2018年1月31日 · In fact, equation (4) defines a double Archimedean spiral (changing $(x,y)$ into $(-x,-y)$ doesn't change this equation). See picture below where the red curve is the Archimedean spiral, strictly speaking, and the magenta curve is its copy through a central symmetry. Equation (4) is less manageable than (3).

2019年1月10日 · Find the area of region inside the "first loop" of the Archimedes spiral (that is, the spiral for $0 \le ...

Control Equation of equidistantly sine wave on Archimedean Spiral Hot Network Questions zsh: History is skipping commands and integers+, how to correct this?

$\begingroup$ It would be nice if you could supply a formula for an Archimedean spiral. You might like to change the question tag: arc-length is very much differential geometry, it requires integrating certain derivatives.

2014年12月23日 · Differentiate Archimedes's spiral. Ask Question Asked 10 years, 1 month ago. Modified 7 years, 10 months ago.

To place objects equidistantly on an Archimedean (arithmetic) spiral, the arc length of the spiral has to increase linearly between the objects. This is what I have so far: The length of a spiral is determined by $$ l = \frac{a}{2}\left[\varphi\cdot\sqrt{1+\varphi^2}+\ln \left(\varphi+\sqrt{1+\varphi^2} \right)\right] $$ I presume that solving ...

2024年8月7日 · I recently discovered mathematical contents about (Archimedean) spiral and I find it fascinating. Gauss Wantzel's theorem ensures that we can neither perform an angle trisection nor construct $\pi$ only with a ruler and a compass, …

2017年6月25日 · I've already looked at some of answers to similar questions - there's this one, where the formula proposed in the question is already far beyond my understanding; and this one for which the answer seems to use a unit spiral rather than an absolute spiral. A more easy-to-understand solution would be very appreciated.

2019年2月7日 · Anyway, the Archimedes spiral seems to throw an additional wrench into the works, and it's got me very confused. The Archimedes spiral is $ r = a\theta $ If we do the test for symmetry under reflection through the vertical axis, we replace $(r, \theta)$ with $(-r, -\theta)$. Doing this we get $-r = a(-\theta) \\ r = a \theta$

2022年7月31日 · I was wondering, if you have a simple Archimedean spiral, defined as : $ r=b\times \theta $, with the angle $ \theta $ going from $ 0 $ to $ \Theta$ If you consider the spiral as a succession of infintesimally small circle arcs, couldn't you then just get the total length of the spiral by calculating the following integral :