Juan Carlos Ponce Campuzano
Independent Mathematics Educator
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
Rotating circles
By Juan Carlos Ponce Campuzano
Rotating circles
Construction of rotating circles.
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