Radial visualization & spring forces

[jst]

Hi, I am taking a Data Mining course and have this problem in radial visualization. To complete this problem, we use some physics formulas, which I really do not understand.

Derive formulas for radial visualization of:

a. 3-dimensional samples

b. 8-dimensional samples

Homework Equations

Here's what our notes have:
http://glomawr.com/radialvisualization.jpg

The Attempt at a Solution

I thought it would be very similar to the example in the notes so for part A I did:

x = (K1 - K3 - K5 - K7)/(K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8)
y = (K2 - K4 - K6 - K8)/(K1 + K2 + K3 + K4 + K5 + K6 + K7 + K8)

and part B, I did:

x = (K1 - K3)/(K1 + K2 + K3)
y = (K2)/(K1 + K2 + K3)

I am told: "sqrt (2)/2, or sqrt(3)/2 or 1/2 are in the final expressions because of trig functions: sin and cos for angles of 30, 45 or 60 degrees."

I'm really lost, our book doesn't have what I need it sort of assumes that this example was enough, but I'm lost. I really don't need the solution, as much as an explanation of how to do it myself.

Thanks a lot,

Jason

 

[member 553083]

The formulas for radial visualization of 3-dimensional and 8-dimensional samples can be derived using the trigonometric functions cosine (cos) and sine (sin).For 3-dimensional samples, the formula is:x = cos(α) * K1 - sin(α) * K3y = sin(α) * K1 + cos(α) * K3where α is the angle measured in radians and K1 and K3 are constants. To derive the formula for an 8-dimensional sample, we use the same principles but with different constants. The formula is:x = cos(α) * (K1 - K3 - K5 - K7) - sin(α) * (K2 - K4 - K6 - K8)y = sin(α) * (K1 - K3 - K5 - K7) + cos(α) * (K2 - K4 - K6 - K8)where again, α is the angle measured in radians and K1, K2, K3, K4, K5, K6, K7, and K8 are constants. The constants are determined by the sample data points.The constants sqrt (2)/2, or sqrt(3)/2 or 1/2 are in the final expressions because of the trigonometric functions. As the angle increases, the sine and cosine values approach these constants, which allows us to simplify our equations. For example, if the angle is 45 degrees (1.57 radians), then cos(1.57) = 0.707 and sin(1.57) = 0.707, so sqrt (2)/2 is used in the final expression.