Overview
Point-line distance measures the perpendicular separation between a point and an infinite line, usually by projecting the point onto the line or using a determinant formula.
FoundationSource checked
Point-Line Distance
Computes the shortest distance from a point to a line.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Geometry
- Difficulty
- Beginner
- Family
- Distance Query
- Tier
- Foundation
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 4
- Code example
- Ready
- Visualization
- Ready
Core Concepts
perpendicular distanceprojectionline equationdegenerate line
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(1)
Average
O(1)
Worst
O(1)
Space
O(1)
When to Use
- Use it for geometric proximity to an infinite line.
- Use it inside collision, fitting, or distance-to-feature calculations.
- Use segment-distance formulas instead when endpoints matter.
When Not to Use
- Avoid it for line segments unless the projection is clamped to the segment.
- Avoid it with a zero-length line definition.
- Avoid exact comparisons without tolerance in floating-point geometry.
What It Does
- Measures the perpendicular distance from a point to an infinite line
- Uses perpendicular distance and cross product to guide the work.
- Produces geometric relation, simplified shape, or constructed geometry.
Use Cases
- Collision and intersection tests
- Shape analysis
- Geometry processing
Inputs
- Points, segments, polygons, or meshes
- Query shape or tolerance
Outputs
- Geometric relation, simplified shape, or constructed geometry
Key Formulas (1)
Point-line distance
distance = |cross(b − a, p − a)| / ||b − a||
distance = |cross(b - a, p - a)| / ||b - a||Measures perpendicular distance from a point to a line.
Pseudocode
- 1.Prepare the inputs and state needed for Point-Line Distance.
- 2.Apply the perpendicular distance step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return geometric relation, simplified shape, or constructed geometry.
Canonical Example
- Input
- point = (0, 2), line = y = 0
- Output
- distance = 2
- Explanation
- The shortest path from the point to the x-axis is the vertical segment of length 2.
Implementation Notes
- Represent the line consistently, such as two points or ax + by + c = 0.
- Check for degenerate line definitions before dividing by length.
- Use squared distances when only comparisons are needed.
Common Pitfalls
- Using infinite-line distance when segment distance is required.
- Failing to handle coincident line points.
- Comparing floating-point distances for equality without tolerance.
Interactive Mechanism Walkthrough
Spatial Primitive & Sweep-Line CanvasDraw the input shapes, active tests, and generated geometric result.
Step 1 of 4: Prepare the inputs and state needed for Point-Line Distance.
geometry-mechanismNotes
For segments and rays, compute the projected parameter and clamp it to the allowed interval.
Sources (1)
What each reference supports for this entry.
Computational Geometry
BookComputational geometry textbook reference for the predicates or constructions used by Point-Line Distance.
Explore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Used byRamer-Douglas-Peucker
Geometry · Core
RDP uses point-to-segment distance to decide which vertices to keep.
Compare with thisConceptual neighborLine Segment Intersection
Geometry · Foundation
Both are basic geometric predicates over points and lines.
Compare with thisRelationships
View full relationship pageConceptual neighborLine Segment Intersection
Both are basic geometric predicates over points and lines.
Used byRamer-Douglas-Peucker
RDP uses point-to-segment distance to decide which vertices to keep.
Related Algorithms
Tags
Computational GeometryGeometryDistance MetricLinear Algebra