Can bi-cubic surfaces be class A?...< 10 years of focus on shape G1 construction of high quality...

Can bi-cubic surfaces be class A?

Kestutis Karciauskas Jorg Peters

Vilnius University University of Florida

NSF CCF-0728797

K. Karciauskas, J. Peters (VU, UF) SGP 2015 1 / 31

Context

37 years since Catmull & Clark suggested a simple way to smooth quadmeshes by an infinite sequence of nested rings of bi-3 splines.CC surfaces are an important modeling tool (often the last stage of thegeometry processing pipeline) butWe know that CC surfaces have shape deficiencies.25 years of geometric continuity: joining (a finite number of) surfacepieces by change of variables1990s focused on algebraic challenges to keep polynomial degree low –shape secondary< 10 years of focus on shapeG1 construction of high quality (class A) of degree bi-5NEW SGP 2015: bi-3 high quality and sacrifice smoothness (a bit).→ alternative to CC: better shape, few bi-3 pieces.

Context

Outline

1 What are class A surfaces?

2 Smoothness and low degree – class A?

3 Algorithm

4 Choices and ExamplesChallenging InputValence n = 3 and n > 7

5 Discussion

What are class A surfaces?

Class A in the blogosphere

aesthetic visible surfaceno undesirable waviness

gap at join < 0.01mm (∗SGP 2015 will be watertight)angle between normals < 0.1 degrees

Surface Filling

where do gaps come from?

Autodesk user examples

no unwanted sharp edgesangle between normals < 0.1 degrees not visible

Autodesk user examples

no unwanted sharp edgesangle between normals < 0.1 degrees not visible

Class A in wikipedia

automotive designsurfaces of high quality (= curvature and tangency alignment)aesthetical reflection quality

Reflection lines – highlight lines

reflection lines highlight lines [BC94]

Reflection lines – highlight lines

reflection lines highlight lines [BC94]

Reflection lines → highlight lines

Beier,Chen 94: place zebra stripes above surface;surface color := color reached by extending point’s normal

highlight lines highlight lines

Smoothness and low degree – class A?

3 Algorithm

5 Discussion

rectangular tensor grid → univariate

→ focus on irregularities

n = 5 net

n = 6 net bi-3 ring + cap

rectangular tensor grid → univariate → focus on irregularities

n = 5 net

n = 6 net bi-3 ring + cap

n = 5 net n = 6 net bi-3 ring

n = 5 net n = 6 net bi-3 ring + cap

Can smooth bi-cubic surfaces be class A?

n = 6 net

[GZ94]

Gregory patches [LSNC09] 7 bi-3 pieces per sector

n = 6 net [GZ94]

Gregory patches [LSNC09]

7 bi-3 pieces per sector

n = 6 net [GZ94]

Can smooth bi-cubic surfaces be class A? NO

n = 6 net [GZ94]

Can C0 bi-cubic surfaces be class A?

best bi-3 G1 surface

C0 surface

Can C0 bi-cubic surfaces be class A? YES!

best bi-3 G1 surface C0 surface

Algorithm

3 Algorithm

5 Discussion

Algorithm

Smooth (G1) Bi-5 reference cap

Dag14 [KP]: G1 surface construction with good curvature distribution:

Gray ⇒ G1 connected bi-5 sectors.Red ⇒ well-defined curvature at extraordinary point.Green ⇒ G1 (not C1 as in most constructions) connection to input data.

Algorithm

Pipeline

C0 G1 C1

Step 1: deCasteljau split ⇒ four bi-5 pieces.

Step 2: bi-5 pieces are transformed to bi-3 patches ⇒ C0 junctionsStep 3: repair internal cap smoothness

Algorithm

Pipeline

C0 G1 C1

Algorithm

Pipeline

C0 G1 C1

Step 2: bi-5 pieces are transformed to bi-3 patches ⇒ C0 junctions

Step 3: repair internal cap smoothness

Algorithm

Pipeline

C0 G1 C1

Algorithm

Step 3

G1 between sectors restored by setting circle points.Red ⇒ well-defined curvature at extraordinary point.Brown: deviation of normals < 0.1◦.

