Bridges, Seoul, Aug. 15, 2014 Physically Correct Rendering of Pacioli ’s Rhombicuboctahedron

Carlo H. Séquin and Raymond Shiau University of California, Berkeley

“

Ritratto di Frà Luca Pacioli

”

(1495)

Museo e Gallerie di Capodimonte in Naples, Italy

RCO = Rhombi-cuboctahedron

RCO “Leonardo style”  Archimedean solid: 8 triangles, 18 square faces

Praise of the Pacioli RCO

 … a masterpiece of reflection, refraction, and perspective.

[George Hart]  Its surface reflects a view out an open window, showing the Palazzo Ducale in Urbino.

[Margaret Davis]

How Accurate Is the Rendering ?

 … 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.  … the way the images are formed doesn ’t seem very realistic at all.

[Joost Rekveld]

Comparison to a Physical Model

 Quite different surroundings !

Model by Claude Boehringer Photo by Herman Serras

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

 Limiting the depth of the ray-trace tree: M=1 R=2

Mi

M=1 M=8 R=2 R=8

Prism

Glass / Water Interface

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

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

Set-up Slightly exaggerated tinting

Final Comparison

1495 Painting 2014 Computer Rendering

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