IV
which are the points on AC and FE, and have overlapping projections in H-plane
For AC and FE, we need to judge their above-below relation
According to the principle of lengthwise alignment, we find that the point on AC is above, and the point on FE is below
The point 3 is in a bracket
AC is above, and FE is below, so the overlapping part of KE is drawn as a dashed line
while this part of KF is drawn as a solid line
Using the points with overlapping projections, we have finished the determination of the visibility
In this lesson, we learned the steps to draw the intersection point of the general-position line and general-position plane
It is divided into four steps
Step one, make an auxiliary plane perpendicular to the projection plane and including the line
Step two, solve the intersection line of the auxiliary plane and the known plane
Step three, solve the intersection point of the intersection line and the known line
Step four, determine the visibility
OK, that’s the end of this lesson.