Hello, everyone
We will start to learn the relative positions of the two lines
The content is about the straight lines
Objectives and requirements
We need to master the relative position of two straight lines
There are three relative positions of two lines in space
Parallel, intersect and cross
And also there is a special case in intersection and noncoplanar
That's vertical. Let's call it type four
vertical intersection or vertical noncoplanar
First, parallelism
If two lines are parallel in space
The projections of the two lines must also be parallel to each other
On the other hand, if the projection of two lines on the same plane is parallel to each other
So these two lines must also be parallel to each other in space
If two lines are parallel lines of the projection plane
When using two different view projections
We have to get a projection on a parallel plane
The horizontal lines for example
The picture on the left shows three parallel projections
two lines parallel to each other
And the side horizontal line in the right picture
Side projection is not parallel
So the conclusion is that AB is not parallel to the CD in space
Obviously it doesn't intersect either
So this is crossing
And the side horizontal line
The front and the horizontal projections are the same in both pictures
They all look parallel to each other
But the third projection
This is the side projection of the solid
it's parallel on the left and not parallel on the right
The conclusion is determined by the side projection
So when we have parallel lines of the plane
We have to get the projection on the parallel plane
Otherwise we would have to use the three-sided projection
So when the line is the general position line
Only need the two projections to be parallel to distinguish
Intersection
Two lines intersect in space
Their projections must also intersect in the same plane
And the projection of intersection points conforms to the projection law of points
In other words, the projection between points satisfies the following rules
The equality of length, height and width
If the projection intersects
And the intersection satisfies the projection law of the point
The conclusion is
Two lines in space intersect
If one of the two intersecting lines is a parallel line of the projection plane
the projection must include the projection of the parallel projection plane
In that case, they are parallel
For example, there is a side horizontal line AB
From the front and horizontal projection of the side horizontal line
It looks like they intersect at a point where the length is aligned
But when we see the side projection
that the height line is uneven and the width is not equal
So even though the three projections intersect
the intersection does not satisfy the projection law of the points
So it doesn't intersect
This is because it is not enough to by only one horizontal line
We have to get a side projection
We can't do that with just two projections
In this picture
It is the intersection of two general position lines
In the three plane projection they all intersect
And the intersection point conforms to the projection rule of the point
it intersects
If it's the general position line
You can actually get it directly by the projection on both sides
And what we find is that when we discriminant parallel and intersecting
If parallel lines are involved
We have to get a projection of a parallel plane
Otherwise we need three projections
And then the third case, this is the noncoplanar
If two lines in space are neither parallel nor intersect
it's called a cross line
Just like we saw AB and CD
They are neither parallel nor intersecting
How about their projection
It could either look like its intersecting
or it could look like it's parallel
It might look like they're all intersecting
But the intersection does not satisfy the projection law of the points
So a line like this, we can check that it's an intersecting line
We’re going to use the exclusion method to identify intersecting lines from the projection
That is, maybe our projection looks parallel
But the third projection is not parallel
So we can't draw parallel conclusions
And of course it doesn't intersect
It is neither parallel nor intersecting
So it has to be an intersection
Maybe all of projections are intersecting
But the intersection does not conform to the projection law of the point
So we can't draw an intersection
Not parallel, obviously
It's neither intersecting nor parallel, it's intersecting
So test the cross, we're using elimination
If we exclude both non-parallel and disjoint, just left with the intersection
And in a projection where two lines cross,
We're going to find some points that look like intersection
A point like this is called an overlapping point
3 and 4 in the front projection of this picture
1 and 2 in the horizontal projection
These are all overlapping points on the graph
1 and 2 are two different points
It's the overlapping points in the up and down directions of AB and CD
3 and 4 are the overlapping points in the front and rear directions of AB and CD
For the overlapping point, we're going to check its visibility
to test of the visibility of the overlapping point
We need to know which one point is in front of another
Which one point is up
From the front projection, 1 and 2 are up and down relationship
2 is below, so 2 has to be bracketed in the horizontal projection
3 and 4 in the front projection
From the horizontal projection, 3 and 4 are before and after relationship
3 is behind, So 3 is going to be in parentheses
This is the visibility test for overlapping points
Let’s do a couple of examples
please check the relative positions of the two lines
example one, what are the relative positions of the two lines
It's definitely not parallel. Is it intersecting?
As we say
If we have a horizontal line here
It's not enough just to project the front and the horizontal projection
We have to get a side projection
Is there a horizontal line here
Yes, there is. CD is the horizontal line
So we need to make a side projection
Okay, we have our navigation line
Then another 4 points are got by the law of equality
AB and CD do not intersect in the side projection
So AB does not intersect CD
It's clearly not parallel, so it has to be intersecting
Ok, same problem
If we don't want to draw a side projection
Can we judge the position of two lines?
Only by their frontal and horizontal projections
In this case, we can assume
That they intersect in this point
like1, that's the intersection
If it intersects, it's a point on AB
And also the point on CD
So, if 1 is a point on CD
What can be done? Thinking about the proportionality
Ok, the ratio from the front projection is same in the horizontal projection
If point 1 is a point on CD, its horizontal projection would be right there
Obviously, it's not the same as the intersection of the known conditions
According to the isometric property of 1 point
we can check that AB and CD are not actually intersecting
So the conclusion is intersection
Today, we talked about
1. Correlation between two lines
Including parallel, intersection, crossing and their special cases (vertical)
2. Determine the relative positions of two lines in space according to their projections
3. Determine the mutual position of the two straight lines according to the characteristics of the intersection of the two straight lines or the overlapping point
Ok, that's all