When solving geometric problems in space, the problem of determining the distance between a plane and a point often arises. In some cases, this is necessary for a complete solution. This value can be calculated by finding the projection onto the point plane. Consider this issue in more detail in the article.
Equation to describe a plane
Before proceeding to the consideration of the question of how to find the projection of a point on a plane, you should familiarize yourself with the types of equations that define the latter in three-dimensional space. More details - below.
A general equation that defines all points that belong to a given plane is the following:
A * x + B * y + C * z + D = 0.
The first three coefficients are the coordinates of the vector, which is called the guide for the plane. It coincides with the normal for it, that is, it is perpendicular. This vector is denoted by n¯ (A; B; C). The free coefficient D is uniquely determined from the knowledge of the coordinates of any point belonging to the plane.
Further in the article we will use the written equation. It is required to find the projection of a point onto a plane.
The concept of the projection of a point and its calculation
Suppose that a certain point P (x 1 ; y 1 ; z 1 ) and a plane are given. It is defined by the equation in general form. If we draw a perpendicular line from P to a given plane, then it is obvious that it will intersect the latter at one certain point Q (x 2 ; y 2 ; z 2 ). Q is called the projection of P onto the plane under consideration. The length of the segment PQ is called the distance from the point P to the plane. Thus, the PQ itself is perpendicular to the plane.
How can I find the coordinates of the projection of a point on a plane? This is not difficult to do. First you need to make an equation of a line that will be perpendicular to the plane. The point P will belong to it. Since the normal vector n¯ (A; B; C) of this line must be parallel, the equation for it in the corresponding form can be written as follows:
(x; y; z) = (x 1 ; y 1 ; z 1 ) + λ * (A; B; C).
Where λ is a real number, which is usually called the parameter of the equation. By changing it, you can get any point on the line.
After the vector equation is written for the perpendicular plane of the line, it is necessary to find a common intersection point for the considered geometric objects. Its coordinates will be the projection P. Since they must satisfy both equalities (for the line and for the plane), the problem reduces to solving the corresponding system of linear equations.
The concept of projection is often used in the study of drawings. They depict lateral and horizontal projections of the part on the zy, zx, and xy planes.
Calculation of the distance from the plane to the point
As noted above, knowing the coordinates of the projection onto the point plane allows you to determine the distance between them. Using the notation introduced in the previous section, we find that the desired distance is equal to the length of the segment PQ. To calculate it, it suffices to find the coordinates of the vector PQ¯, and then calculate its modulus using the well-known formula. The final expression for the d distance between the P point and the plane takes the form:
d = | PQ¯ | = √ ((x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 + (z 2 - z 1 ) 2 ).
The obtained value of d is presented in units in which the current Cartesian coordinate system xyz is specified.
Task example
Suppose there is a point N (0; -2; 3) and a plane, which is described by the following equation:
2 * x - y + z + 4 = 0.
You should find the projection points on the plane and calculate the distance between them.
First of all, we will draw up the equation of a straight line that intersects the plane at an angle of 90 o . We have:
(x; y; z) = (0; -2; 3) + λ * (2; -1; 1).
Writing this equality explicitly, we arrive at the following system of equations:
x = 2 * λ;
y = -2 - λ;
z is λ + 3;
2 * x - y + z + 4 = 0.
Substituting the coordinate values from the first three equalities into the fourth, we obtain the value of λ, which determines the coordinates of the common point of the line and plane:
2 * (2 * λ) - (-2 - λ) + λ + 3 + 4 = 0 =>
6 * λ + 9 = 0 =>
λ = 9/6 = 3/2 = 1.5.
We substitute the found parameter into the equation of the line and find the coordinates of the projection of the starting point on the plane:
(x; y; z) = (0; -2; 3) + 1.5 * (2; -1; 1) = (3; -3.5; 4.5).
To calculate the distance between the geometric objects specified in the problem condition, we apply the formula for d:
d = √ ((3 - 0) 2 + (-3.5 + 2) 2 + (4.5 - 3) 2 ) = 3.674.
In this problem, we showed how to find the projection of a point on an arbitrary plane and how to calculate the distance between them.