How To Determine The Distance From A Point To A Plane, Set The Following | The Science

The Science

# How to determine the distance from a point to a plane, set the following

One fairly common problems encountered in the initial courses of higher mathematics IHE, is to determine the distance from the arbitrary point to a plane. As a rule, the plane defined by the equation in one form or another. But there are other methods of determining planes. For example, the following.

### You will need:

- data traces plane; - Coordinates of the point.

## Instruction how to determine the distance from a point to a plane, set the following

Step 1:

If the initial conditions do not contain the coordinates of points, which are at the intersections of the plane with the axes of the coordinate system (tracks can be set in a similar way), define them. If you must define arbitrary pairs of points belonging to the planes XY, XZ, the YZ, make a straight equation (in the data plane), containing the respective segments. Solving the equation, get the coordinates of the intersections of the traces to the axes. Let it be the point A (X1, Y1, Z1), B (X2, Y2, Z2), C (X3, Y3, Z3).

Step 2:

Proceed to finding the equation of the plane defined by the original tracks. Make a determinant form: (X-X 1) (Y-Y1) (Z-Z1) (X2-X1) (Y2-Y1) (Z2 - Z1) (X3-X1) (Y3-Y1) (Z3 - Z1) where X1, X2, X3, Y1, Y2, Y3, Z1, Z2, Z3 - coordinate values ​​of points a, B, C, results in the previous step, X, Y and Z - the variables appearing in the resulting equation. Note that the elements of the bottom two rows of the matrix will eventually contain constant values.

Step 3:

Calculate the determinant. Equate to zero the resulting expression. This will be the equation of the plane. Please note that the determinant of the form (n11) (n12) (n13) (n21) (n22) (n23) (n31) (n32) (n33) can be calculated as: n11 * (n22 * n33 - n23 * n32) + n12 * (n21 * n33 - n23 * n31) + n13 * (n21 * n32 - n22 * n31). Since the value of n21, n22, n23, n31, n32, n33 - constant, and the first line contains the variables X, Y, Z, the resulting equation is of the form: AX + in Y + CZ + D = 0.

Step 4:

Determine the distance from the point to the plane defined by the original tracks. Let the coordinates of this point are the values ​​Xm, Ym, Zm. With these values, and the coefficients A, B, C, and a free term in the D, obtained in the previous step, use the formula of the form: P = | AHm + + VYm SZm + D | / √ (A² + V² + s²) for the calculation of the resulting distance.