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

Contents Motivation Mathematical Tool Formula Another Method Iterative Methods

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

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

Motivation 2-2 Linear Mapping Range Space Domain Space Eig=-0.37,5.37 Det(A)=-2 Eig. Vec =[-0.82 0.56] , [ 0.41, 0.90]

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

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

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

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

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

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

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

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

Mathematical Tool 4 Simplify

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

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

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

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 )

Mathematical Tool 9 Notification : 2

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

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

Formula 2 Final Formula

Formula 3

Another Method 1 Rule 1 Rule 2 Rule 3

Another Method 2 WARMING UP1 WARMING UP2 Final Formula

Iterative Method 1

Iterative Method 2

Iterative Method 3

Iterative Method 4

Iterative Method 5

Iterative Method 6

Iterative Method 7

Thank You !