Split Packing: An Algorithm for Packing Circles with up to Critical - PowerPoint PPT Presentation

Circles in a right/obtuse triangle b a c Condition: a 2 + b 2 ≤ c 2 � − ( a − b − c )( a + b − c )( a − b + c ) π < 53 . 91% Critical density: ( a + b + c ) 3 Approximation factor: Larger than 1.8552 Sebastian Morr Split Packing 2016-06-09 30 / 57

Circles in a right/obtuse triangle: Examples Sebastian Morr Split Packing 2016-06-09 31 / 57

Circles in a thick isosceles triangle b c c Condition: √ ≤ b ≤ c 2 √ 4 b 2 − c 2 ) 2 π Critical density: 48 . 60% < ( c − 2 b + √ < 53 . 91% 4 b 2 − c 2 2 c Approximation factor: Between 1.8552 and 2.0576 Sebastian Morr Split Packing 2016-06-09 32 / 57

Circles in a thick isosceles triangle: Examples Sebastian Morr Split Packing 2016-06-09 33 / 57

The problem with acute triangles Sebastian Morr Split Packing 2016-06-09 34 / 57

The problem with acute triangles Sebastian Morr Split Packing 2016-06-09 34 / 57

The problem with acute triangles Sebastian Morr Split Packing 2016-06-09 34 / 57

Circles in a long rectangle h w √ Condition: w ≥ 2 + 3 2 h ≈ 1 . 5607 h 4 Critical density: π h 4 w < 50 . 33% Approximation factor: 4 w π h > 1 . 9870 Sebastian Morr Split Packing 2016-06-09 35 / 57

Circles in a long rectangle: Examples Sebastian Morr Split Packing 2016-06-09 36 / 57

Outline Packing circles in a square 1 Other container types 2 Other object types 3 Future work 4 Sebastian Morr Split Packing 2016-06-09 37 / 57

Other shapes in a square? Sebastian Morr Split Packing 2016-06-09 38 / 57

Other shapes in a square? Sebastian Morr Split Packing 2016-06-09 38 / 57

Other shapes in a square? Sebastian Morr Split Packing 2016-06-09 38 / 57

Rubies! Sebastian Morr Split Packing 2016-06-09 39 / 57

Rubies! Sebastian Morr Split Packing 2016-06-09 39 / 57