BIZOMAT MAKEOS · Geometry
Straight Lines: Intersection, Angle and Distance
Where do these two lines cross, at what angle, and how far is a point from the first line?
Analytical geometry of straight lines each defined by two points: slopes and intercepts, the intersection point, the angle between the lines, and the perpendicular distance and foot of the perpendicular from a point to the first line.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Line 1 | ||
| x₁ | mm | First point on line 1. |
| y₁ | mm | First point on line 1. |
| x₂ | mm | Second point on line 1. |
| y₂ | mm | Second point on line 1. |
| Line 2 | ||
| x₃ | mm | First point on line 2. |
| y₃ | mm | First point on line 2. |
| x₄ | mm | Second point on line 2. |
| y₄ | mm | Second point on line 2. |
| Point | ||
| Point x | mm | Point whose distance from line 1 is wanted. |
| Point y | mm | Point whose distance from line 1 is wanted. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Line 1
What this means: x₁
- What it is
- The horizontal coordinate of the first point that defines line 1, measured from your chosen origin.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- All eight line coordinates and the point must use the same origin and axes.
What this means: y₁
- What it is
- The vertical coordinate of the first point that defines line 1, measured from the same origin as the x values.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Coordinates below or left of the origin are negative.
What this means: x₂
- What it is
- The horizontal coordinate of the second point on line 1. The line runs through both points and extends beyond them.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The second point must differ from the first, or line 1 is undefined.
What this means: y₂
- What it is
- The vertical coordinate of the second point on line 1, which with the first point fixes the line’s slope.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A vertical line has equal x values; that is allowed and reported as vertical.
Line 2
What this means: x₃
- What it is
- The horizontal coordinate of the first point that defines line 2.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- If line 2 is parallel to line 1 there is no intersection; the tool says so.
What this means: y₃
- What it is
- The vertical coordinate of the first point that defines line 2.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same sign convention as line 1: up positive.
What this means: x₄
- What it is
- The horizontal coordinate of the second point on line 2, which with the first fixes the direction of line 2.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The two points on line 2 must not coincide.
What this means: y₄
- What it is
- The vertical coordinate of the second point on line 2.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The intersection may lie outside the segments you drew; the lines are treated as endless.
Point
What this means: Point x
- What it is
- The horizontal coordinate of a separate point whose perpendicular distance from line 1, and the foot of that perpendicular, are wanted.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Distance is measured to line 1 only, not to line 2.
What this means: Point y
- What it is
- The vertical coordinate of the separate point whose perpendicular distance from line 1 is wanted.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A point on line 1 itself gives a distance of zero.
Method and formulas
- Each line in the form A x + B y = C from its two points; the intersection solves both (none when parallel).
- Angle between lines from their direction vectors.
- Distance from a point = |A x₀ + B y₀ − C| ÷ √(A² + B²).
- Example: the lines cross at (53.333, 26.667) at 71.565°.
Limitations
- Plane geometry; each line needs two distinct points.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find a theoretical sharp corner from two drawn edges
- Check a CNC contour intersection
- Measure the distance of a hole from an inclined edge
Who it is for
- CNC programmers
- Toolmakers
- Design engineers
What you get
- Intersection point
- Angle between the lines
- Distance from the point to line 1
Interpreting the result
- Use the foot of the perpendicular as the gauging point when inspecting an edge position.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 40–46. Topic reference; no source tables or recommended settings are reproduced.
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Open the Straight Lines: Intersection, Angle and Distance calculator