Line Segment Intersection Calculator
Easily find the intersection point of two line segments in a 2D plane with our Line Segment Intersection Calculator. Enter the coordinates and see the result and visual representation.
Calculate Intersection
Denominator (D): –
Parameter t: –
Parameter u: –
Input Coordinates
| Line | Point A (x, y) | Point B (x, y) |
|---|---|---|
| Line 1 | (10, 10) | (100, 100) |
| Line 2 | (10, 100) | (100, 10) |
Visual Representation
What is a Line Segment Intersection Calculator?
A Line Segment Intersection Calculator is a tool used to determine if two line segments in a two-dimensional plane intersect, and if they do, to find the coordinates of their intersection point. Line segments are finite portions of lines defined by two endpoints. Unlike lines that extend infinitely, segments have a start and an end, which makes the intersection calculation more specific – the intersection must occur within the bounds of both segments.
This calculator is useful for various fields, including computer graphics, game development, geographic information systems (GIS), physics simulations, and general geometry problems. It helps visualize and mathematically locate the point where two defined segments cross. If the segments don't cross but their extending lines do, the Line Segment Intersection Calculator will indicate no intersection between the segments themselves.
Common misconceptions include thinking that if the lines containing the segments intersect, the segments must also intersect. However, the segments only intersect if the intersection point lies between the endpoints of *both* segments. Our Line Segment Intersection Calculator specifically checks this condition.
Line Segment Intersection Calculator Formula and Mathematical Explanation
To find the intersection of two line segments, we represent each segment using parametric equations. Let the first line segment (L1) be defined by points P1(x1, y1) and P2(x2, y2), and the second line segment (L2) by points P3(x3, y3) and P4(x4, y4).
Parametric equation for L1:
x = x1 + t * (x2 – x1)
y = y1 + t * (y2 – y1) (where 0 ≤ t ≤ 1)
Parametric equation for L2:
x = x3 + u * (x4 – x3)
y = y3 + u * (y4 – y3) (where 0 ≤ u ≤ 1)
For the segments to intersect, there must be values of t and u between 0 and 1 (inclusive) such that the (x, y) coordinates are the same:
x1 + t * (x2 – x1) = x3 + u * (x4 – x3)
y1 + t * (y2 – y1) = y3 + u * (y4 – y3)
Rearranging these equations, we get a system of two linear equations with two variables, t and u:
t * (x2 – x1) – u * (x4 – x3) = x3 – x1
t * (y2 – y1) – u * (y4 – y3) = y3 – y1
We can solve for t and u. Let:
Dx = x2 – x1, Dy = y2 – y1
Dx' = x4 – x3, Dy' = y4 – y3
rx = x3 – x1, ry = y3 – y1
t * Dx – u * Dx' = rx
t * Dy – u * Dy' = ry
The denominator for solving this system is D = Dx * (-Dy') – Dy * (-Dx') = Dx * Dy' – Dy * Dx' = (x2 – x1)(y4 – y3) – (y2 – y1)(x4 – x3).
If D (Denominator) is zero, the lines are parallel or coincident. If D is non-zero, we can find t and u:
t = (rx * (-Dy') – ry * (-Dx')) / D = ((x3 – x1)(y4 – y3) – (y3 – y1)(x4 – x3)) / D
u = (Dx * ry – Dy * rx) / D = ((x2 – x1)(y3 – y1) – (y2 – y1)(x3 – x1)) / D
The segments intersect if and only if 0 ≤ t ≤ 1 and 0 ≤ u ≤ 1. If these conditions are met, the intersection point (Ix, Iy) is:
Ix = x1 + t * (x2 – x1)
Iy = y1 + t * (y2 – y1)
If D is very close to zero, the lines are nearly parallel, and we treat them as parallel to avoid division by a small number. If D is zero, we check for collinearity and overlap.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| (x1, y1), (x2, y2) | Coordinates of endpoints for Line 1 | Units (e.g., px, cm) | Any real numbers |
| (x3, y3), (x4, y4) | Coordinates of endpoints for Line 2 | Units (e.g., px, cm) | Any real numbers |
| D | Denominator in the solution for t and u | Unit² | Any real number |
| t, u | Parameters along the line segments | Dimensionless | 0 to 1 for intersection |
| (Ix, Iy) | Coordinates of the intersection point | Units (e.g., px, cm) | Real numbers within the segment range |
Practical Examples (Real-World Use Cases)
Let's see how the Line Segment Intersection Calculator works with examples.
