Mysterious
Rotating
Circles
Centers
C1=?
C_1 = \;?
C2=?
C_2 = \;?
⋮
\vdots
Cn=?
C_n = \;?
C4=?
C_4 = \;?
C3
C_3
=?
= \;?
⋮
\vdots
Radii
r1=?
r_1 =\;?
r2=?
r_2 =\; ?
r3=?
r_3 = \;?
r4=?
r_4 =\; ?
rn=?
r_n =\; ?
C1P1
=r1
C1P1=r1
C2P2=r2
=r1s
C1P1=r1
C2P2=r2
C3P3=r3
=r1s
=r1s2
=r2s
=(r1s)s
C1P1=r1
C2P2=r2
C3P3=r3
Here 0<s<1
⋮
CnPn=rn=r1sn−1
=r1s
=r1s2
=r2s
=(r1s)s
r1=C1C2+C2P1,
⇒C1C2=r1−C2P1
=r1−r1s
=r1(1−s)
r1=C1C2+C2P1,
⇒C1C2=r1−C2P1
r2=C2C3+C3P2,
⇒C2C3=r2−C3P2
=r1−r1s
=r1(1−s)
=r1s−r1s2
=r1s(1−s)
r1=C1C2+C2P1,
⇒C1C2=r1−C2P1
r2=C2C3+C3P2,
⇒C2C3=r2−C3P2
=r1−r1s
=r1(1−s)
=r1s−r1s2
=r1s(1−s)
C3C4=r1s2(1−s)
r1=C1C2+C2P1,
⇒C1C2=r1−C2P1
⋮
CnCn+1=r1sn−1(1−s)
r2=C2C3+C3P2,
⇒C2C3=r2−C3P2
=r1−r1s
=r1(1−s)
=r1s−r1s2
=r1s(1−s)
C3C4=r1s2(1−s)
Again 0<s<1!
C1C2=r1(1−s)
⋮
CnCn+1=r1sn−1(1−s)
C2C3=r1s(1−s)
C3C4=r1s2(1−s)
Lenght of segments
Now let's find the coordinates of each point
Cn=(?,?)
C2=(r1(1−s)cos(θ),r1(1−s)sin(θ))
C_2 = \bigg( r_1(1-s) \cos(\theta), r_1 (1-s)\sin(\theta) \bigg)
θ
\theta
0≤θ≤2π
0\leq \theta\leq 2 \pi
C1C2=r1(1−s)
\overline{C_1C_2} =r_1(1-s)
C1=(0,0)
C_1 = (0, 0)
Set
C2=(r1(1−s)cos(θ),r1(1−s)sin(θ))
C_2 = \bigg( r_1(1-s) \cos(\theta), r_1 (1-s)\sin(\theta) \bigg)
=(r1(1−s);θ)
= \bigg( r_1(1-s) ; \theta \bigg)
θ
\theta
0≤θ≤2π
0\leq \theta\leq 2 \pi
C1=(0,0)
C_1 = (0, 0)
Set
←Polar form
C1=(0,0)
C_1 = (0, 0)
C2=(r1(1−s);θ)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
Set
C3
C_3
=(r1(1−s);θ)
= \bigg( r_1(1-s) ;\theta \bigg)
C1C2=r1(1−s)
\overline{C_1C_2} =r_1(1-s)
C2C3=r1s(1−s)
\overline{C_2C_3} =r_1s(1-s)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C1=(0,0)
C_1 = (0, 0)
C2=(r1(1−s);θ)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
Set
C3
C_3
=(r1(1−s);θ)
= \bigg( r_1(1-s) ;\theta \bigg)
C1C2=r1(1−s)
\overline{C_1C_2} =r_1(1-s)
C2C3=r1s(1−s)
\overline{C_2C_3} =r_1s(1-s)
θ
\theta
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C1=(0,0)
C_1 = (0, 0)
C2=(r1(1−s);θ)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
Set
C3
C_3
=(r1(1−s);θ)
= \bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C1C2=r1(1−s)
\overline{C_1C_2} =r_1(1-s)
C2C3=r1s(1−s)
\overline{C_2C_3} =r_1s(1-s)
2θ
2\theta
(r1(1−s);θ)
\bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C1=(0,0)
C_1 = (0, 0)
C2=(r1(1−s);θ)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
Set
C3
C_3
=(r1(1−s);θ)
= \bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C4=
C_4=
C1C2=r1(1−s)
\overline{C_1C_2} =r_1(1-s)
C2C3=r1s(1−s)
\overline{C_2C_3} =r_1s(1-s)
C3C4=r1s2(1−s)
\overline{C_3C_4} =r_1s^2(1-s)
(r1(1−s);θ)
\bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C1=(0,0)
C_1 = (0, 0)
C2=(r1(1−s);θ)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
Set
C3
C_3
=(r1(1−s);θ)
= \bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
(r1(1−s);θ)
\bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C4=
C_4=
+(rs2(1−s);3θ)
+\bigg(rs^2(1-s);3\theta \bigg)
C1C2=r1(1−s)
\overline{C_1C_2} =r_1(1-s)
C2C3=r1s(1−s)
\overline{C_2C_3} =r_1s(1-s)
C3C4=r1s2(1−s)
\overline{C_3C_4} =r_1s^2(1-s)
C1=(0,0)
C_1 = (0, 0)
C2=(r1(1−s);θ)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
Set
C3
C_3
=(r1(1−s);θ)
= \bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
C1C2=r1(1−s)
\overline{C_1C_2} =r_1(1-s)
C4=
C_4=
C2C3=r1s(1−s)
\overline{C_2C_3} =r_1s(1-s)
C3C4=r1s2(1−s)
\overline{C_3C_4} =r_1s^2(1-s)
3θ
3\theta
(r1(1−s);θ)
\bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
+(r1s2(1−s);3θ)
+\bigg(r_1s^2(1-s);3\theta \bigg)
C1=(0,0)
C_1 = (0, 0)
C2=(r1(1−s);θ)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
Set
C3
C_3
=(r1(1−s);θ)
= \bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
⋮
\vdots
Cn=k=0∑n−2(r1sk(1−s);(k+1)θ)
C_n = \sum_{k=0}^{n-2}\bigg( r_1s^k(1-s) ;(k+1)\theta \bigg)
C4=
C_4=
(r1(1−s);θ)
\bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
+(r1s2(1−s);3θ)
+\bigg(r_1s^2(1-s);3\theta \bigg)
0≤θ≤2π
0\leq \theta \leq 2\pi
0<s<1
0\lt s\lt1
Radii
Centers
r1=r1s0
r_1 = r_1 s^0
r2=r1s1
r_2 = r_1 s^1
r3=r1s2
r_3 = r_1 s^2
r4=r1s3
r_4 = r_1 s^3
⋮
\vdots
rn=r1sn−1
r_n = r_1 s^{n-1}
C2=(r1(1−s);θ)
C_2 = \bigg( r_1(1-s) ;\theta \bigg)
C3
C_3
=(r1(1−s);θ)
= \bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
⋮
\vdots
Cn=k=0∑n−2(r1sk(1−s);(k+1)θ)
C_n = \sum_{k=0}^{n-2}\bigg( r_1s^k(1-s) ;(k+1)\theta \bigg)
C4=
C_4=
(r1(1−s);θ)
\bigg( r_1(1-s) ;\theta \bigg)
+(r1s(1−s);2θ)
+ \bigg( r_1s(1-s); 2\theta \bigg)
+(r1s2(1−s);3θ)
+\bigg(r_1s^2(1-s);3\theta \bigg)
C1=(0,0)
C_1 = (0, 0)
0≤θ≤2π
0\leq \theta \leq 2\pi
0<s<1
0\lt s\lt1
Radii
Centers
r1=s0
r_1 = s^0
r2=s1
r_2 = s^1
r3=s2
r_3 = s^2
r4=s3
r_4 = s^3
⋮
\vdots
rn=sn−1
r_n = s^{n-1}
r1=1
r_1=1
C2=((1−s);θ)
C_2 = \bigg( (1-s) ;\theta \bigg)
C3
C_3
=((1−s);θ)
= \bigg( (1-s) ;\theta \bigg)
+(s(1−s);2θ)
+ \bigg( s(1-s); 2\theta \bigg)
⋮
\vdots
Cn=k=0∑n−2(sk(1−s);(k+1)θ)
C_n = \sum_{k=0}^{n-2}\bigg( s^k(1-s) ;(k+1)\theta \bigg)
C4=
C_4=
((1−s);θ)
\bigg( (1-s) ;\theta \bigg)
+(s(1−s);2θ)
+ \bigg( s(1-s); 2\theta \bigg)
+(s2(1−s);3θ)
+\bigg(s^2(1-s);3\theta \bigg)
C1=(0,0)
C_1 = (0, 0)
s = Slider(0.1, 1.5, 0.01) t = Slider(0, 2pi, 0.01) n = Slider(0, 50, 1) Ln = 0...n LR = Zip(s^k, k, Ln) LP = Join({(0, 0)}, Zip((s^k * (1 - s); (k+1) * t), k, Ln)) LS = Zip(Sum(LP, k), k, Ln+1) LC = Zip(Circle(P, r), P, LS, r, LR)
GeoGebra Script
Check Ben Sparks' construction
using the Spreadsheet in GeoGebra
Link in the description
Mathematical topics
- Geometry:
- Rotations, Dilations, Tangency of circles
- Analytic geometry:
- Cartesian and polar coordinates
- Recursive Sequences and Series
There is also a conexion with
Complex Numbers!
