Bridges, Seoul, Aug. 15, 2014 Carlo H. Séquin and Raymond Shiau University of California, Berkeley Physically Correct Rendering of Pacioli’s Rhombicuboctahedron.

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Presentation transcript:

Bridges, Seoul, Aug. 15, 2014 Carlo H. Séquin and Raymond Shiau University of California, Berkeley Physically Correct Rendering of Pacioli’s Rhombicuboctahedron

“Ritratto di Frà Luca Pacioli” (1495) Museo e Gallerie di Capodimonte in Naples, Italy

RCO = Rhombi-cuboctahedron u Archimedean solid: 8 triangles, 18 square faces RCO “Leonardo style”

Praise of the Pacioli RCO u … a masterpiece of reflection, refraction, and perspective. [George Hart] u Its surface reflects a view out an open window, showing the Palazzo Ducale in Urbino. [Margaret Davis]

How Accurate Is the Rendering ? u … do the reflections show that the window is high up towards the left? Or rather towards the bottom right? Both do not really make sense. u … the way the images are formed doesn’t seem very realistic at all. [Joost Rekveld]

Comparison to a Physical Model Model by Claude Boehringer Photo by Herman Serras u Quite different surroundings !

Do a Computer Graphics Rendering! Which of these pictures is correct? There are many parameters to adjust! Difficulties with the glass/water interface.

Matching the Viewing Geometry Front view Side view

Depicting Surface Reflections 0° 10° 34° A line reflected across a non-flat dihedral angle results in a “sheared” discontinuity.  Pasted-on rectangular “sticker” is wrong!

Bad Window Reflection! 34° A line reflected across a non-flat dihedral angle results in a “sheared” discontinuity.  Pasted-on rectangular “sticker” is wrong!

Simple Refraction in Water Painting no refraction with refraction  Painter did not observe any refraction! (probably did not have a water-filled model)

Effect of Refraction in Water Computer rendering Boehringer model  Painter did not have a water-filled model !

Multiple Refraction- & Mirror- Events u Limiting the depth of the ray-trace tree: M=1 R=2 M=1 M=8 R=2 R=8 Mi Prism

Glass / Water Interface Water half-cube behind Glass half-cube Water/gap/Glass Water/Glass (fused) view

Water Half-cube behind Glass Half-cube Tracing ray-bundles Properly fused interface

Water Half-cube behind Glass Half-cube Ray-bundles Properly fused interface Ray-bundles

The Complete RCO The various materials and interfaces: Opaque filler (O), Water (W), Glass (G), G/W-interface (I)

Modeling the Whole RCO In the striped Cornell Box

Modeling the Whole RCO M=1 R=3

Modeling the Whole RCO R=4 M=1, R=4

Modeling the Whole RCO M=1, R=6 M=R=10 (max) = “ground truth” ?

RCO in Pacioli’s Room Slightly exaggerated tinting Set-up

Final Comparison 1495 Painting 2014 Computer Rendering

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