Example 1: Clear Intersection
Suppose Line 1 goes from (10, 10) to (100, 100) and Line 2 goes from (10, 100) to (100, 10).
Inputs: x1=10, y1=10, x2=100, y2=100, x3=10, y3=100, x4=100, y4=10
Using the Line Segment Intersection Calculator, we find:
Denominator (D) = (100-10)(10-100) – (100-10)(100-10) = 90*(-90) – 90*90 = -8100 – 8100 = -16200
t = ((10-10)(10-100) – (100-10)(100-10)) / -16200 = (0 – 90*90) / -16200 = -8100 / -16200 = 0.5
u = ((100-10)(100-10) – (100-10)(10-10)) / -16200 = (90*90 – 0) / -16200 = 8100 / -16200 = -0.5 (Error in manual u calc, let's re-use formula)
u = -((100-10)(100-10) – (100-10)(10-10)) / -16200 using numerator u = -((x2-x1)(y3-y1) – (y2-y1)(x3-x1))
Numerator u = -((90)(90) – (90)(0)) = -8100
u = -8100 / -16200 = 0.5
Since 0 ≤ 0.5 ≤ 1 and 0 ≤ 0.5 ≤ 1, the segments intersect.
Intersection x = 10 + 0.5 * (100 – 10) = 10 + 0.5 * 90 = 10 + 45 = 55
Intersection y = 10 + 0.5 * (100 – 10) = 10 + 0.5 * 90 = 10 + 45 = 55
Result: Intersection at (55, 55).
Example 2: No Intersection Between Segments
Suppose Line 1 goes from (0, 0) to (50, 50) and Line 2 goes from (60, 0) to (100, 40).
Inputs: x1=0, y1=0, x2=50, y2=50, x3=60, y3=0, x4=100, y4=40
The lines containing these segments might intersect, but let's check the segments using the Line Segment Intersection Calculator:
D = (50-0)(40-0) – (50-0)(100-60) = 50*40 – 50*40 = 2000 – 2000 = 0
Wait, D = 50*40 – 50*40 = 0? No. (x2-x1)(y4-y3) – (y2-y1)(x4-x3) = 50*40 – 50*40. That's for parallel. My x4,y4 are different.
D = (50)(40) – (50)(40) = 2000 – 2000 = 0. They are parallel? No. x4=100, y4=40, x3=60, y3=0 -> x4-x3 = 40, y4-y3=40. x2-x1=50, y2-y1=50.
D = 50*40 – 50*40 = 0. The lines have the same slope (1). They are parallel.
The Line Segment Intersection Calculator would report "Lines are parallel". If they were collinear, it would check for overlap.
Example 3: Lines Intersect, Segments Don't
Line 1: (0,0) to (5,5). Line 2: (0,10) to (10,0).
x1=0, y1=0, x2=5, y2=5, x3=0, y3=10, x4=10, y4=0
D = 5*(-10) – 5*10 = -50 – 50 = -100
t = (0*(-10) – 10*10) / -100 = -100 / -100 = 1
u = – (5*10 – 5*0) / -100 = -50 / -100 = 0.5
t=1 is at the edge, u=0.5 is within range. Intersection at x=5, y=5, which is an endpoint of L1. So, they intersect at (5,5).
Line 1: (0,0) to (4,4). Line 2: (0,10) to (10,0).
x1=0, y1=0, x2=4, y2=4, x3=0, y3=10, x4=10, y4=0
D = 4*(-10) – 4*10 = -40 – 40 = -80
t = (0*(-10) – 10*10) / -80 = -100 / -80 = 1.25
Since t > 1, the intersection point lies on the line of L1 but outside the segment. Result: No intersection within segments.
