Post by NobodyPost by b***@coolfone.comze.comIs it possible to use a Bezier curve with one or two control points to
represent a perfectly rounded corner?
I know this is after the fact, but... This can be *approximated* quite
well with either a Quadratic (3 total points) or Cubic (4 total points)
Bezier. The Quadratic [I meant Cubic] version is in my free on-line book (On
Interpolation), in the appendix (Site below). I see I neglected to get
the Quadratic version into the book.
It's unfortunate that your book doesn't discuss rational curves, as they
allow circular arcs (and, in fact, any conic section) to be represented exactly,
are directly supported by 3D graphics APIs such as OpenGL and DirectX
(as a direct consequence of using homogeneous coordinates), and are
among the most widely-used curves (NURBS are probably the most common
curved primitives in 3D modelling software).
Steve Adds:
Not really. I addressed fundamentals and explain my rationale in the preface as well as a “Why I did it” document on my site. I provided understanding of the math concepts, not instructions for any specific computer language or graphics engine capability.
While all you say is true, an understanding of NURBS was beyond my need and I believe there are (or were) already enough sources describing NURBS. I saw a need for a simplified path to understanding the basics and the reader could, then, go on, if desired, to the more advanced topics with a basic understanding under his belt.
There had to be others like me that only needed a better path to understand the basics. In addition, doing a good job explaining NURBS would require considerable effort that I had no need for and never did try to understand. Writing well took considerable time as it was. For Laser light show graphics, approximations work very well. In fact some of the simplifications in motion control are alarmingly trivial, yet extremely effective on the screen. The motion of a bouncing ball with only 3 different speeds (six velocities) looks completely smooth.
The quest for the perfect circle is a prevalent one in postings and the literature. It is an admirable goal, but my original graphics engine ran with a clock speed of just under 1 MHz. (yes ONE MEGA) and computational efficiency was first priority. I was, in fact, counting processor cycles and *not* using some of the processors (6809) advanced indexing instructions because other methods of doing it myself proved faster. Page 144 in my Appendix shows a cubic half circle with 0.5% radial error and it is quite indistinguishable from a true circle. A cubic quarter circle has 0.02% radial error. For on-screen graphics that will be viewed, there is no need for a perfect fit in shape nor motion.
Regards, Steve
I just posted a "Shameless Plug" that has links to the book.