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«Interacting With Surfaces In Four Dimensions Using Computer Graphics TR93-011 March, 1993 David Banks Department of Computer Science University of ...»

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Interacting With Surfaces In Four

Dimensions Using Computer Graphics


March, 1993

David Banks

Department of Computer Science

University of North Carolina at Chapel Hill

Chapel Hill, NC 27599-3175

UNO is an Equal Opportunity/Affirmative Action Institution.

Interacting With Surfaces In Four Dimensions

Using Computer Graphics

A dissertation submitted to the faculty of the University ofNonh Carolina in partial fulfillment of

lhe requirements for the degree of Doctor of Philosophy in lhe Department of Computer Science.

David Banks Chapel Hill, 1993 ~ Approved by: /.

~~ 'X~C 4 /~~ ~~ David 0. v. Beard ©1993 David Cotton Banks


ii DAVID CorroN BANKs. Interacting With Surfaces in Four Dimensions Using Computer Graphics (Under the direction of STEPHEN M. PIZER.) ABSTRACT High-speed, high-quality computer graphics enables a user to interactively manipulate surfaces in four dimensions and see them on a computer screen. Surfaces in 4-space exhibit properties that are prohibited in 3-space. For example, non-orientable surfaces may be free of self-intersections in 4-space. Can a user actually make sense of the shapes of surfaces in a larger-dimensional space than the familiar 3D world? Experiment shows he can. A prototype system called Fourphront, running on the graphics engine Pixel-Planes 5 (developed at UNC-Chapel Hill) allows the user to perform interactive algorithms in order to determine some of the properties of a surface in 4-space. This dissertation describes solutions to several problems associated with manipulating surfaces in 4-space. It shows how the user in 3-space can control a surface in 4-space in an intuitive way. It describes how to extend the common illumination models to large numbers of dimensions. And it presents visualization techniques for conveying 4D depth information, for calculating intersections, and for calculating silhouettes.

iii Acknowledgements Many minds, many hands, and many pockets contributed to this research. My sincerest thanks go to Brice Tebbs and Greg Turk for teaching me about computer graphics and showing me how to program Pixel-Planes;

Howard Good (callback functions, one-sided polygon picking), Marc Olano (conditional executes, parallel picking), David Ellsworth (animated cursor, parallel picking), and Andrew Bell (multipass transparency) for contributing code to improve the Fourphront system;

Trey Greer ("Front"), Howard Good ("Pphront"), and Vicki Interrante ("Xfront") for developing the 3D progenitors of Fourphront;

David Beard, Gary Bishop, Raben Bryant, Steve Pizer, and Jim Stasheff for serving on my committee;

Oliver Steele for helping to create 4D datasets;

Nelson Max for suggesting that intersection curves be highlighted;

Trey Greer for teaching me how to write an X-window interface;

Ross Whitaker for suggesting the use of the shape operator in determining fixed-width silhouettes;

David Eberly for solving the differential equations for constant curvature in 4-space;

Raben Bryant for suggesting I use Morse Theory and investigate quatemion rotations;

Russell Taylor for helping to pon Fourphront to run with the head-mounted display;

David Harrison for teaching me how to edit videotape;

Howard Lander and Data General Corporation for contract support;

Myron Banks and Warren Herron for their generosity;

Digital Equipment Corponation for scholarship funding;

Terry Yoo and the NSF (Gang of Five) for travel and production support;

Elizabeth Banks for her tremendous love, patience, and encouragement.

