Intersection of straight plane: Calculate |

In this article, you’ll learn everything you need to know about calculating the intersection of a line and a plane. Whenever there is a straight line and a plane in space that are not parallel to each other, then they intersect. This cut can be either a Just be that lies in the plane or a single point that also puncture point is called. You can find out how to calculate this point in this article.

Straight and level basic knowledge

First of all, you should read this section carefully to clarify the basics and requirements. It is important to know how a straight line in three-dimensional space is mathematically correctly defined.

A line in three-dimensional space is in parametric form if it satisfies the following equation:

where is the support vector, the direction vector and any real number.

Then you should also know how a plane is defined in three-dimensional space.

A plane in three-dimensional space is in coordinate equation or coordinate form if this satisfies the following equation:

where a, b, c and d are real numbers.

A plane in three-dimensional space is in parametric equation or parametric form if it satisfies the following equation:

where r, s are real numbers, the support vector and , the direction vectors of the plane

Positional relationship straight plane intersection

There is three different caseshow a straight line can relate to a plane in three-dimensional space. A straight line can level intersect at one point, lie in the plane or parallel to the plane get lost. Graphically you can imagine this as follows:

intersection

Figure 1: The straight line intersects the plane

Just lies in the plane

Figure 2: Line lies in the plane

Line is not in the plane, but line is parallel to the plane

Figure 3: Line runs parallel to the plane and does not intersect it

If this was too fast for you, then you should read the article on the mutual position of straight lines and planes!

In this article we will look at the first case, so that the straight line intersects the plane and the so-called intersection point or point of intersection is to be calculated.

Calculate the intersection of the straight plane

Below you will find examples of how you can always proceed when calculating the intersection between a straight line and a plane in three-dimensional space. We also look at the different cases in which a plane can exist!

Intersection Straight plane Coordinate form

If the plane is given in coordinate form, then calculating the point of intersection is relatively easy. 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.

Task 1

Calculate the intersection of the line g with the plane E

solution

1st step: Set up the equation of the straight line as a linear system of equations based on their coordinates.

2nd step: Substitute the coordinates into the coordinate equation of the plane.

3rd step: Simplify the resulting equation and solve for lambda.

4th step: Now insert lambda into the equation of the straight line g and use it to determine the point of intersection S.

The point where the line g and the plane E intersect is S(-1|-1|2).

You can visualize the task like this. The light blue plane intersects the orange line at point S.

Figure 4: Intersection of line g with plane E

Intersection straight plane parametric form

If the plane is given in parametric form, then you have two options to calculate the point of intersection.

  1. Transform the plane in parameter form to coordinate form and do as above
  2. Calculate the point of intersection directly

First, let’s look at the second way. You will notice that the calculation steps are a bit more complex than in the first calculation. But if you don’t have any problems with that, you can always proceed like this when calculating the point of intersection.

exercise 2

Calculate the intersection of the line g with the plane E.

solution

1st step: Represent the two forms of representation in their coordinate form.

2nd step: Now the respective variables can be equated.

3rd step: Now you can plug s and r into the third equation.

4th step: You can simplify this equation and solve for lambda!

5th step: Now insert lambda into the equation of the straight line g and use it to determine the point of intersection S.

The point where the line g and the plane E intersect is S(-1|-1|2).

You may have noticed that this is the same intersection as the previous example, and it’s intentional! The two levels are one and the same.

Since the plane was the same as in the first example, the mapping and intersection is the same as above.

Figure 5: Intersection of line g with plane E

If you don’t want to remember the different paths or you find it easier to reshape the layers, then the following approach is for you!

To do this, you convert the plane in parameter form into a plane in coordinate form.

To do this, you first write the plane in the corresponding coordinates as a linear system of equations.

Now plug the first two equations into the second equation.

You then rearrange this equation after the number, so the variables are on one side.

The steps of calculating the point of intersection are now the same as in the first example!

Now you have learned everything about the intersection of a straight line and a plane! Take a look at the associated flashcards to intensify your knowledge directly!

Intersection straight plane – the most important thing

  • If the direction vector is not parallel to the plane, then the line intersects the plane.
  • Calculating the point of intersection corresponds to solving a system of linear equations.
  • The calculation is easiest when the plane is given in the form of coordinates.