List the points for the path in order of sequence. Write them down in an organized fashion or use computer software. Write the points in coordinate form; that is, (x, y), where “x” refers to the coordinate showing location on the x-axis and “y” refers to the coordinate displaying the location on the y-axis. You should end up with a set a points such as (3, 33); (21, 8); (44, 0).
Calculate the distance vectors for the paths between each sequential set of points. Use the calculation for vector for a specific pair of points (x1, y1) and (x2, y2); <x2-x1, y2-y1>. Subtract the origin point from the destination point. For example, the distance vector between points (2, 9) and (7, 15) is <7-2, 15-9> or <5, 6> after simplifying.
Write the vectors in sequential order. Write them in an organized form, such as <3, 9>; <4, 0>; <44; 3>. This sequence of vectors is the path between points for your geometric data.
Calculate the centroid of the data if you do not already have a pre-calculated centroid. Each point of the data has three values associated with it--one for each of the x- y- and z-axes. Sum the x-values for each point and divide by the number of points. Call this value “xm.” Do the same for the y- and z- values; call these values “ym” and “zm,” respectively. The centroid is the 3-D point (xm, ym, zm). The centroid represents the center of the sphere on which the data lies.
Convert the points into spherical coordinates. For each point, convert the x- y- and z-values to r, theta and psi values. Calculate r with the equation r = sqrt(x^2 + y^2 + z^2), where “sqrt” represents the square root function. Calculate theta with the equation theta = arccos(z/r), where “arccos” represents the arc cosine function. Calculate psi with the equation psi = arctan(y/x), where “arctan” represents the arctan function. Your new points will be in the form (r, theta, psi).
List the points in the path in sequential order.
Find the distances between each point in spherical vector form. For a pair of points (r1, theta1, psi1) and (r2, theta2, psi2), calculate the vector <r2-r1, theta2-theta1, psi2-psi1>. Do this for all pairs of points in your list.
Write the vectors in sequential order. This is the path between points.