Visualization of Multidimensional Functions Team Members: Mrinmayee Kulkarni Reenal Mahajan Priya Shastri

Motivation Difficulty in visualizing functions of higher dimensions. Inspiration from Sanjini’s work.

Our Approach Focal point rays (n-1) dimensional hypercube +X -X-Z -Y+Y +Z +X +Y +Z Terms used: Rays, Focal Point, Range Range of x,y and z = 40 (-20 to 20) Radius = 400

Approach in detail. Function to screen mapping Steps involved Step 1 Calculate the number of slices using the formula No. of Slices(n)=2*Dimensions Step 2 Angle of each slice(theta) = 360/No.of slices Step 3 No. of rays(r) = 400*2Pi/2n

Step 4 Each (n-1) dimensional face of the hypercube is divided into smaller cubes using the formula 40^(n-1)= r * x^(n-1) where “x” is the dimension of each hypercube. Step 5 Order the hypercubes row-wise and later layer-wise. Step 6 Divide the line joining the center of the hypercube and the focal point into 400 points and compute the function values for each. Step 7 Plot the function points on the screen with intensity computed using the formula intensity = function value * 255/maxvalue of function

Stage 1 First line drawn All 400 points.

Stage 2 +X Surface Co-ordinates 400,400 Angle theta

Stage 3 + X Surface White portion which was due to out of range color values

Stage 4 Background Changed To Black for Better effect. Out of range color values removed

Current stage

Some interesting Functions

Screen to function mapping Step 1 Store the intensity values calculated for each point (a,b) plotted in 2 arrays green[800][800] and red[800][800] Step 2 Get screen co-ordinates (a,b) on mouse click Step 3 Search in green,red arrays and find intensity at (a,b) Step 4 Shift coordinate axis to focal point [Here (400,400) ] Step 4 Shift coordinate axis to focal point [Here (400,400) ]

Step 5 Find angle and quadrant of shifted coordinates;hence find surface which they map to in hypercube Step 6 Substitute known values in the intensity function and get the value of function Step 7 Sign of function is obtained from plotted color (red- negative; green – positive)

Green[576][264]=intensity value Function > =0 Value= Red[576][264]=0

Pros and Cons Pros Visualization of functions of higher dimensions. Ability to scale up to 20 dimensions. Ordering of cubes can be altered to accommodate more rays. Easier to classify the type of function (symmetric, asymmetric). Cons More processing time. Aggregation of function points. Limited interaction.

Future Work Addition of mouse interactions. Enable the user to change focal point. Display function values at different points. Change the range of display of function values. Use of a different method to order the rays on the face of cube. Make a generalized module to extend to n- dimensions.( part of the work going on).

Acknowledgements Thanks to Dr. North & Sanjini for their help Questions???