Choices and Examples Challenging Input

3 Algorithm

5 Discussion

Claim: < 0.1 degree mismatch

not strictly true, but

Basic functions satisfy < 0.1 degrees (for n < 60)Obstacle course of challenging nets satisfy < 0.1 degrees

Basic functions satisfy < 0.1 degrees (for n < 60)

Obstacle course of challenging nets satisfy < 0.1 degrees

Basic functions satisfy < 0.1 degrees (for n < 60)Obstacle course of challenging nets satisfy < 0.1 degrees

Joining two crossing beams

n = 6 net

ACC [LS08] Catmull-Clark

SGP2015

n = 6 net ACC [LS08]

Catmull-Clark

SGP2015

n = 6 net ACC [LS08] Catmull-Clark

SGP2015

n = 6 net ACC [LS08] Catmull-Clark

SGP2015

Benefits of one Catmull-Clark-subdivision step

n = 6 net

no CC one CC-step

one CC step multiple CC steps

n = 6 net no CC

one CC-step

n = 6 net no CC one CC-step

one CC step

multiple CC steps

Choices and Examples Valence n = 3 and n > 7

3 Algorithm

5 Discussion

Valence n = 3 case

n = 3 net

Gaussian curvature: [GZ94] SGP2015

Valence n = 3 case

n = 3 net

Gaussian curvature: [GZ94]

SGP2015

Valence n = 3 case

n = 3 net

Gaussian curvature: [GZ94] SGP2015

Valences n > 7

n = 9 net highlight shading mean curvature

’Platonic’ surfaces from 5-sided patches

n = 5 nets layout highlight lines

mesh n = 5 nets

layout highlight lines

mesh n = 5 nets layout

highlight lines

mesh n = 5 nets layout highlight lines

Discussion

3 Algorithm

5 Discussion

Discussion

When C0 beats G1

Mismatch of normals: < 0.1◦ is acceptable in industry.One local CC step improves shape for difficult input.Quad meshes from reverse engineering typically do not require a CCstep (neither for mismatch nor for shape)C0 continuity along cap boundary.C0 continuity between sectors does not work! The deviation of normalsbecomes large.The guiding bi-5 cap is essential for deriving A class surface.Method is a finite alternative to CC subdivision.

Thank You!

Discussion

When C0 beats G1

Thank You!

Discussion

When C0 beats G1

Thank You!

Discussion

When C0 beats G1

Thank You!

Discussion

When C0 beats G1

Thank You!

Discussion

When C0 beats G1

Thank You!

Discussion

When C0 beats G1

Thank You!

Discussion

When C0 beats G1

Thank You!

Discussion

Magnified view

SGP 2015, no CC

one CC step + SGP 2015 bi-5 guide,no CC

A- A AA

(Standard & Poor’s rating ;-)

Discussion

Magnified view

SGP 2015, no CC one CC step + SGP 2015

bi-5 guide,no CC

A- A AA

Discussion

Magnified view

SGP 2015, no CC one CC step + SGP 2015 bi-5 guide,no CC

A- A AA

Discussion

Magnified view

A- A AA

Discussion

Magnified view

SGP 2015, no CC

one CC step + SGP 2015 bi-5 guide,no CC

A- A AA

Discussion

Magnified view

SGP 2015, no CC one CC step + SGP 2015

bi-5 guide,no CC

A- A AA

Discussion

Magnified view

A- A AA

Discussion

Magnified view

A- A AA

Discussion

Design with sketches

sketch

highlight shading mean curvature

Discussion

sketch net

Discussion

sketch net

Can bi-cubic surfaces be class A?...< 10 years of focus on shape G1 construction of high quality...

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