Plot all the centers
and join them
k=0∑nzk
\sum_{k=0}^{n}z^k
for z=x+iy
Geometrical representation
of the geometric series
=(reiθ)k
=\left(re^{i\theta}\right)^k
zk
z^k
=rkei(kθ)
=r^ke^{i (k \theta)}
k=0∑nzk
\sum_{k=0}^{n}z^k
=k=0∑nrkei(kθ)
=\sum_{k=0}^{n}r^k e^{i(k\theta)}
z=x+iy
z=x+iy
=reiθ
=re^{i\theta}
Geometrical representation of the
Geometric series
Geometric series
Cn=k=0∑n−2(sk(1−s);(k+1)θ)
C_n = \sum_{k=0}^{n-2}\bigg( s^k(1-s) ;(k+1)\theta \bigg)
k=0∑nzk
\sum_{k=0}^{n}z^k
=k=0∑nrkei(kθ)
=\sum_{k=0}^{n}r^k e^{i(k\theta)}
Relationship between the Geometric series and the points Cn
k=0∑nzk
\sum_{k=0}^{n}z^k
Cn=k=0∑n−2(sk(1−s);(k+1)θ)
C_n = \sum_{k=0}^{n-2}\bigg( s^k(1-s) ;(k+1)\theta \bigg)
=k=0∑nrkei(kθ)
=\sum_{k=0}^{n}r^k e^{i(k\theta)}
Relationship between the Geometric series and the points Cn
k=0∑nzk
\sum_{k=0}^{n}z^k
Cn
C_n
=k=0∑n(rkcos(kθ),rksin(kθ))
= \sum_{k=0}^{n}\bigg(r^k\cos(k\theta), r^k\sin(k\theta) \bigg)
=k=0∑n−2(sk(1−s)cos((k+1)θ),sk(1−s)sin((k+1)θ))
= \sum_{k=0}^{n-2}\bigg( s^k(1-s) \cos((k+1)\theta), s^k(1-s) \sin((k+1)\theta) \bigg)
Re-write both in cartesian form!
Relationship between the Geometric series and the points Cn
k=0∑nzk
\sum_{k=0}^{n}z^k
=k=0∑n(rkcos(kθ),rksin(kθ))
= \sum_{k=0}^{n}\bigg(r^k\cos(k\theta), r^k\sin(k\theta) \bigg)
Cn=(1−s)k=0∑n−2(skcos((k+1)θ),sksin((k+1)θ))
C_n = (1-s) \sum_{k=0}^{n-2}\bigg( s^k \cos((k+1)\theta), s^k \sin((k+1)\theta) \bigg)
Factorize (1−s)
Relationship between the Geometric series and the points Cn
k=0∑nzk
\sum_{k=0}^{n}z^k
=(r0cos(0),r0sin(0))+(r1cos(θ),r1sin(θ))+⋯+(rncos(nθ),rnsin(nθ))
= \bigg( r^0\cos(0), r^0\sin (0) \bigg) + \bigg( r^1\cos(\theta), r^1\sin (\theta) \bigg)+\cdots + \bigg( r^n\cos(n\theta), r^n\sin (n\theta) \bigg)
=k=0∑n(rkcos(kθ),rksin(kθ))
= \sum_{k=0}^{n}\bigg(r^k\cos(k\theta), r^k\sin(k\theta) \bigg)
Cn=(1−s)k=0∑n−2(skcos((k+1)θ),sksin((k+1)θ))
C_n = (1-s) \sum_{k=0}^{n-2}\bigg( s^k \cos((k+1)\theta), s^k \sin((k+1)\theta) \bigg)
Relationship between the Geometric series and the points Cn
k=0∑nzk
\sum_{k=0}^{n}z^k
=(r0cos(0),r0sin(0))+(r1cos(θ),r1sin(θ))+⋯+(rncos(nθ),rnsin(nθ))
= \bigg( r^0\cos(0), r^0\sin (0) \bigg) + \bigg( r^1\cos(\theta), r^1\sin (\theta) \bigg)+\cdots + \bigg( r^n\cos(n\theta), r^n\sin (n\theta) \bigg)
=(1−s)[(s0cos(θ),s0sin(θ))+(s1cos(2θ),s1sin(2θ))+
= (1-s)\bigg[\big( s^0\cos(\theta),s^0 \sin(\theta) \big) + \big(s^1 \cos(2\theta), s^1\sin(2\theta) \big)+\bigg.
