Introduction to Coffee Cup Light Patterns

• Demonstration of shapes appearing on coffee/tea surface when light shines through kitchen window or spotlight
• Shadow of rim visible, plus additional curved shapes (B-like, heart-like patterns)
• Shapes change as light source moves
• Phenomenon called "caustics" - from word meaning "burning" (like magnifying glass effect)

Physics of Light Reflection

• Light coming sideways bounces off circular wall of mug
• Most mug remains in shadow, only reflected light creates patterns
• Circular wall geometry causes specific light behavior
• Light travels in straight lines but overlaps create brighter areas where multiple rays converge

Simulation and Visualization

• Crude model: light source, mug, rays showing reflection patterns
• Sun modeled as distant source creating parallel light rays
• 176 rays initially, cranked up to 1000+ for realistic wall of light
• Transparent overlapping lines technique reveals curve emergence
• Cusp formation - mathematical term for pointed shape
• Curve doesn't actually exist - artifact of overlapping straight lines
• Lines are tangent to invisible curve

Mathematical Connection - Curve Stitching

• Circle with numbered points (1, 2, 3, etc.)
• Rule: join each number to 3× itself (1→3, 2→6, 3→9, etc.)
• Creates familiar curved shapes from straight lines
• Different multipliers (2×, 3×, 4×) create different curves
• Parallel to coffee cup phenomenon - curves emerging from straight lines

Curve Classification

• Nephroid: kidney-shaped curve with two cusps
• Cardioid: heart-shaped curve with one cusp
• Both are epicycloids (made by rolling circles around other circles)
• Parametric equations for nephroid:
  • x = 3cos(t) + cos(3t)
  • y = 3sin(t) + sin(3t)
  • Parameter t ranges from 0 to 2π
  • Describes circle of radius 4

Tangent Line Mathematics

• Derived tangent equation: y·sin(2t) + x·cos(2t) = 4cos(t)
• Each point on curve has tangent line following this equation
• Simulation shows tangent line moving around curve edge
• Critical for understanding light reflection patterns

Physics and Reflection Analysis

• Quarter-circle mug model with center point
• Parallel light rays from distant source
• Point on circle defined by trigonometry: (cos θ, sin θ) scaled by radius 4
• Angle of incidence = angle of reflection
• Gradient analysis using angle θ
• Reflection ray equation derived: y·sin(2θ) + x·cos(2θ) = 4cos(θ)

Parameter Comparison and Results

• Nephroid parameter 't' vs reflection angle parameter 'θ'
• Both equations identical in form, confirming nephroid shape
• θ limited to 0→π (hitting back half of mug only)
• Explains why only half the kidney shape visible in cup
• Light source distance affects curve type:
  • Distant source → nephroid
  • Closer source → cardioid
  • Source on circle edge → different epicycloid

Multiple Bounces Analysis

• Second bounce creates more heart-shaped pattern (colored red)
• Third bounce even more heart-shaped (colored green)
• Each bounce creates fainter but more defined heart shapes
• Practical demonstration with reflective wedding ring
• Multiple reflections possible but typically washed out by first bounce

Practical Demonstration

• Wedding ring experiment in darkness with phone torch
• Successfully observed second bounce pattern
• Confirmed theoretical predictions with real-world observation