–  –  –



i Overview

1.1 Goal of the Research

1.2 Why Surfaces in 4-space are Interesting

1.2.1 About Large Dimensions

1.2.2 About 4-space

1.2.3 About Surfaces

1.3 Related Work

1.3.1 Compu1er Gr aphics

1.3.2 Mathematics

1.3.3 Pedagogy

2 The Basics of Graphics In 4-space

2.1 How to Map Input From User Space to Object Space

2.1.1 Mapping 2D Input to 3D Transformations

2.1.2 Mapping 3D Input to 4D Transformations

2.1.3 Simple Approximations of Rotation Matrices

2.2 Illumination

2.2.1 Review of Illuminating Surfaces in 3-space

2.2.2 Illumination Independent of Dimension

2.2.3 Why it Mal1ers

2.3 Depth Cues

2.3.1 Occlusion and Shadows

2.3.2 Texture and Color

2.3.3 Focus and Transparency

2.3.4 Perspective and Parallax

2.4 Projecting From Object Space to Screen Space

2.4.1 Stereopsis and Parallax

2.4.2 Mixing Rotations, Projections, and Translations

v 3 Visualization Tools

3.1 Ribboning

3.2 Clipping

3.3 Transparency

3.4 Paint

3.5 Silhouettes

3.5.1 Varying-width Silhouette Curves

3.5.2 Fixed-width Silhouette Curves

3.6 Intersections

3.6.1 Varying-width Intersection Curves

3.6.2 Fixed-width Intersection Curves

4 Interactive AlgorHhms

4.1 Intersections in 4-space

4.2 OrientabiiHy

4.3 Genus

5 Observations of Interactive Sessions

5.1 Intersections

5.2 OrientabiiHy

5.3 Genus

5.3.1 Recognizing the Shape of a Surface

5.3.2 Applying Morse Theory to a Surface

5.3.3 Finding Cmical Points at Pinch Points

5.4 Links and Knots

5.4.1 Knotted Torus

5.4.2 Linked Spheres

5.4.3 Knotted Sphere

5.5 Constant Curvature in 4-space

5.6 Quaternion Rotations

5.7 lntuHion

6 Conclusions and Future Work

6.1 User Interface

6.2 Illumination

6.3 Depth Cues

6.4 Visualization Tools

6.4.1 Seeing Into a Surface

6.4.2 Applying Paint

6.4.3 Locating Intersections and Silhouettes in Screen-space

VI 6.5 Results from Interactive Sessions

Appendix A Pixel-Planes 5

A.1 Hardware Archnecture

A.2 Renderer Instructions

A.3 An


Programming Model of the Renderers


Appendix 8: Fourphront

8.1 System Capabilities

8.2 User Interface

8.3 System Performance


–  –  –

Figure 1-1. The six faces on this octagonal figure correspond to the six axial planes of fourdimensional SIBCe.---··--··--·····-------··---··-------·-------········-······-····-····-·-························13 Figure 2-1. A user in 3-space manipulates a surface in 4-space, which projects to 3-space and then onto the screen

Figure 2-2. These are three of the six axial planes in xyzw-space, defmed by the axis pairs xw, yw, and zw. The other three axial planes (xz, yz, and xy) lie in the 3dimensional xyz-subspace

Figure 2-3. At the bottom point of this circular trajectory, the mouse's velocity is purely hOOzontal, while its acceleratioo is purely vertical

Figure 2-4. The mappings of 3D input space to 4D world space that promote kinesthetic sympathy

Figure 2-5. The 3D joystick rotates in the x'z', y'z', and x'y' planes, which can produce a momentary translation in the x and the y directions. In the input space coordinates, x' is rightward, y' is forward, and z' is vertical

Figure 2-6. The mappings of spaceball and joystick input that promote kinesthetic sympathy in 4D world space

Figure 2-7. Moving the joystick produces a rotation. The velocity of rotation is proportional to the amount that the joystick is displaced from its centered, "rest" position................ 39 Figure 2-8. Diffuse and specular illumination at a point on a surface are governed by the surface noonaJ N, the light vector L, and the view vector V

Figure 2-9. If the codirnension is 2, the normal space at a point on the surface will be a plane N

Figure 2-10. Opacity can be used as a depth cue in thew-direction without significantly interfering with the' usnal depth cues in the z-direction. For example, points on the surface near the eye are rendered nearly transparent, while points far away in w are nearly opaque. Fourphront uses a linear opacity ramp that is clamped at controllable near aod far values

Figure 2-11. The (x y z w)-axes (left) project in thew-direction to the (x y z)-axes (middle), which project in the z-direction to the (x y)-axes of the image plane.............. 55 Figure 2-12. When there are two eye positions involved in projecting an image, either of them can produce parallax. In this figure, spheres A and B project from 3-space onto a 2-dimensional plane as disks. The disks project to a 1-dimensionalline as segments. By tilting the page obliquely, you can see what the second eye sees.

