Theory of Accuracy: Perfect Circles & Real Life

[Zuryn]

If you were to create a seemingly perfect circle on your computer and then zoom in on it, wouldn't it have line segments rather than all curves? Of course, with the computer it is a matter of pixels. But what about in real life?

 

[Tide]

Obviously, you cannot have a "perfect" circle (is there any other kind?) on a computer because you are limited by the pixel resolution and aspect ratio. More to the point, whether your circle consists of pixel resolution or straight line segments depends entirely on the algorithm one uses to produce the circles.

Mathematically, the best you can do is to approximate the circle because your computer can only deal with rational values of the coordinates and the set of points that get plotted has finite cardinality.

 

[Zuryn]

Obviously, you cannot have a "perfect" circle (is there any other kind?) on a computer because you are limited by the pixel resolution and aspect ratio. More to the point, whether your circle consists of pixel resolution or straight line segments depends entirely on the algorithm one uses to produce the circles.

Mathematically, the best you can do is to approximate the circle because your computer can only deal with rational values of the coordinates and the set of points that get plotted has finite cardinality.

I was just using the computer as a visual perspective as to what I mean.

 

[Rocko]

i do not believe pefect anything exists, especially not a percect circle, recently i recall hearing they have found pi to some extraordinary number which i have forgot but its huge and there is no limit to how precise you can get pi it goes on forever. perhaps we need a new way of thinking and a new number system to properly describe it but i can't contrive it.

 

[rcgldr]

Drawing circles with computers is a lost art.

Very early graphical CRT's had the ability to draw arcs. The parameters included radius, initial direction, arc length (angle of arc), and I think the rate of chane on the radius, but since those early days, it was decided to draw everything on screens and plotters by using straight line segments. The commands to draw arcs were dropped on later graphical screens and plotters.

I'm not sure why this was done. Plotters had the hardware to be able to draw arcs, just no means of instructing them to do this. Would have greatly sped up the time it took to plot curves.

The old electronic analog computers (differential equation solvers) also had displays that could draw arcs. If a shape could be described as a differential equation, the analog computer could draw it (or at least a portion of it, limited range, like -100 volts to +100 volts).

To generate a sine or cosine wave, you just fed -x into 2nd derivative of x (x double dot), and setup initial parameters (for x (position) and x dot (slope)), then let the analog computer run. Use a cosine wave to drive x, and a sine wave to drive y, on the CRT and you got a circle drawn on the CRT. IIRC, there was a program to let you apply forces to a satellite orbiting around a planet in a 2d world. You could rotate the satellite and fire rockets, which changed the shape of the orbit (including transitioning into a hyperbola). Sort of a pre-cursor to the lunar lander programs.

Nowdays, numerical integration is used instead. Analog computing is a lost art. I'm 52 years old and just barely had a brief encounter with an analog computer, and only because the college I went to had kept their ancient beast up and running at the time.

 

[rcgldr]

But what about in real life?

The quantum mechanical physicists tell us that nothing is continous, not even straight lines. Rather than matter moving smoothly, stuff just dissappears, then reappears in it's next location. There's a concept of a smallest distance, but I don't know if it's value is known yet. Same goes for levels of energy, electronic charge, gravity, all have a smallest amount of increment.

 

[Chrono]

Obviously, you cannot have a "perfect" circle (is there any other kind?) on a computer because you are limited by the pixel resolution and aspect ratio. More to the point, whether your circle consists of pixel resolution or straight line segments depends entirely on the algorithm one uses to produce the circles.

I beg to differ. My math professor always draws perfect circles. Not only that, he also draws perfect parallelograms, bowls, saddles, planes, and cones. We always tend to agree with him.

 

[Gokul43201]

Mathematically, perfect circles exist. [itex]|\mathbf r-\mathbf C]|=R[/itex] defines a perfect circle.

If you try to draw a circle in real life on a meterial background, you will at the least be limited by the resolution offered by atomic spacings.

What about drawing circles in space ? How about trying to trace a circle as the locus of the center of mass of some celestial object ? In general, this will not give you a circle; even in the inertial frame of some reference body, the path of an orbiter will be an ellipse. And if you try to engineer it, you run into limitations of accuracy.

 

[Rocko]

I can imagine only 2 different perfect circles in the physical world which may seem kind of like the same idea:
1. 2 of the smallest physical particles (not sure if there is a clear definition what this is) in a completely empty space, by that i mean nothing else exist besides them and set 1 as the inertial reference frame and watch the other complete perfectly circular orbits around it unpractical since our world obviously conatins more than 2 particles.

2. then imagine the big bang right at the release. I believe, would it not move out at first in concentric perfect circles, er spheres i guess? Basically suggesting the edge of the universe is a perfect circle.