Whenever two levels are in space that are not parallel to each other, then they intersect. However, this intersection is not only a point but always a straight line. You can find out how to calculate this straight line in this article.
Basic knowledge level
Like straight lines can also levels in space represented mathematically by vectors. A plane is uniquely spanned by two non-parallel vectors and a point. The screen you are currently using to read this article also represents a plane in three-dimensional space. Can you think of other objects that represent planes in everyday life?
Solution ideas: A piece of paper, a photo, a poster on the wall, the room door and much more. You can also abstract your body as a plane in the coordinate system.
Any plane in the coordinate system could look like this:
Figure 1: A turquoise plane in the three-dimensional coordinate system
Positional relationship of two planes
Two planes E and F can take exactly three different positions to each other. The levels can identical be, parallel to each other be or intersect in a straight line g. Graphically you can imagine this as follows:
identical
parallel
line of intersection
If this was too fast for you, then you should read the article on the positional relationship of two planes.
In the following you will learn how to calculate the third case – the Line of intersection of two planes. When two planes intersect, all points that lie on the line of intersection of the two planes lie in both the first and second plane. Otherwise, the planes have no other points in common.
Calculate the intersection of two planes
If one of the two levels in coordinate form and the other in parametric form is given, then the calculation is relatively simple. Below is an example with explanations. You can use this example to orient yourself, since the steps involved in the calculation are always the same.
First, it will be clarified again what is meant by a coordinate form or parameter form, since we need this knowledge in the following.
A plane in three-dimensional space is in coordinate equation or. coordinate formif it satisfies the following equation:
Here is the normal vector
and a,b,c and d are real numbers, i.e. .
A plane in three-dimensional space is in parametric equation or. parametric formif it satisfies the following equation:
Here r,s are real numbers, der support vector and the direction vectors the level.
Now you can look at an example. One level is given in coordinate form and the other level in parameter form.
Task 1
Determine the line of intersection of the planes E and F:
solution 1
1st step: First you determine the coordinates of F
2nd step: Now you plug the coordinates of F into the plane equation of E.
3rd step: Rearrange the obtained equation for one variable.
4th step: Substitute the variable in the parametric equation and solve
The line g that has now been set up is the line of intersection of the planes E and F.
We can visually check the solution of the task. The turquoise plane corresponds to plane E, the orange plane corresponds to plane F and the straight line g is drawn in dark blue. You can see here that the straight line lies in both plane E and plane F.
Figure 2: Graphic of the line of intersection of the two planes
Line of intersection of two planes Coordinate form
If the planes are given in coordinate form or you have brought the planes into coordinate form, then you will find below an example of how the calculation of the intersection line works. In this case, too, the calculation is relatively simple and short.
exercise 2
Determine the line of intersection of planes E and F:
solution 2
1st step: Set up a system of linear equations and simplify as far as you can.
2nd step: Subtract the first equation from the second equation.
3rd step: Rearrange the obtained equation for one variable.
4th step: Replace a variable with a new variable. We vote:
5th step: By substituting, you can calculate the remaining variable:
6th step: Now you put your values into the equation of the straight line you are looking for and simplify.
The line g that has now been set up is the line of intersection of the planes E and F.
Again, a visualization of what has just been calculated. Level E – shown here in turquoise – and level F shown in orange. The line of intersection g is the line drawn in dark blue.
Figure 3: Graphic of the line of intersection of the two planes
Line of intersection of two planes Parametric form
If both planes are specified in parameter form, then you have the option of first converting one of the two planes into a coordinate equation and proceeding as already explained. Otherwise you can find the line of intersection g of the two planes by equating both equations. We explain this again with an example:
task 3
Determine the line of intersection of planes E and F:
solution 3
1st step: Since both layers are in the same shape, you can equate them.
2nd step: Now all three equations are simplified!
3rd step: Then the linear system of equations with four unknowns has to be solved.
After all the unknowns have been placed on one side, the first and second lines are swapped. Then we multiply the first row by -1.
We add 1.5 times the second row to the third row.
Then we bring the coefficients of the preceding variables to 1.
From the last equation you can see that s is arbitrary. Then you can represent all equations in terms of s.
4th step: Now you can proceed as in the 1st example. The variable is substituted into the plane equation that replaced the other variables – in our case it is s.
The straight line g is the line of intersection of the planes E and F.
We can visually check the solution of the task. The turquoise plane corresponds to plane F, the orange plane corresponds to plane E and the line g is drawn in dark blue. You can see here that the straight line lies in both plane E and plane F.
Figure 4: Graphic of the line of intersection of the two planes
Intersection of two levels – the most important things at a glance
- Two planes in three-dimensional space can either be identical, parallel to each other or intersect.
- This cut is always a straight line.
- You can calculate the line of intersection in different ways.
- The principle of the calculation is close to solving a linear system of equations.