MrW1-Going-off-of-Q12883-How-many-unique-angles-angles-in-Z-3-What-about-Z-n-

Question Number 13031 by FilupS last updated on 12/May/17

MrW 1

Going off of Q 12883

How many unique angles angles in Z^{3} ?

What about Z^{n} ?

Answered by mrW1 last updated on 12/May/17

F o r 1 ⩽ x_{i} ⩽ p_{i} w i t h 1 ⩽ i ⩽ n a n d x_{i} \in P^{n}

t h e n u m b e r o f u n i q u e a n g l e s f o r p o i n t s

P (x_{1,} x_{2}, \cdot \cdot \cdot, x_{i}, \cdot \cdot \cdot, x_{n}) i s

Φ = \underset{i = 1}{\prod^{n}} p_{i} - \underset{x_{1} = 1}{\sum^{p_{1}}} \underset{x_{2} = 1}{\sum^{p_{2}}} \cdot \cdot \cdot \underset{x_{i} = 1}{\sum^{p_{i}}} \cdot \cdot \cdot \underset{x_{n} = 1}{\sum^{p_{n}}} s i g n [G C D (x_{1}, x_{2}, \cdot \cdot \cdot, x_{i}, \cdot \cdot \cdot, x_{n}) - 1]

Commented by mrW1 last updated on 12/May/17

F o r - r_{i} ⩽ x_{i} ⩽ p_{i} w i t h 1 ⩽ i ⩽ n a n d x_{i} \in Z^{n}

t h e s o l u t i o n i s \dots

i n t h i n k i n g \dots h o w t o w r i t e t h e f o r m u l a

Commented by FilupS last updated on 12/May/17

why is x_{i} \in P^{n} ?

Commented by mrW1 last updated on 12/May/17

I w a n t e d t o s o l v e t h i s s i m p l e c a s e a t

f i r s t : 1 ⩽ x_{i} ⩽ p_{i} .

T h e r e s u l t f o r t h e c a s e - r_{i} ⩽ x_{i} ⩽ - 1 i s

t h e s a m e a s f o r 1 ⩽ x_{i} ⩽ r_{i} .

I t m e a n s t h e p r o b l e m f o r x_{i} \in Z^{n} w i l l

b e s o l v e d a s a c o u p l e o f p r o b l e m s

f o r x_{i} \in P^{n .}

Commented by mrW1 last updated on 12/May/17

E x a m p l e : t h e s o l u t i o n f o r

- 127 ⩽ x ⩽ 128

- 128 ⩽ y ⩽ 127

i s :

Φ = 128 \times 127 - \underset{i = 1}{\sum^{128}} \underset{j = 1}{\sum^{127}} s i g n (G C D (i, j) - 1)

+ 127 \times 127 - \underset{i = 1}{\sum^{127}} \underset{j = 1}{\sum^{127}} s i g n (G C D (i, j) - 1)

+ 127 \times 128 - \underset{i = 1}{\sum^{127}} \underset{j = 1}{\sum^{128}} s i g n (G C D (i, j) - 1)

+ 128 \times 128 - \underset{i = 1}{\sum^{128}} \underset{j = 1}{\sum^{128}} s i g n (G C D (i, j) - 1)

+ 4

w i t h l i n e 1 f o r q u a d r a n t 1 a n d

l i n e 2 f o r q u a d r a n t 2 a n d

l i n e 3 f o r q u a d r a n t 3 a n d

l i n e 4 f o r q u a d r a n t 4 a n d

l i n e 5 f o r p o i n t s o n a x e s

Commented by FilupS last updated on 12/May/17

This is all so fascinating and gives

me so many more questions to explore .

For example :

for u ⩽ x ⩽ m and v ⩽ y ⩽ n, x, y \in Z

What is the average angle ?

Commented by mrW1 last updated on 12/May/17

{L e t}^{'} s s a y - u ⩽ x ⩽ m a n d - v ⩽ y ⩽ n .