How to Use This Line Segment Intersection Calculator
Using our Line Segment Intersection Calculator is straightforward:
- Enter Coordinates for Line 1: Input the x and y coordinates for the starting point (x1, y1) and ending point (x2, y2) of the first line segment.
- Enter Coordinates for Line 2: Input the x and y coordinates for the starting point (x3, y3) and ending point (x4, y4) of the second line segment.
- Calculate: The calculator automatically updates as you type, or you can click the "Calculate" button.
- View Results: The primary result will tell you if the segments intersect, are parallel, collinear, or don't intersect within their bounds. If they intersect, the coordinates of the intersection point (Ix, Iy) are displayed.
- Intermediate Values: You can see the calculated Denominator (D), and parameters t and u. These help understand where the intersection falls relative to the segments' lengths.
- Visual Representation: The canvas shows the two line segments and the intersection point (if it exists within the segments and the auto-scaled view).
- Reset: Click "Reset" to return to the default values.
- Copy Results: Click "Copy Results" to copy the main result, intermediate values, and input coordinates to your clipboard.
The visualization helps confirm the numerical result. If t or u are outside the [0, 1] range, the intersection point shown on the canvas (if the lines intersect) might be outside one or both drawn segments, but the primary result will correctly state "No intersection within segments". We only draw the intersection point if 0<=t<=1 and 0<=u<=1.
Key Factors That Affect Line Segment Intersection Results
Several factors determine whether and where two line segments intersect:
- Coordinates of Endpoints: The x and y values of the four endpoints (x1,y1, x2,y2, x3,y3, x4,y4) define the position and orientation of the segments.
- Slopes of the Lines: If the slopes are identical, the lines are parallel (D=0). If they are also collinear, they might overlap. The Line Segment Intersection Calculator checks this.
- Relative Positions: Even if the infinite lines intersect, the segments themselves might be far from the intersection point.
- Segment Lengths: The lengths of the segments, determined by their endpoints, constrain the region where an intersection is possible (0≤t≤1 and 0≤u≤1).
- Collinearity: If all four points lie on the same line (collinear), the segments might not intersect, touch at an endpoint, or overlap over a range. The calculator checks for overlap if collinear.
- Numerical Precision: When the denominator D is very close to zero, we treat the lines as parallel due to floating-point precision limits. Our Line Segment Intersection Calculator uses a small epsilon for this comparison.
Frequently Asked Questions (FAQ)
Q1: What does it mean if the Line Segment Intersection Calculator says "No intersection within segments"?
A1: It means that although the infinite lines containing the segments might cross, the point where they cross does not fall within the boundaries defined by the endpoints of *both* segments (i.e., either t or u, or both, are outside the 0-1 range).
Q2: What if the calculator shows "Lines are parallel"?
A2: This means the two line segments have the same slope and will never intersect, no matter how far they are extended, unless they are collinear.
Q3: What does "Segments are collinear and overlapping" mean?
A3: This indicates that all four endpoints lie on the same straight line, and the segments share more than just a single endpoint; they overlap over a portion of their lengths.
Q4: Can the Line Segment Intersection Calculator handle vertical lines?
A4: Yes, the parametric form used by the Line Segment Intersection Calculator works correctly even for vertical lines (where x1=x2 or x3=x4).
Q5: What are the parameters 't' and 'u'?
A5: 't' and 'u' are parameters that indicate how far along each line segment the intersection point lies. A value of 0 corresponds to the start point (P1 or P3), and 1 corresponds to the end point (P2 or P4). For segments to intersect, both 't' and 'u' must be between 0 and 1 inclusive.
Q6: What if the segments just touch at an endpoint?
A6: If the intersection occurs at an endpoint of one or both segments, the calculator will still report an intersection and give the coordinates of that endpoint, as long as t and u are within [0,1].
Q7: How accurate is the Line Segment Intersection Calculator?
A7: The calculator uses standard floating-point arithmetic. It's accurate for most practical purposes, but very small differences due to precision might occur, especially when lines are nearly parallel.
Q8: Can I use negative coordinates?
A8: Yes, the Line Segment Intersection Calculator accepts negative and decimal values for coordinates.