In heavy civil and mining environments, surveyors and engineers constantly operate where the natural world collides with human design. One of the most complex spatial problems to solve in the field is figuring out exactly where a naturally dipping geological feature (like a coal seam, rock fault, or ore vein) daylights against a designed engineering surface (like a steep highwall, batter, or pit ramp).
Mathematically, this requires calculating the Line of Intersection of Two Planes. Doing this by hand involves complex 3D vector cross-products—math that is incredibly easy to mess up while sitting in a dusty truck. To eliminate the guesswork, we built the 3D Intersection of Two Planes Calculator at sitemath.net.
Intersection of Two Planes
Surface / Plane 1
Surface / Plane 2
Set-Out Extents
Define the running distances from the calculated Base Point to generate your Start and End set-out coordinates.
Intersection Line Geometry
Vector Orientation
Calculated Base Point
Set-Out Coordinates
Start Point
End Point
Interactive 3D Solution
Sample Data
Scenario: Geological Seam intersecting a Highwall Batter
Plane 1: The Geological Seam
This plane dips gently to the East and slightly to the North.
Point 1.1: E: 5000.000, N: 10000.000, Z: 150.000
Point 1.2: E: 5100.000, N: 10000.000, Z: 140.000 (Drops 10m over 100m East)
Point 1.3: E: 5000.000, N: 10100.000, Z: 145.000 (Drops 5m over 100m North)
Plane 2: The Design Batter
This plane represents a steep, flat-faced wall climbing upward toward the East.
Point 2.1: E: 5000.000, N: 10000.000, Z: 100.000
Point 2.2: E: 5100.000, N: 10000.000, Z: 150.000 (Rises 50m over 100m East)
Point 2.3: E: 5000.000, N: 10100.000, Z: 100.000 (Flat along the North axis)Versatile 3D Inputs for the Field
Because surveyors get data from different disciplines, a standard “3-point” calculator isn’t enough. Our tool allows you to define each of your two planes using the format you actually have on hand:
- 3 Known Points: Perfect for field surveyors who have just shot three random points on a rock face or concrete slab.
- Point + Dip Direction & Angle: The gold standard for geologists. If the geo hands you a strike and dip for a fault line, you can enter it directly alongside your design batter.
- Point + Normal Vector (i, j, k): For the structural engineers and CAD technicians working strictly in mathematical vectors.
From Infinite Math to Actionable Pegs. The problem with planar geometry is that planes are mathematically infinite, which means the line where they intersect is also infinite. An infinite line doesn’t help a surveyor who needs to pound a wooden peg into the dirt.
To solve this, our calculator features a Set-Out Extents engine. You simply tell the calculator how far along the line you want to peg (e.g., from 0.000m to 100.000m). The algorithm instantly calculates the exact Start (X,Y,Z) and End (X,Y,Z) coordinates so you can upload them straight to your datalogger and stake the line.
Instant Visual QA/QC Trusting blind math can be dangerous. That is why every calculation instantly generates an Interactive 3D Plot. You can rotate, pan, and zoom around your solution to visually verify that your geological seam is slicing through your highwall exactly as expected before you ever pick up a prism pole.
Frequently Asked Questions (FAQs)
What is the line of intersection of two planes? In 3D geometry, when two flat, non-parallel surfaces (planes) meet, they slice through each other along a single, perfectly straight continuous line. The vector of this line is calculated using the cross-product of the normal vectors of the two original planes.
Can two planes not have a line of intersection? Yes. If two planes have the exact same dip and direction (meaning they are perfectly parallel), they will never intersect. Our calculator automatically checks for this and will alert you if your inputted planes are parallel.
How does Dip and Dip Direction work in this calculator? Dip Direction (Azimuth) is the compass bearing (0-360°) that the plane is sloping towards. Dip Angle is the steepness of that slope measured downward from the horizontal (0-90°). If a geologist gives you a “Strike,” simply add 90 degrees to find the Dip Direction (assuming the Right-Hand Rule).
Why do I need a Start and End distance? Because the mathematical line where two planes meet goes on forever in both directions. By inputting a Start and End distance, the calculator generates hard, physical XYZ coordinates along that vector so a surveyor can physically stake the line in the real world.