1. VOLUME FORMULA OF N-DIMENSIONAL HYPERSPHERE (ITERATIVE METHOD)
WCU 정보통신사업단 Colloquium
VOLUME FORMULA OF N-DIMENSIONAL HYPERSPHERE (ITERATIVE METHOD) 
August 7, 2009
Prof. Woonchul Ham

2. Contents Motivation Mathematical Tool Formula Another Method
Iterative Methods

3. Motivation 1 Linear Mapping Range Space Domain Space Nonlinear Mapping
Function2
Shape2
Area/Volume2
Domain Space
Function1
Shape1
Area/Volume1
Nonlinear Mapping

4. Motivation 2-1 Linear Mapping IDEAL CASE Range Space Domain Space EASY
Function2 (similar)
Shape2 (similar)
Area/Volume2
Domain Space
Function1
Shape1
Area/Volume1
IDEAL CASE
EASY
MATRIX
DETERMINANT >1 Expansion
<1 Compression

5. Motivation 2-2 Linear Mapping Range Space Domain Space Eig=-0.37,5.37
Det(A)=-2
Eig. Vec
=[ ]
, [ 0.41, 0.90]

6. Motivation 3-1 Nonlinear Mapping REAL SYSTEM Range Space Domain Space
Function2 (differ)
Shape2 (differ)
Area/Volume2
Domain Space
Function1
Shape1
Area/Volume1
REAL SYSTEM
Complicate
LINEARIZED MATRIX
(Locally True)
DETERMINANT >1 Expansion
<1 Compression

7. Motivation3-2 Nonlinear Mapping Range Space Domain Space
Deformed Image
Reflected Image
Domain Space
Image of Woman
Concave Mirrored Image

8. Motivation 4-1
2- Dimensional Mapping
Square :
Circle:

9. Motivation 4-2
3- Dimensional Mapping
Cube :
Sphere:

10. n- Dimensional Mapping
Motivation 4-3
4- Dimensional Mapping
HyperCube :
HyperSphere:
n- Dimensional Mapping

11. Mathematical Tool 1 Random Variables : Probability Density Function
1/2
: Uniformly Distributed RV … -1 +1
Q1 : What is the pdf of
Q2 : What is the pdf of …
Qn : What is the pdf of

12. Mathematical Tool 2 Answer of Q1 : PDF of x PDF of z 1/2 Rule -1 +1 -1

13. Addition of Random Variables
Mathematical Tool 3
Answer of Q2 :
PDF of
Rule
Addition of Random Variables
Convolution of PDFs -1 +1 -1

14. Mathematical Tool 4
Simplify

15. Mathematical Tool 5
Next Question : What is the Application of the PDF of x2
Consider Random Picking of a point in a Square.
What is the probability P of
that point is within the Circle?
P= = pi/4 1 x1
P= Prob. of { 0< z=(x1^2 +x2^2) <1 }
=Area of Circle / Area of Square
Therefore,
The Area of Circle ( case r=1) =4*(pi/4)=pi

16. Addition of Random Variables
Mathematical Tool 6
Next Question :
What is the PDF of x1^2+x2^2+x3^2 ?
Rule
Answer :
Addition of Random Variables
Convolution of PDFs
Simplify
Similarly

17. Mathematical Tool 7
Next Question : What is the Application of the PDF of
Consider Random Picking of a point in a Cube.
What is the probability P of
that point is within the Sphere?
P= Prob. of
{ 0< z=(x1^2 +x2^2+x3^2) <1 }
=Vol. of Sphere / Vol. of Cube P=
= (pi/4)(2/3)=pi/6
Therefore,
The Volume of Sphere =8*(pi/6)=(4/3)pi
( case r=1) , where Volume of Cube =8

18. Mathematical Tool 8 Notification : 1
We only need the function shape of in the domain z between 0 and 1 . ( PDF of x1^2 + x2^2 + …. + xn^2 )

19. Mathematical Tool 9
Notification : 2

20. Mathematical Tool 10
Next Question : What is the Application of the PDF of
Consider Random Picking of a point in a 4D-HyperCube.
What is the probability P of
that point is within the 4D-Hyper Sphere? P=
= (pi/4) (pi/4)(1/2)=(pi*pi)/32 ?
Therefore,
The Volume of Sphere =16*(pi*pi/32)=(1/2)pi*pi
( case r=1) , where Volume of 4D-HyperCube =16

21. Formula 1 Vcn : Hyper volume of the Unit HyperCube
Vsn : Hyper volume of the Unit HyperSphere

22. Formula 2
Final Formula

23. Formula 3

24. Another Method 1
Rule 1
Rule 2
Rule 3

25. Another Method 2
WARMING UP1
WARMING UP2
Final Formula

26. Iterative Method 1

27. Iterative Method 2

28. Iterative Method 3

29. Iterative Method 4

30. Iterative Method 5

31. Iterative Method 6

32. Iterative Method 7

33. Thank You !