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Circle-Line Intersection

Finds where a line crosses, touches, or misses a circle.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Geometry
Difficulty: Intermediate
Family: Geometric Intersection
Tier: Core
Status: Verified
Review: Source Checked
Sources: 2
Formulas: 1
Relationships: 4
Code example: Ready
Visualization: Ready

Overview

Circle-line intersection substitutes a line equation into the circle equation and solves the resulting quadratic, or equivalently projects the circle center onto the line and compares distance to the radius.

Core Concepts

circle and line intersectiongeometric intersectiongeometric primitivesgeometry

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 when the line and circle are represented with compatible coordinate units.
Use when tangent and near-tangent cases can be handled carefully.

When Not to Use

Avoid when a segment or ray is needed but the implementation only handles infinite lines.
Avoid when floating-point tolerance has not been defined for tangent cases.

What It Does

Computes distance from the circle center to the line
Classifies no intersection, tangent, or two intersections
Solves for intersection points when they exist

Use Cases

Collision and visibility tests
Geometric construction routines
Ray or segment intersection preprocessing

Inputs

Points, segments, polygons, or geometric primitives
Query or construction parameters

Outputs

Geometric decision, measurement, or constructed structure

Key Formulas (1)

Quadratic substitution

||p₀ + t·d − c||² = r²

||p0 + t*d - c||^2 = r^2

Solves for line parameters where the line reaches the circle radius.

Pseudocode

1.Represent the line as point p plus direction d
2.Project circle center onto the line
3.Compute squared distance from center to the projection
4.Compare it with radius squared
5.Return zero, one, or two points using the offset along d

Canonical Example

Input: circle center (0, 0), radius 1, line y = 0
Output: intersection points (-1, 0) and (1, 0)
Explanation: Substituting y = 0 into x^2 + y^2 = 1 leaves x^2 = 1.

Implementation Notes

Normalize the line direction or account for its squared length in formulas.
Use a tolerance around the discriminant or distance comparison.
Test no-hit, tangent, and two-hit cases.

Common Pitfalls

Treating a tangent as two distinct intersections.
Forgetting to clamp solutions when the input is a segment rather than a line.
Using a tolerance that is too large can create false intersections.

Interactive Mechanism Walkthrough

Spatial Primitive & Sweep-Line Canvas

Show substituting a line into a circle equation and solving the resulting quadratic on a small circle and line intersection example.

Step 1 of 5: Represent the line as point p plus direction d

geometry-mechanism

Notes

Clarify whether the input is an infinite line, ray, or segment before using the intersection result.

Sources (2)

What each reference supports for this entry.

Computational Geometry: Algorithms and Applications

Book

Computational geometry textbook support for Circle-Line Intersection's geometric model, data structures, and robustness concerns.

Open source

Real-Time Collision Detection

Book

Geometric collision reference for Circle-Line Intersection's primitive tests, bounding volumes, and numerical edge cases.

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