CG ECSE-4750-01 and 6964-01 Computer Graphics, Fall 2018, Rensselaer Polytechnic Institute (Posts about class)

CG Class 30, Wed 2018-12-12

W Randolph Franklin, RPI — Tue, 11 Dec 2018 05:00:00 GMT

Review before final exam.

CG Class 29, Mon 2018-12-10

W Randolph Franklin, RPI — Sun, 09 Dec 2018 05:00:00 GMT

Presentation of student videos

Review?

Do you want a review in the Wed lab session?

CG Class 28, Thurs 2018-12-06

W Randolph Franklin, RPI — Wed, 05 Dec 2018 05:00:00 GMT

Table of contents

1 SIGGRAPH 2018 Videos

1 SIGGRAPH 2018 Videos

CG Class 27, Mon 2018-12-03

W Randolph Franklin, RPI — Sun, 02 Dec 2018 05:00:00 GMT

Table of contents

1 OpenGL in Guha vs WebGL in Angel

Guha, the text I used before Angel, uses OpenGL. Angel uses WebGL.
OpenGL has a C API; WebGL uses Javascript.
That OpenGL is the obsolete version 2; WebGL is based on the current OpenGL 3.
OpenGL 2 has an immediate mode design: you draw things and they are forgotten.

In WebGL you send buffers to the GPU and then draw them.
OpenGL has compute shaders and geometry shaders. They fill in points along Bezier curves and draw trimmed NURBS.
A NURBS surface is a 2D parametric surface in 3D (or 4D if homogeneous).

A trim line is a 1D parametric curve in the the 2D parameter space of the surface.

The trim lines cut around the outside of the desired region and also cut out holes.

This is a powerful technique.

2 Chapter 15 slides

You do not need to learn most of those slides. Later I'll summarize what you need.
15_1 Global rendering.
15_2 Ray tracing.
15_3 What's next?.

3 Introduction to GPUs

from UT Computer Science

CG Class 26, Thurs 2018-11-29

W Randolph Franklin, RPI — Thu, 29 Nov 2018 05:00:00 GMT

Table of contents

1 Chapter 14 slides ctd

2 Old OpenGL in C and with NURBS

Before Angel, I used another book, Guha. It used old OpenGL in C. However therefore it could show NURBS. Some sample programs are in

https://wrf.ecse.rpi.edu/wiki/ComputerGraphicsFall2013/guha/Code

Look at bezierCurves, which shows moving control points.

trimmedBicubicSplineSurface shows NURBS.

3 WebGL query

Victor Calvert writes,

The WebGL report shows current-environment specifics, including the maximum number of textures, maximum framebuffer resolution, and other information, retrieved via the WebGL API.

CG Class 25, Mon 2018-11-26

W Randolph Franklin, RPI — Sun, 25 Nov 2018 05:00:00 GMT

Table of contents

1 ACM SIGSPATIAL GIS Cup 1st place

Here.

In 2016, we won 2nd. We also won 2nd in 2015.

2 Chapter 13 slides ctd

13_4 Display issues.

We've seen some of this.

3 Chapter 14

Curves are the next chapter of Angel. WebGL does this worse than full OpenGL. Here is a summary. Big questions:
1. What math to use?
2. How should the designer design a curve?
3. My notes on Bezier curves.
Partial summary:
1. To represent curves, use parametric (not explicit or implicit) equations.
2. Use connected strings or segments of low-degree curves, not one hi-degree curve.
3. If the adjacent segments match tangents and curvatures at their common joint, then the joint is invisible.
4. That requires at least cubic equations.
5. Higher degree equations are rarely used because they have bad properties such as:
  1. less local control. Changing one control point of a hi-degree curve changes the whole curve. Parts of the curve distant from that point may move a lot. This makes designing a desired curve impossible.
  2. numerical instability. Small changes in coefficients cause large changes in the curve, even if computations are exact.
  3. roundoff error. Computations are not exact.
6. See my note on Hi Degree Polynomials.
7. One 2D cartesian parametric cubic curve segment has 8 d.f. in 2D (12 in 3D).
  
