Line of intersection of two planes calculator

The Intersection Calculator is used to calculate the intersection point between two lines. The two lines are the linear equations with degree 1. The calculator computes the x and y coordinates of the intersecting point in a 2-D plane.

The calculator takes the linear equations for the two lines as input and outputs the intersecting point or the solution of both lines. The two equations are the function of x and y.

If the variable z is entered in one or both of the two equations, the calculator computes only the x-coordinate of the intersecting point and gives another equation which is a function of y and z.

The three-variable equation requires three equations to compute the complete coordinates of the intersection point. The two equations are not enough for the calculator to compute the numerical values of x, y, and z coordinates of the intersection point.

So, the calculator gives the numerical values for the intersection point only for two-variable equations.

What Is an Intersection Calculator?

The Intersection Calculator is an online tool that is used to calculate the intersection point of two linear equations or lines in a 2-D plane.

The intersection point is the point where the two lines meet or cross each other, giving the x and y coordinates.

So the intersecting point is the common point (x,y) between the two lines. At this point, the x-coordinate and y-coordinate for both the lines are the same.

How To Use the Intersection Calculator

The Intersection Calculator can be used by following the steps given below: 

Step 1

First, the user enters the first linear equation of the two equations in the input block against the title, Intersection of. The linear equation is a two-variable equation.

The calculator shows the first equation by default as follows:

y = 3x + 2 

The default variables used are x and y. The equation is a function of y in terms of x.

The two variables can be any alphabet such as (a,b) depending upon the user’s requirement.

Step 2

Enter the second linear equation in the second input tab of the Intersection Calculator. It is entered in the block titled against and. The user should use the same two variables as used for the first linear equation for correct results.

The second linear equation set by default by the calculator is:

y = 2x – 1 

If a third variable is entered in any of the two equations, the calculator gives the value for a single coordinate such as x and gives another equation in the result window.

This calculator does not support the 3-D system.

Step 3

After entering both the equations, the user should press Submit button for the calculator to compute the intersection point. If the user forgets to enter one of the two equations, the calculator displays Not a valid input; please try again.

Output

The calculator processes the two equations and shows the output in the two windows.

Input Interpretation

This window shows the interpreted input by the calculator. It shows the two equations for which the intersection point is required. This helps the user to confirm the input for correct results.

Result

This window shows the x and y coordinates of the intersection point of the two lines. The calculator computes the intersection point by the substitution and elimination method.

The intersection point is the point common in both the lines. It is also known as the solution for both the lines as both the equations satisfy the intersection point.

For the default equations y = 3x + 2 and y = 2x – 1 set by the calculator, the intersection point displayed in the result’s window is as follows:

\[ x = – \ 3 \]

\[ y = – \ 7 \]

The Result window also shows the option of viewing a detailed solution of the problem labeled as Need a step-by-step solution for this problem? By pressing on it, the user can acquire all the mathematical steps needed to calculate the displayed result by the calculator.

Solved Examples

Here are some solved examples for the Intersection Calculator.

Example 1

For the two linear equations,

x + y = 3

3x – 2y = 4 

Calculate the point of intersection between the two lines.

Solution

The user enters the two linear equations in the input window one by one. The user presses “Submit” for the calculator to compute the intersection point.

The calculator displays “intersections” with the two equations in the input interpretation window. The equations are the same as entered by the user.

In the Result window, it shows the x and y coordinates for the intersection point of the two lines. The calculator uses the elimination and substitution method and computes the result as follows:

x = 2 

 y = 1 

Hence, the point of intersection for the linear equations x + y = 3 and 3x -2y = 4 is (2,1).

Example 2

Compute the intersecting point of the two linear equations given as:

4x – 3y = 1 

x – 2y = – 6 

Solution

At first, the user enters the equations for the two lines for which the intersection point is required. To get the result, the user submits the input equations and the calculator starts computing the x and y coordinates for the point of intersection.

The input interpretation window shows the input equations assumed by the calculator. The user can verify the input equations from this window.

The Result window shows the intersection point in terms of two variables x and y. Both the equations satisfy the result given by the calculator. The (x,y) coordinates of the intersection point are the same for both equations.

The result displayed by the calculator for the above linear equations is as follows:

x = 4 

 y = 5 

So the intersection point for the two line 4x – 3y = 1 and x – 2y = – 6 is (4,5).

