cross ratio: meaning, definition, pronunciation and examples
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Quick answer
What does “cross ratio” mean?
A specific numeric value calculated from four collinear points or four concurrent lines in projective geometry, which remains invariant under projective transformations.
Audio
Pronunciation
Definition
Meaning and Definition
A specific numeric value calculated from four collinear points or four concurrent lines in projective geometry, which remains invariant under projective transformations.
In mathematics, particularly projective geometry, the cross ratio (or anharmonic ratio) is a fundamental invariant. It can be extended to complex analysis (the ratio of four complex numbers) and has applications in relativity and computer vision.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling of related words (e.g., 'behaviour' vs. 'behavior') may follow regional conventions in surrounding text.
Connotations
Purely technical, no affective connotations in either variety.
Frequency
Equally rare and specialized in both varieties, confined to advanced mathematics contexts.
Grammar
How to Use “cross ratio” in a Sentence
the cross ratio of [four points/lines]the cross ratio ([point A], [point B]; [point C], [point D])Vocabulary
Collocations
Examples
Examples of “cross ratio” in a Sentence
verb
British English
- [No verb form exists]
American English
- [No verb form exists]
adverb
British English
- [No adverb form exists]
American English
- [No adverb form exists]
adjective
British English
- [No dedicated adjective form. Use attributive noun: 'cross ratio invariance']
American English
- [No dedicated adjective form. Use attributive noun: 'cross ratio property']
Usage
Meaning in Context
Business
Not used.
Academic
Primary context. Used in advanced geometry, complex analysis, and physics lectures and papers.
Everyday
Never used.
Technical
Core context. Used in mathematical proofs, computer vision algorithms, and theoretical physics.
Vocabulary
Synonyms of “cross ratio”
Strong
Neutral
Weak
Vocabulary
Antonyms of “cross ratio”
Watch out
Common Mistakes When Using “cross ratio”
- Using 'cross-ratio' (with hyphen) inconsistently; the solid form 'cross ratio' is standard in many style guides. Confusing it with a simple ratio of two quantities.
FAQ
Frequently Asked Questions
It is a compound noun typically written as two separate words ('cross ratio'), though hyphenated ('cross-ratio') is also seen.
It is a number you get from four points on a line. Even if you project that line onto another line (like a shadow), this specific number stays the same for those four points.
It is the most basic projective invariant. If the cross ratios of two sets of four points are different, the sets cannot be mapped onto each other by a projective transformation.
Yes, many computer vision and geometry libraries (e.g., OpenCV, geometric libraries for Python/Matlab) include functions to compute the cross ratio for point tuples.
A specific numeric value calculated from four collinear points or four concurrent lines in projective geometry, which remains invariant under projective transformations.
Cross ratio is usually technical in register.
Cross ratio: in British English it is pronounced /ˈkrɒs ˌreɪ.ʃi.əʊ/, and in American English it is pronounced /ˈkrɔːs ˌreɪ.ʃi.oʊ/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “[No idioms exist for this technical term]”
Learning
Memory Aids
Mnemonic
Think of four points on a line. The CROSS RATIO is the RATIO of ratios, CROSS-multiplied to give a special, unchanging number.
Conceptual Metaphor
A FINGERPRINT FOR FOUR POINTS (an invariant signature that identifies the projective arrangement of four elements).
Practice
Quiz
In which field is the term 'cross ratio' primarily used?