  $x(t) = \sum_{i=0}^3 a_i t^i$,
  
  $y(t) = \sum_{i=0}^3 b_i t^i$, for $0\le t\le1$.
8. Requiring the graphic designer to enter those coefficients would be unpopular, so other APIs are common.
9. Most common is the Bezier formulation, where the segment is specified by 4 control points, which also total 8 d.f.: P0, P1, P2, and P3.
10. The generated curve starts at P0, goes near P1 and P2, and ends at P3.
11. The curve stays inside the control polygon, the convex hull of the control points. A flatter control polygon means a flatter curve. Designers like this.
12. A choice not taken would be to have the generated curve also go thru P2 and P3. That's called a Catmull-Rom-Oberhauser curve. However that would force the curve to go outside the control polygon by a nonintuitive amount. That is considered undesirable.
13. Instead of 4 control points, a parametric cubic curve can also be specified by a starting point and tangent, and an ending point and tangent. That also has 8 d.f. It's called a Hermite curve.
14. The three methods (polynomial, Bezier, Hermite) are easily interconvertible.
15. Remember that we're using connected strings or segments of cubic curves, and if the adjacent segments match tangents and curvatures at their common joint, then the joint is invisible.
16. Matching tangents (called $G^1$ or geometric continuity) is sufficient, and is weaker than matching the 1st derivative ($C^1$ or parametric continuity), since the 1st derivative has a direction (tangent) and a length. Most people do $C^1$ because it's easier and good enough. However $G^1$ gives you another degree of freedom to use in your design.
17. Similarly, matching the radius of curvature ($G^2$ or geometric continuity) is weaker than matching the 2nd derivative ($C^2$ or parametric continuity), but most people do parametric continuity.
18. Parametric continuity reduces each successive segment from 8 d.f. down to 2 d.f.
19. This is called a B-spline.
20. From a sequence of control points we generate a B-spline curve that is piecewise cubic and goes near, but probably not thru, any control point (except perhaps the ends).
21. Moving one control point moves the adjacent few spline pieces. That is called local control. Designers like it.
22. One spline segment can be replaced by two spline segments that, together, exactly draw the same curve. However they, together, have more control points for the graphic designer to move individually. So now the designer can edit smaller pieces of the total spline.
23. Extending this from 2D to 3D curves is obvious.
24. Extending to homogeneous coordinates is obvious. Increasing a control point's weight attracts the nearby part of the spline. This is called a rational spline.
25. Making two control points coincide means that the curvature will not be continuous at the adjacent joint.
  
  Making three control points coincide means that the tangent will not be continuous at the adjacent joint.
  
  Making four control points coincide means that the curve will not be continuous at the adjacent joint.
  
  Doing this is called making the curve (actually the knot sequence) Non-uniform. (The knots are the values of the parameter for the joints.)
26. Putting all this together gives a non-uniform rational B-spline, or a NURBS.
27. A B-spline surface is a grid of patches, each a bi-cubic parametric polynomial.
28. Each patch is controlled by a 4x4 grid of control points.
29. When adjacent patches match tangents and curvatures, the joint edge is invisible.
30. The surface math is an obvious extension of the curve math.
  1. $x(u,v) = \sum_{i=0}^3\sum_{j=0}^3 a_{ij} u^i v^j$
  2. $y, z$ are similar.
  3. One patch has 48 d.f. for Cartesian points, or 64 d.f. for homogeneous points, although most of those are used to establish continuity with adjacent patches.
My extra enrichment info on Splines.
The program I showed earlier is robotArm is Chapter 9.
To run program figure there, you may first need to fix an error in figure.html. Change InitShaders to initShaders.

Many of the textbook programs have errors that prevent them from running. You can see them in the console log.

4 Chapter 14 slides

14_1 Curves and surfaces.
Programs drawing the Utah teapot.

CG Class 24, Mon 2018-11-19

W Randolph Franklin, RPI — Sun, 18 Nov 2018 05:00:00 GMT

Table of contents

1 Thanksgiving trivia questions

When the native American Squanto greeted the Pilgrims in March 1621, what language did he use?
Where had he learned it?

2 Nice site on visualizing quaternions

Lessons by Grant Sanderson, Technology by Ben Eater.

here

3 Chapter 13 slides ctd

13_2 Polygon rendering. Includes clipping polygons, hidden surface algorithms.
13_3 Rasterization.