Math Calculator List

The Intersection Calculator option allows you to quickly and accurately calculate:

• The line of intersection of two planes
• The plane which passes through two lines

Line From Two Planes

To calculate the intersection of two planes:

1. Select Tools > Intersection Calculator > Line from Two Planes
2. On the stereonet graphically enter the location of two planes.
3. You will see the Intersection Calculator dialog, with the orientation coordinates of the graphically entered planes, and the resulting intersection line. If necessary you can edit the plane orientations in the dialog.
4. Click OK to add the planes and intersection point to the stereonet.

Plane from Two Lines

To calculate the plane which passes through two lines:

1. Select Tools > Intersection Calculator > Plane from Two Lines
2. On the stereonet graphically enter the location of two lines.
3. You will see the Intersection Calculator dialog, with the trend/plunge coordinates of the graphically entered lines, and the resulting intersection plane. If necessary you can edit the line orientations in the dialog.
4. Click OK to add the lines and intersection plane to the stereonet.

Application: Convert apparent dip and dip direction at two points into true dip and dip direction:

If you locate two points on the stereonet using the Trend/Plunge Convention for the apparent dip and dip direction, then use the Add User Plane option to graphically fit a great circle through these two points, and this will give you the plane with true dip and dip direction.

Results

The results of the intersection calculation are displayed as a popup tooltip if you hover the mouse over the intersection line or plane.

While this problem has a great textbook answer, as @walcher explained, I don't think it's very elegant. This is because, the solution depends on picking an arbitrary point, which lacks geometric intuition. Ideally, we'd like this point to have some meaning, such as being close to the planes, or the line or etc.

For that, I'd like to remind you of a solution by John Krumm, which remains unnoticed by many. Let $\mathbf{p}=\{p_x,p_y,p_z\}$ and $\mathbf{n}=\{n_x,n_y,n_z\}$ compose the plane $\mathbf{P}=\{\mathbf{n},\mathbf{p}\}$. Let there be two planes $P_1$ and $P_2$, for which we'd like to compute the intersecting line $\mathbf{l}$. It's trivial to compute the direction as the cross-product: $$\mathbf{l}_d=\mathbf{n}_1 \times \mathbf{n}_2$$

If we additionally desire that the resulting point $\mathbf{p}$ is as close to the chosen point $\mathbf{p}_0$ as possible, we could write a distance : $$\lVert \mathbf{p}-\mathbf{p}_0 \rVert = (p_x-p_{0x})^2 + (p_y-p_{0y})^2 + (p_z-p_{0z})^2$$ Incorporating the other points in the similar fashion, and writing this constraint using Lagrange multipliers into an objective function results in: $$J=\lVert \mathbf{p}-\mathbf{p}_0 \rVert+\lambda(\mathbf{p}-\mathbf{p}_1)^2 + \mu(\mathbf{p}-\mathbf{p}_2)^2$$

Using the standard Lagrange framework (omitting the details), one establishes a nice matrix, in the form: $$ \mathbf{M}= \left[ {\begin{array}{ccc} 2 & 0 & 0 & n_{1x} & n_{2x}\\ 0 & 2 & 0 & n_{1y} & n_{2y}\\ 0 & 0 & 2 & n_{1z} & n_{2z} \\ n_{1x} & n_{1y} & n_{1z} & 0 & 0\\n_{2x} & n_{2y} & n_{2z} & 0 & 0 \end{array} } \right] $$ This matrix can now be used in a system of linear equations to solve for the unknown point, $\mathbf{p}$, as well as the Lagrange multipliers, $\{\lambda, \mu\}$.: $$ \mathbf{M}\left[ {\begin{array}{c} p_x \\ p_y \\ p_z \\ \lambda \\ \mu \end{array} } \right] = \left[ {\begin{array}{c} 2p_{0x} \\ 2p_{0y} \\ 2p_{0z} \\ \mathbf{p}_1 \cdot \mathbf{n}_1 \\ \mathbf{p}_2 \cdot \mathbf{n}_2 \end{array} } \right] $$ While the multipliers are not of particular interest they would be interesting for understanding configuration of points, or for different parameterizations.

I think this is a pretty neat approach giving a nice and simple method, with a geometrically interpretable results. I post the MATLAB code at my blog.