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- Basic definitions: circle, center, radius, diameter, chord, arc, circumference - Fundamental measurements: circumference (C = 2πr) and area (A = πr²) with derivations - Parts of a circle: chords, arcs (minor/major/semicircle), sectors, and segments with formulas - Circle equations: standard form, parametric form, and general form - Angles in circles: central angles, inscribed angles, and angle relationships - Common theorems: chord properties, tangent properties, and Thales' theorem - Applications: geometry, trigonometry, engineering, physics, and real-world uses - All mathematical notation uses LaTeX (degree signs as ^\circ) - Comprehensive MathWorld links for all key terms - Follows same documentation pattern as Triangles and Quadrilaterals chapters
- Added alternative circle definition via three non-collinear points (circumcircle) - Added comprehensive Angle Measurement section explaining radians and degrees - Defined radian as natural angular unit (arc length = radius) - Added conversion formulas between radians and degrees - Added centered HTML table with common angle conversions (90°, 180°, 270°, 360°) - Reorganized Arc Length as its own subsection for better structure - All degree symbols use LaTeX notation (^\circ)
- Add Circles section with step-by-step derivation of circle through three points
- Use standard notation: [h,k] for center, [x,y] for points
- Add aligned equations for clear step-by-step algebra
- Fix Symbolics.jl numeric evaluation using build_function with expression=Val{false}
- Document alternative method using Symbolics.value with unwrap parameter
- Preserve both methods for learning purposes
- Update all substitute() calls to include fold=true parameter - Ensures proper evaluation of numeric expressions - Completes the Symbolics.jl numeric evaluation fixes
- Convert π from Irrational type to Float64 for numeric display - Ensures the value prints as decimal approximation rather than symbol
- Include radius calculation for r>1 case - Complete the polygon area calculation examples
- Reorganize into clearer sections (Overview, Core Architecture, Critical Patterns) - Add explicit code examples for computational/plotting separation pattern - Document both Symbolics.jl numeric evaluation methods (substitute+value and build_function) - Expand documentation patterns section with comprehensive style guidelines - Add mathematical concept documentation structure guidelines - Include MathWorld linking conventions and visual element guidance - Streamline CI/CD pipeline descriptions - Improve overall clarity and actionability
- Add practical pizza example demonstrating circle through three points - Fix markdown linting issues in Circles documentation - Calculate pizza center and radius from three edge points - Verify circle equation with all three points
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Summary
Comprehensive improvements to mathematical notebooks and copilot instructions, including fixes for Symbolics.jl breaking changes, detailed circle derivations, and practical examples.
Key Changes
Symbolics.jl Numeric Evaluation Fixes
Symbolics.value()no longer evaluates transcendental functions numericallybuild_function(expr, vars, expression=Val{false})for compiled functionsSymbolics.value(expr, unwrap=true)for direct evaluationfold=Val(true)to allsubstitute()callsFloat64()conversionCircle Mathematics
alignenvironments for equation formattingDocumentation Improvements
Testing
Files Changed
notebooks/Basics.ipynb: Symbolics fixes, circle derivation, pizza example.github/copilot-instructions.md: Comprehensive restructuring and expansiondocs/src/Geometry/04 Circles.md: Linting fixes