Is a circle really a circle?

[sb4]

I'm wondering -- when we create a circular spline by lathing a line into a cylinder and deleting the CPs at one end, is the resulting geometry a true circle or do splines merely provide a very good approximation of a circle?

 

-SB

[Rodney]

As much as an circle can be a circle it is a true circle when lathed in A:M.

 

Having said that... there have been times when the math (magnatude) has been off and lathing a sphere hasn't always resulted in the optimal circular spline.

When you add other splines to that one it can also cause the original spline's magnitude to adjust (try holding the SHIFT key down while adding the new spline to keep the old one in place)

 

If you are lathing with less than 8 Control Points you may want to increase that number to ensure you have enough CPs to maintain that spline.

[robcat2075]

I'm wondering -- when we create a circular spline by lathing a line into a cylinder and deleting the CPs at one end, is the resulting geometry a true circle or do splines merely provide a very good approximation of a circle?

 

-SB

 

 

It is an approximation. You can test that by reducing the lathe sections to three to see the roughest circle that A:M will make. Still not bad however.

 

Of course, a true perfect circle is achievable by neither man nor machine, we are all making approximations.

[sb4]

I'm wondering -- when we create a circular spline by lathing a line into a cylinder and deleting the CPs at one end, is the resulting geometry a true circle or do splines merely provide a very good approximation of a circle?

 

-SB

 

 

It is an approximation. You can test that by reducing the lathe sections to three to see the roughest circle that A:M will make. Still not bad however.

 

Of course, a true perfect circle is achievable by neither man nor machine, we are all making approximations.

 

Ok. I was thinking in the sense that if I use the circle equation x^2 + y^2 = r^2, for a given radius r I can solve to some precision for the circle points and plot the pixels. If I plot enough of them (say subresolution of my screen resolution), then in effect I have drawn a perfect circle as far as my screen can display it.

 

On the other hand, with splines, each spline has a formula that I can similarly plot to my screen resolution. The questions is: do the spline formulas mathematically reduce to the formula of a circle, or are they mathematically an approximation? An example would be a polynomial series expansion of a function -- it will exactly fit any functions that are polynomials of equal or lesser order, but not exact for other functions, although possibly very close.

 

When we lathe a circle, it is made up of splines, so unless splines can mathematically reduce to the formula of a circle, there will be an approximation in the points between the CPs; and unless we use enough splines so that the pixels computed are exactly the same as the circle formula pixels, I'd say it is visually an approximation of a circle.

 

-SB

[robcat2075]

When we lathe a circle, it is made up of splines, so unless splines can mathematically reduce to the formula of a circle, there will be an approximation in the points between the CPs; and unless we use enough splines so that the pixels computed are exactly the same as the circle formula pixels, I'd say it is visually an approximation of a circle.

 

I think we can presume the CPs are placed as accurately as digital measurement will permit and the splines between them are a fair approximation of circledom.

 

I woudl say the approximation is good enough to pass any scrutiny that would be incurred in story telling animation, which is all A:M is intended for. It is certainly superior to any polygon approximation of a circle.

[robcat2075]

Watch

in action!

[sb4]

When we lathe a circle, it is made up of splines, so unless splines can mathematically reduce to the formula of a circle, there will be an approximation in the points between the CPs; and unless we use enough splines so that the pixels computed are exactly the same as the circle formula pixels, I'd say it is visually an approximation of a circle.

 

I think we can presume the CPs are placed as accurately as digital measurement will permit and the splines between them are a fair approximation of circledom.

 

I woudl say the approximation is good enough to pass any scrutiny that would be incurred in story telling animation, which is all A:M is intended for. It is certainly superior to any polygon approximation of a circle.

 

Yes, as long as it fools most of the people most of the time

. I must have been wondering about making mechanical machines with physical constraints, if parts would jam, like an axel or something, or ball bearings -- nothing I'll be playing with anytime soon .

 

That guy drawing a circle is something -- like a good card trick.

 

-SB