Demonstrating the Pythagorean theorem through circle drawing.
- Author
- Sukesh Ashok Kumar
This example visualizes one of the most famous theorems in mathematics - the Pythagorean theorem - by using it to draw a perfect circle. A circle is simply the set of all points at a fixed distance (radius) from a center point.
Mathematical Concepts:
- Circle equation: x² + y² = r²
- Pythagorean theorem: a² + b² = c²
- Distance formula: d = √(x² + y²)
- Radius: constant distance from center to any point on circle
- Center point: (centerX, centerY)
- Cartesian to polar relationship
Mathematical Derivation: For a circle centered at origin with radius r:
- Any point (x,y) on the circle satisfies: distance = r
- Distance = √(x² + y²)
- Therefore: √(x² + y²) = r
- Squaring both sides: x² + y² = r²
Programming Concepts:
- Nested loops for 2D scanning
- Distance calculation using sqrt() from math.h
- Tolerance range for floating-point comparison
- Relative coordinates vs absolute screen coordinates
- Bounding box optimization
What you'll learn:
- How the Pythagorean theorem defines circles
- Using the distance formula in graphics
- Why we need tolerance (±1) when comparing floating-point values
- Converting between relative and absolute coordinates
- Efficient circle drawing by limiting search space
- Including and linking the math library (-lm flag)
Algorithm Explanation:
- Search only within a square bounding box (-radius to +radius)
- For each point (x,y), calculate distance from origin
- If distance ≈ radius (within tolerance), it's on the circle
- Translate relative coordinates to screen position
Why Tolerance? We check if distance is between (radius-1) and (radius+1) because:
- Pixels are discrete (integers), but math uses continuous (floating-point)
- Rounding can cause exact matches to miss
- Tolerance creates a thicker, more visible circle outline
Compile:
gcc m-circle.c gfx/simplegfx.c -o output/m-circle -lX11 -lm
Note: -lm links the math library for sqrt() function
Run:
1.9.1