4 My commentaries

4.1 Clipping

Many of these algorithms were developed for HW w/o floating point, where even integer multiplication was expensive.
Efficiency is now less important in most cases (unless you're implementing in HW).
The idea of clipping with a 6-stage pipeline is important.
Jim Clark, a prof at Stanford, made a 12-stage pipeline using 12 copies of the same chip, and then left Stanford to found SGI.
1. Later he bankrolled Netscape and 2 other companies.
2. More recently he had the world's 4th largest yacht.

4.2 Polygon rendering

My note on Bresenham Line and Circle Drawing. Jack Bresenham, then at IBM invented these very fast ways to draw lines and circles with only integer addition and subtraction. My note gives step-by-step derivations by transforming slow and clear programs to fast and obscure programs.
My note on Two polygon filling algorithms.

4.3 Visibility methods

Here's my summary of problems with the main methods:

Painters:
1. The painter's algorithm is tricky when faces are close in Z.
2. Sorting the faces is hard and maybe impossible. Then you must split some faces.
3. However sometimes some objects are always in front of some other objects. Then you can render the background before the foreground.
Z-buffer:
1. Subpixel objects randomly appear and disappear (aliasing).
2. Artifacts occur when objects are closer than their Z-extent across one pixel.
3. This happens on the edge where two faces meet.
BSP tree:
1. In 3D, many faces must be split to build the tree.
The scanline algorithm can feed data straight to the video D/A. That was popular decades ago before frame buffers existed. It became popular again when frame buffers are the slowest part of the pipeline.
A real implementation, with a moving foreground and fixed background, might combine techniques.
References: wikipedia.

5 Textbook programs

Chapter 10 Mandlebrot showing serious computation in the fragment shader.

CG Class 23, Mon 2018-11-12

W Randolph Franklin, RPI — Mon, 12 Nov 2018 05:00:00 GMT

Table of contents

1 Upcoming classes

No class Wed 11/14.
No class Thurs 11/15 because Pres Jackson scheduled a faculty meeting then.
Next class Mon 11/19

2 Chapter 12 slides ctd

12_3 Graphical Objects and Scene Graphs.
12_4 Graphical Objects and Scene Graphs 2.
12_5 Rendering overview.

At this point we've learned enough WebGL. The course now switches to learn the fundamental graphics algorithms used in the rasterizer stage of the pipeline.

3 Chapter 13 slides

13_1 Clipping.

A lot of the material in the clipping slides is obsolete because machines are faster now. However perhaps the rendering is being done on a small coprocessor.

Big idea (first mentioned on Oct 20): Given any orthogonal projection and clip volume, we transform the object so that we can view the new object with projection (x,y,z) -> (x,y,0) and clip volume (-1,-1,-1) to (1,1,1) and get the same image. That's a normalization transformation'.

4 Textbook programs

Chapter 10 Mandlebrot showing serious computation in the fragment shader.

5 Fun stuff for Thanksgiving

CG Class 22, Thu 2018-11-01

W Randolph Franklin, RPI — Thu, 01 Nov 2018 04:00:00 GMT

Table of contents

1 Chapter 12 slides

2 Textbook programs

Chapter 9 simple hierarchical models
Chapter 10 Mandlebrot showing serious computation in the fragment shader.

CG Class 21, Wed 2018-10-31

W Randolph Franklin, RPI — Wed, 31 Oct 2018 04:00:00 GMT

Table of contents

1 DARPA

I've mentioned DARPA several times; they funded research into computer graphics starting in the 1970s. So here's some enrichment material on DARPA and autonomous vehicles. This shows why the US is the best in the world at R&D.

Autonomous vehicles make a marvelous example of a successful engineering project. Many different parts all have to work to make the total project succeed. They include laser rangefinding, image recognition, a road database accurate to the specific lane, and GPS navigation. This is also a model for government - university - industry interaction.

DARPA (The Defense Advanced Research Projects Agency) started this concept with a contest paying several $million in prizes. (DARPA started connecting computers in different cities with phone lines in 1968. This was the ARPAnet. They funded computer graphics in the 1970s and some early steps in virtual reality in the 1980s.)

In the 1st contest, the best vehicle failed after about 10 miles. Five vehicles completed the 130 mile course in the 2nd contest. The winning project leader was Sebastian Thrun, a Stanford CS prof. He quit and moved to Google, who has now been funding this for several years.

Finally, Dr Tony Tether, Director of DARPA when this happened, is an RPI BS EE grad. (He went to Stanford for his PhD).