=k=0∑n(rkcos(kθ),rksin(kθ))
= \sum_{k=0}^{n}\bigg(r^k\cos(k\theta), r^k\sin(k\theta) \bigg)
Cn=(1−s)k=0∑n−2(skcos((k+1)θ),sksin((k+1)θ))
C_n = (1-s) \sum_{k=0}^{n-2}\bigg( s^k \cos((k+1)\theta), s^k \sin((k+1)\theta) \bigg)
+⋯+(sn−2cos((n−2)θ),sn−2sin((n−2)θ))]
\bigg. +\cdots + \big( s^{n-2} \cos((n-2)\theta), s^{n-2}\sin((n-2)\theta) \big) \bigg]
Relationship between the Geometric series and the points Cn
r0sin(0)+r1sin(θ)+r2sin(2θ)+⋯+rnsin(nθ))
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
s0sin(θ)+s1sin(2θ)+s2sin(3θ)+⋯+sn−2sin((n−2)θ))]
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
(s0cos(θ)+s1cos(2θ)+s2cos(3θ)+⋯+sn−2cos((n−2)θ),
\bigg .\bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
Cn=
C_n =
(1−s)[
(1-s) \bigg [ \bigg.
k=0∑nzk=
\sum_{k=0}^{n}z^k=
(r0cos(0)+r1cos(θ)+r2cos(2θ)+⋯+rncos(nθ),
\bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
Relationship between the Geometric series and the points Cn
k=0∑nzk=
\sum_{k=0}^{n}z^k=
(r0cos(0)+r1cos(θ)+r2cos(2θ)+⋯+rncos(nθ),
\bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
r0sin(0)+r1sin(θ)+r2sin(2θ)+⋯+rnsin(nθ))
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
(s0cos(θ)+s1cos(2θ)+s2cos(3θ)+⋯+sn−2cos((n−2)θ),
\bigg .\bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
s0sin(θ)+s1sin(2θ)+s2sin(3θ)+⋯+sn−2sin((n−2)θ))]
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
Cn=
C_n =
(1−s)[
(1-s) \bigg [ \bigg.
Relationship between the Geometric series and the points Cn
k=0∑nzk=
\sum_{k=0}^{n}z^k=
(r0cos(0)+r1cos(θ)+r2cos(2θ)+⋯+rncos(nθ),
\bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
r0sin(0)+r1sin(θ)+r2sin(2θ)+⋯+rnsin(nθ))
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
(s0cos(θ)+s1cos(2θ)+s2cos(3θ)+⋯+sn−2cos((n−2)θ),
\bigg .\bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
s0sin(θ)+s1sin(2θ)+s2sin(3θ)+⋯+sn−2sin((n−2)θ))]
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
Cn=
C_n =
(1−s)[
(1-s) \bigg [ \bigg.
