UCF EXCEL Dr. Niels Lobo Computer Science Using the Geometric Sequence Macro in ImageJ

UCF EXCEL Outline Geometric Sequence Using the GP Macro GP in ImageJ

UCF EXCEL Geometric Sequence A geometric sequence (also know as geometric progression) is a sequence in which each term after the first is found by multiplying the pervious term by a constant, r, called the common ratio. Example: 2, 6, 18, 54,... Has the common ratio of 3 A n+1 =rA n A 0 –scale factor r – common ratio Note: In your own time check Wikipedia.org and check that Geometric Progression and Geometric Sequence are the same things.

UCF EXCEL Geometric Sequence Question Question: What is the common ratio in this geometric sequence: 5, 25, 125, 625, 3125 … A. 3 B. 5 C. 15

UCF EXCEL Geometric Sequence Practical application – photographer prints pictures. She charges families $10 per square inch of the photo paper used, because she only has limited supply of paper.

UCF EXCEL Area as a Geometric Sequence Suppose the available sizes are: 2x2,4x4,8x8, etc. Both sides increase geometrically in length What about the area ? 2x2=4, 4x4=16, 8x8=64, 16x16=256 The area also grows geometrically ! Wow.

UCF EXCEL Area as a Geometric Sequence For those of you mathematically inclined: A n+1 =pA n B n+1 =qB n A n+1 B n+1 =(pq) A n B n The common ratio of the area is pq, where p and q are common ratios of the sides ! Wow.

UCF EXCEL Area as a Geometric Sequence Homework: Suppose the sizes are 2x2, 6x4, 18x8, 54x16,… A n+1 =3A n B n+1 =2B n What is the common factor of A n B n ?

UCF EXCEL Perimeter as a Geometric Sequence Suppose now our photographer also charges $100 per one inch of the photo frame (she is a perfectionist and an expensive one to work with !) The charge is proportional to the Perimeter of the photo. Available photo sizes still form a geometric sequence: 2x2,4x4,8x8. What about the perimeter ?

UCF EXCEL Perimeter as a Geometric Sequence Sizes are: 2x2, 4x4, 8x8,16x16, Perimeters are: = 8, =16, =32, =64 Perimeter also forms geometric sequence. Wow !

UCF EXCEL Perimeter as a Geometric Sequence Perimeter grows geometrically with sides. What is the common factor ? A,B – sides p – common factor P - perimeter A n+1 =pA n B n+1 =pB n P n+1 = 2(A n+1 + B n+1 )=2pA n + 2pB n =pP n The common factor of the perimeter is the same as the common factor of the sides. Wow !

UCF EXCEL Using the GP Macro Open ImageJ Go to plugins> macros> run…

UCF EXCEL Using the GP Macro Select 111GP.txt This will take you to My Pictures, select any picture you like.

UCF EXCEL Using the GP Macro Now, press the space bar, and notice how the picture increases size:

UCF EXCEL Using the GP Macro Notice also that the area of the picture that is highlighted increases as well:

UCF EXCEL Using the GP Macro As you keep clicking the space bar, it will eventually reach the end of the geometric sequence shown, and will prompt you for the area of the next number in the geometric sequence :

UCF EXCEL Using the GP Macro It is important to note that the cost of the paper increases, as the area of the image increases as a geometric sequence; when the length and width of the image is multiplied by the constant.

UCF EXCEL GP in ImageJ In these equations, the new width and height of the larger picture is calculated:

UCF EXCEL GP in ImageJ Question In this picture What is the area equal to?