I n q u a d r a n t 1 w e h a v e

θ_{i j} = \tan^{- 1} (\frac{y}{x}) = \tan^{- 1} (\frac{j}{i}) = \tan^{- 1} (\frac{j^{'}}{i^{'}})

S_{1} = \underset{i = 1}{\sum^{m}} \underset{j = 1}{\sum^{n}} θ_{i j} = \underset{i = 1}{\sum^{m}} \underset{j = 1}{\sum^{n}} \tan^{- 1} (\frac{j}{i})

I n q u a d r a n t 2 w e h a v e

θ_{i j} = \tan^{- 1} (\frac{y}{x}) = π - \tan^{- 1} (\frac{j}{i}) = π - \tan^{- 1} (\frac{j^{'}}{i^{'}})

S_{2} = \underset{i = 1}{\sum^{u}} \underset{j = 1}{\sum^{n}} θ_{i j} = u n π - \underset{i = 1}{\sum^{u}} \underset{j = 1}{\sum^{n}} \tan^{- 1} (\frac{j}{i})

I n q u a d r a n t 3 w e h a v e

θ_{i j} = \tan^{- 1} (\frac{y}{x}) = π + \tan^{- 1} (\frac{j}{i}) = π + \tan^{- 1} (\frac{j^{'}}{i^{'}})

S_{3} = \underset{i = 1}{\sum^{u}} \underset{j = 1}{\sum^{v}} θ_{i j} = u v π + \underset{i = 1}{\sum^{u}} \underset{j = 1}{\sum^{v}} \tan^{- 1} (\frac{j}{i})

I n q u a d r a n t 4 w e h a v e

θ_{i j} = \tan^{- 1} (\frac{y}{x}) = 2 π - \tan^{- 1} (\frac{j}{i}) = 2 π - \tan^{- 1} (\frac{j^{'}}{i^{'}})

S_{4} = \underset{i = 1}{\sum^{m}} \underset{j = 1}{\sum^{v}} θ_{i j} = 2 m v π - \underset{i = 1}{\sum^{m}} \underset{j = 1}{\sum^{v}} \tan^{- 1} (\frac{j}{i})

O n + x a x i s : m p o i n t s w i t h a n g l e 0,

O n + y a x i s : n p o i n t s w i t h a n g l e \frac{π}{2},

O n - x a x i s : u p o i n t s w i t h a n g l e π,

O n - y a x i s : v p o i n t s w i t h a n g l e \frac{3 π}{2},

S_{5} = m \times 0 + n \times \frac{π}{2} + u π + v \times \frac{3 π}{2} = \frac{(n + 2 u + 3 v) π}{2}

S u m o f a l l a n g l e s :

S = S_{1} + S_{2} + S_{3} + S_{4} + S_{5}

= \underset{i = 1}{\sum^{m}} \underset{j = 1}{\sum^{n}} \tan^{- 1} (\frac{j}{i})

+ u n π - \underset{i = 1}{\sum^{u}} \underset{j = 1}{\sum^{n}} \tan^{- 1} (\frac{j}{i})

+ u v π + \underset{i = 1}{\sum^{u}} \underset{j = 1}{\sum^{v}} \tan^{- 1} (\frac{j}{i})

+ 2 m v π - \underset{i = 1}{\sum^{m}} \underset{j = 1}{\sum^{v}} \tan^{- 1} (\frac{j}{i})

+ \frac{(n + 2 u + 3 v) π}{2}

= \frac{(2 u n + 2 u v + 4 m v + n + 2 u + 3 v) π}{2}

+ [\underset{i = 1}{\sum^{m}} - \underset{i = 1}{\sum^{u}}] [\underset{j = 1}{\sum^{n}} \tan^{- 1} (\frac{j}{i}) - \underset{j = 1}{\sum^{v}} \tan^{- 1} (\frac{j}{i})]

T h e a v e r a g e a n g l e i s

\bar{θ} = \frac{S}{(m + u + 1) (n + v + 1)}

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