,2X28ZWZ80SS7.
Sa2X, :aXZ;
227 i ,Sa.
7B . XS.
:0 ; ;Z
Xr . 8i
0. ; S:
8 , ,r
2S;8ZXSSSX0ZXi i XS0aXSSXS02iM
aX2@ :SaX , SaS MSZ,
;SS 8 ZS, . 2S7 8 :Sa
:2: W ;r,iS :2 22
ar W M; M X;
iZ W Z X M ;Z
,a W M . 2i @ B
;; W i; ; W B M
M .a @ . M W W
B 0 M r 0 0 W
0 W 0 . M Z, @
S W :0ZXa0i i , 8 00XS0S Xi W
B ;r :SS2 r@r Ma. iXar X B
:, M ,i,..,,,,i,,,... .MM, .,,,,,. MM .ii,, aB8 ,,i,,,,,i,,.,, B.i.
B Z raS X0 ,M .aSi .Z B
Z 2i 8XX20aXrZa MiXX08XXS; X: W
.Z 8, a , M W M
B 2 M ; .X B W
M M M , 8. W W
7: W S : M B M
Xi W @ : ,X .8 Z
SX W r7 Z M M
.8 W a2 M ,a
2Z W rra, .8 ,W.
SS 0 SS . .S: 0 rS:
Sa;SX :Xa2 , :a2X MiSZ
BZr0aXSSSXZ8S8i i SZ20SXSSSX0a;M.
Z , 7
,Z ; ,Z
0 , r2
2X ; W
.SX , ZX
BS : ;aa
SaS7 XZZ
2S2XZ8Zaa88aXZX7
.
Polar equation: rp = apcos(pΘ)
Sinusoidal spirals can have any rational number p in the formula above. Many standard curves occur as sinusoidal spirals.
Sinusoidal spirals were first studied by Maclaurin, of Maclaurin series fame. They are not, of course, true spirals.
The pedal curve of sinusoidal spirals, when the pedal point is the pole, is another sinusoidal spiral.
The sinusoidal spiral rp = apcos(pΘ) inverts to rp = ap/cos(pΘ) if the centre of inversion is taken at the pole.