Moving an eye to the right will make the farther object seem to move to the right of the nearer object Which sphere looks closer? It depends on which eye does the measuring. A is closer to eye3 than B is. But the projection of B is closer to eye2 than the projection of A is

viii Figure 3-1. Clipping into a torus produces a figure-eight contour. Clipping reveals internal geometry, but complex contows can coofuse the shape.............••••••

A polygon can have different colOIS on its front and back sides. •...•......•••

Figure 3-2.

Figure 3-3. The surface normal is nearly orthogonal to the eye vector in the vicinity of a silhouette curve.------------------------·-·-----···--···-····-·····-·-············ 87 Figure 3-4. Three points, together with their nonnals, can be used to estimate the surface's curvature. The dark: curve represents the silhouette as seen from the eye position........... 88 Figure 3-5. The other two vertices from the surface can be used to produce an orthonormal OOsis lbr the tangent plane at the point p

Figure 3-6. The original surface (white) is approximated by a quadric surface (dark) derived from the shape operator.---·----·-----·-·----·---------····----···-··-················ 90 Figure 3-7. The approximating surface has its own silhouette curve, as seen from the eye position. The angle between the point p and the silhouette curve measures their distance apart wben displayed on the screen.....•••,

Figure 3-8. Now the point p can be compared against a fixed-width threshold based on the quadric approximating surface

Figure 3-9. At their common intersection, two polygons share z-values. The z-values are within some threshold of each other along a thickened intersection curve

Figure 3-10. The goal is to determine whether a point on a surface is within d of the intersection. Depending on the slopes of the two surfaces, the point must lie within some e (in the z-direction) of the other surface

T!'tis is the default n~ge fortl}e 208 bits of per-pixel memory

Figure 3-11.

Figure 3-12. The triangle mesh covers the (x,y)-range of the screen when it is parallel to the screen. When it rotates a quarter turn, it covers the range of the z-buffer

Figure 3-13. Fourphront scavenges bits from the memory-transfer area and from the texture area of the default usage. These bits are used to compute the fixed-width inlelsection curve

Figure 4-1. Two polygons in 4-space (left) intersect a plane which will project to a line in 3-space (right). As a result, the projected polygons will intersect in 3-space................! 09 Figure4-2. A painted polygon P and an unpainted polygon U share a common edge Oeft).

An intervening polygon I intersects P and U along their shared edge, hiding P (middle). Rotation reveals P, but now I hides U (right)

Figure 5-1. A Whitney umbrella. The vertical height function on the surface is critical at the pinch point.

Figure A-1: Two stages of the conventional object-order graphics pipeline

FigureA-2: Architectnre of Pixel-Planes 5. The GPs and renderers at the top embody the transformation and rendering stages of the graphics pipeline of figure A-I. The host workstation connects to the communication network through a special interface. The color display connects to the network via a frame buffer

FlgureA-3: Logical organization of the Pixel-Planes 5 renderer. Each renderer has an expression evaluator and a grid of pixel processors. A single pixel processor (right) has local memory and a 1-bit ALU. Two bits of the local memory are special: the enable bit and the carry bit.

IX FigureA-4: Possible operations within the pixel processor. The soun:e of the operation is represented on the top row. The destination is designated on the left colwnn................ l59 Figure B-1. Performance figures for Fowpbront (triangles per second). The left column describes the attributes that triangles were rendered with. The middle and right columns give the perfonnance for high and low resolution images. These measurements were made using a 3-rack configwation with 40 GP' s and 20 renderers. The dataset had 5000 triangles, and the time includes 4D tnmsformation and rendering·-···-·-·-····-·-·-····-·-····-·-·······-·-·-·-·-··············································169


The torus (cos s, sin s, cos 1, sin t). The extreme values of w occur along the Plale2.2A outer and inner walls of the torus. With the light shining in thew-direction, the diffuse highlights fall along these exttemes of the torus...............,_,..._..,_

Plate 2.2 B A Klein bottle in 4-space with the light still shining in the w-direction.

The highlights again indicate where the extreme values are located on the surtace..................48 The torus (cos s. sins, cost, sin t) with color modulated from black to white Plate2.3 A according to depth in w. The outer wall of the toms, near to the eye in 4-space, is black. The inner wall, far from the eye in 4-space, i.~ light

Plate 2.3 B The torus wilh opacity modulated from transparent to opaque according to depth in w.

In this image, the opacity ramp is centered at w =..0.25 (near the eye in 4space)

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