Let A,B,C,D & E be points and AB,BC,CD & DE be line segments. A is fixed at (0,0) and B at (1,0). Angle ABC, BCD and CDE are always identical in measure. As the angles increase from theta = pi/2 to pi, E moves from (0,0) to (4,0). Is there a good function *either* for y=f(x) or y=f(theta)?Naraht (talk) 17:46, 24 July 2023 (UTC)[reply]

Are the four line segments all the same length (1)? What is x and what is y? Are they the coordinates of E? CodeTalker (talk) 20:32, 24 July 2023 (UTC)[reply]

It's not clear what theta is meant to be, but it's not hard to find the curves traced out by C, D and E in parametric form, assuming the segments are all the same length or at least have known lengths. If θ is the angle measured externally, so θ = 0 represents a straight angle, then C is the point (1+cosθ, sinθ) and the curve traced out is a circle. Next, D is the point (1+cosθ+cos2θ, sinθ+sin2θ) and the curve is a Limaçon trisectrix. If CD is not the same length as BC then the result is a general Limaçon. Either way it's a quartic curve and you can find Cartesian and Polar equations in the respective articles. Finally E is the point (1+cosθ+cos2θ+cos3θ, sinθ+sin2θ+sin3θ) and the curve traced out is a sextic known as Freeth's nephroid. We don't have an article on it because notability is dubious, but you can find more information, including Cartesian and polar equations, at mathcurve.com. --RDBury (talk) 22:36, 24 July 2023 (UTC)[reply]

Let ${\displaystyle \varphi =\pi -\vartheta }$be the supplement of the interior angle ${\displaystyle \vartheta ,}$so ${\displaystyle \varphi =0}$means that the points lie on a straight line. We identify the point ${\displaystyle (x,y)}$with the complex value ${\displaystyle x+iy.}$Assuming all segments have length 1, and putting ${\displaystyle z=\exp(i\varphi ),}$we have ${\displaystyle E=1+z+z^{2}+z^{3}.}$Now ${\displaystyle \exp(i\varphi )=\cos \varphi +i\sin \varphi .}$Using the shorthand notation ${\displaystyle z=c+is,}$we find:

${\displaystyle E=2c(c+1)(2c-1)+i(2sc(2c+1)).}$

When ${\displaystyle \varphi =\pi ,}$${\displaystyle c=0}$and ${\displaystyle s=1,}$so ${\displaystyle E=0.}$When ${\displaystyle \varphi =0,}$${\displaystyle c=1}$and ${\displaystyle s=0,}$so ${\displaystyle E=4.}$Halfway, at ${\displaystyle \varphi =\pi /2,}$${\displaystyle c=s={\sqrt {1/2}},}$so ${\displaystyle E=1+i(1+{\sqrt {2}}).}$ --Lambiam 22:44, 24 July 2023 (UTC)[reply]

OP here. AB, BC, CD and DE are all of length 1. Angles are *interior* , when at Pi/2, A and E are at the same point. When at Pi, the line is straight.Naraht (talk) 00:32, 25 July 2023 (UTC)[reply]