Relationship between the Geometric series and the points Cn
k=0∑nzk
\sum_{k=0}^{n}z^k
Cn
C_n
=(r0cos(0)+r1cos(θ)+r2cos(2θ)+⋯+rncos(nθ),
= \bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
r0sin(0)+r1sin(θ)+r2sin(2θ)+⋯+rnsin(nθ))
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
=(1−s)[(s0cos(θ)+s1cos(2θ)+s2cos(3θ)+⋯+sn−2cos((n−2)θ),
=(1-s) \bigg [ \bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
s0sin(θ)+s1sin(2θ)+s2sin(3θ)+⋯+sn−2sin((n−2)θ))]
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
k=0∑nRkcos(kθ),k=0∑nRksin(kθ)
\sum_{k=0}^{n}\mathcal R^k \cos(k \theta),\quad \sum_{k=0}^{n}\mathcal R^k \sin(k \theta)
Relationship between the Geometric series and the points Cn
k=0∑nzk
\sum_{k=0}^{n}z^k
Cn
C_n
=(r0cos(0)+r1cos(θ)+r2cos(2θ)+⋯+rncos(nθ),
= \bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
r0sin(0)+r1sin(θ)+r2sin(2θ)+⋯+rnsin(nθ))
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
=(1−s)[(s0cos(θ)+s1cos(2θ)+s2cos(3θ)+⋯+sn−2cos((n−2)θ),
=(1-s) \bigg [ \bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
s0sin(θ)+s1sin(2θ)+s2sin(3θ)+⋯+sn−2sin((n−2)θ))]
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
k=0∑nRkcos(kθ),k=0∑nRksin(kθ)
\sum_{k=0}^{n}\mathcal R^k \cos(k \theta),\quad \sum_{k=0}^{n}\mathcal R^k \sin(k \theta)
Relationship between the Geometric series and the points Cn
k=0∑nzk
\sum_{k=0}^{n}z^k
Cn
C_n
=(r0cos(0)+r1cos(θ)+r2cos(2θ)+⋯+rncos(nθ),
= \bigg(r^0 \cos(0) + r^1\cos(\theta) +r^2\cos(2\theta)+ \cdots + r^n \cos(n\theta), \bigg.
r0sin(0)+r1sin(θ)+r2sin(2θ)+⋯+rnsin(nθ))
\bigg. r^0\sin(0) + r^1\sin(\theta) + r^2\sin(2\theta) +\cdots + r^n \sin(n\theta) \bigg)
=(1−s)[(s0cos(θ)+s1cos(2θ)+s2cos(3θ)+⋯+sn−2cos((n−2)θ),
=(1-s) \bigg [ \bigg(s^0 \cos(\theta) + s^1 \cos(2\theta) +s^2 \cos(3\theta) + \cdots + s^{n-2} \cos((n-2)\theta), \bigg. \bigg.
s0sin(θ)+s1sin(2θ)+s2sin(3θ)+⋯+sn−2sin((n−2)θ))]
\bigg. \bigg. s^0 \sin(\theta) + s^1 \sin(2\theta) + s^2 \sin(3\theta) + \cdots+ s^{n-2} \sin((n-2)\theta) \bigg) \bigg]
k=0∑nRkcos(kθ),k=0∑nRksin(kθ)
\sum_{k=0}^{n}\mathcal R^k \cos(k \theta),\quad \sum_{k=0}^{n}\mathcal R^k \sin(k \theta)
k=0∑nzk,
\sum_{k=0}^{n}z^k,
we just need to adjust the expression
To obtain the same result
⎩⎨⎧z=reiθ,0<r<10≤θ≤2π
\left\{
\begin{array}{l}
z= re^{i\theta},\\
0\lt r\lt 1\\
0\leq \theta\leq 2\pi
\end{array}
\right.
with
(r1−1)
\left(\frac{1}{r}-1\right)
⎩⎨⎧z=reiθ,0<r<10≤θ≤2π
\left\{
\begin{array}{l}
z= re^{i\theta},\\
0\lt r\lt 1\\
0\leq \theta\leq 2\pi
\end{array}
\right.
with
k=0∑nzk,
\sum_{k=0}^{n}z^k,
(1−r1)+
\left(1-\frac{1}{r}\right) +
Multiply by
Add this
⎩⎨⎧z=reiθ,0<r<10≤θ≤2π
\left\{
\begin{array}{l}
z= re^{i\theta},\\
0\lt r\lt 1\\
0\leq \theta\leq 2\pi
\end{array}
\right.
with
Challenge: Use this approach to built it in GeoGebra
(r1−1)
\left(\frac{1}{r}-1\right)
k=0∑nzk,
\sum_{k=0}^{n}z^k,
(1−r1)+
\left(1-\frac{1}{r}\right) +
GeoGebra
Desmos
p5.js
Links in the description
Thanks for
watching!
Patreons:
Christopher-Alexander Hermans, Maciej Lasota, Miguel Díaz, bleh, Dennis Watson, Doug Kuhlmann, Newnome Beauton, Adam Parrott, Sophia Wood (Fractal Kitty), pmben, Abei, Edward Huff.
Thanks for
watching!
Patreons:
Christopher-Alexander Hermans, Maciej Lasota, Miguel Díaz, bleh, Dennis Watson, Doug Kuhlmann, Newnome Beauton, Adam Parrott, Sophia Wood (Fractal Kitty), pmben, Abei, Edward Huff.
Mysterious Rotating Circles