Digital Geometry in Image Processing

Distance Functions in Digital Geometry 77

From the above theorems, we state the measures of disks of inverse square

root weighted t-cost norm in 2-D and 3- D in the following corollaries.

Corollary 2.14. In 2-D, the perimeter and area of H

2

isr

(r) are given by

6.6274r and 3.313708r

2

, respectively. H

2

isr

(r) is a regular octagon. 

Corollary 2.15. In 3-D, the surface area and volume of H

3

isr

(r) are given by

14.3319r

2

and 4.7773r

3

, respectively. 

For inverse square root weighted t-cost norm the n-D hyper sphere H

n

isr

(r)

encloses the n-D Euc lide an hypersphere H

n

E

(r). Also H

n

E

(r) touches all the

center points of the faces (hyperplane of dimension n − 1) ...