Ellipses and Circle Strength

[Tarontos]

So random thought I had while working on ellipses, they are stronger where the ellipse curves more than a circle and weak where they curve less than a circle, now noting this I realized that the curvature of a circle is relative,a circle can always have a tighter curve if it is drawn smaller.

So based off this I have a theory that the larger a circle is the weaker it is. This being because of the larger a circle the less it curves. 

So smaller circles should be stronger based off of this reasoning. this continued to lead me to a further question.

"What about Nebrask that has a massive circle and yet contains hoards of wild chalklings."

Theories as to why the Circle of Nebrask is strong enough to contain a constant onslaught please.(assuming of course you agree with my theory.)

otherwise thoughts and/or criticisms of the initial theory about circle strength.

 

Also personal belief as to why this is never brought up, all of the circles we see are roughly the same size, Nebrask is faraway, and, evidently, "classified".

 

Edit: additional thought the wider the ark the closer it is to a line of forbiddance this could theoretically give it some strength if the person got the intent right.

Edited June 5, 2013 by Tarontos

[PhilV]

The book also states that the thickness of a Line of Forbiddance affects its strength. Maybe Lines of Warding (i.e. circles) behave the same.

So the circle in Nebrask might be wicked thick to counteract its shallow curve.

[happyman]

It's also possible that the strength of a line of warding is determined by the ratio between the actual radius of the circle, and the local curvature.  The larger this ratio is, the stronger the line of warding.  After all, the effectiveness of many Rithmatic drawings depends on the global shape.

[Senor Feesh]

It's also possible that the strength of a line of warding is determined by the ratio between the actual radius of the circle, and the local curvature.  The larger this ratio is, the stronger the line of warding.  After all, the effectiveness of many Rithmatic drawings depends on the global shape.

 

I like this. So, if I understand correctly, the more any selected arc on the circle deviates from the perfect circle, the weaker that section will be, which matches what we know - are you also implying a direct correlation between how acute the curve is and the strength? That would explain how ellipses work... I think.

[happyman]

I like this. So, if I understand correctly, the more any selected arc on the circle deviates from the perfect circle, the weaker that section will be, which matches what we know - are you also implying a direct correlation between how acute the curve is and the strength? That would explain how ellipses work... I think.

 

Yes.  Mathematically, for any continuous curve, you can assign it a "local" radius of curvature at each point.  (For the mathematically inclined, this is based on the second derivative of position with respect to distance along the curve at each point).  I suspect that it is the relationship between this local radius of curvature and the global average radius of the curve that determines the strength.  (For other conic sections, if they are even possible, there is an analogous definition for "global radius," but it is clearly more complex since they all go to infinity in the ideal case.)  Specifically, the smaller the local radius of curvature is relative to the global size, the stronger the curve is there.

I suspect that it is a little bit more complicated.  This is the relationship if the circle is *actually* an ideal geometric shape, like either a circle or a very-close-to-circular ellipse.  I suspect that not having perfect geometry also weakens it, but in a different way.

[Ender]

/>The book also states that the thickness of a Line of Forbiddance affects its strength. Maybe Lines of Warding (i.e. circles) behave the same.

So the circle in Nebrask might be wicked thick to counteract its shallow curve.

No, thickness affects the HEIGHT of the the line, not the strength.

[PhilV]

No, thickness affects the HEIGHT of the the line, not the strength.

My mistake. Thickness affects the height of the forcefield, straightness affects the strength (p. 257 hardcover).