Assignment 2
Authors: Grace Pigeau (14sgp, 10187678) and Katherine Baillie (14kb47, 10178845)
Created for: CISC 472 at Queen's University with Professor James Stewart
Contents:
	- Sphere fitting to point cloud
	- Sphere fitting using RANSAC to remove outliers
	- Plotting sphere centres with 9% confidence interval
	
Question 6:
	pivot_calibration_0:
		Sphere:
		centre = [129.0205 -71.8028 -1420.2]
		radius = 188.326261683521
		standard deviation = [9.3383 11.7971 16.4376]
		Sphere with RANSAC:
		centre = [140.6700 -70.9869 -1457.0]
		radius = 219.1667
		standard deviation = [16.3700 24.8709 20.1702]
	pivot_calibration_1:
		Sphere:
		centre = [2.9956 202.4693 -1565.5]
		radius = 158.0867
		standard deviation = [0.3944 1.3904 0.6048]
		Sphere with RANSAC:
		centre = [5.2789 206.5326 -1572.0]
		radius = 168.0851
		standard deviation = [3.7406 2.5975 1.6564]
	pivot_calibration_2:
		Sphere:
		centre = [4.3298 200.6211 -1564.9]
		radius = 159.0181
		standard deviation = [0.3078 0.2865 0.5695]
		Sphere with RANSAC:
		centre = [4.1475 201.1415 -1564.9]
		radius = 158.7846
		standard deviation = [0.3782 0.2958 0.5803]
	
Question 7:
	Differences between collected and provided pivot calibration data?
		Our collected data had fewer outliers than the provided data. This is most likely because data collection was only started once the stylus was in place on the knee model and so the points  - which can be seen in the provided data - from moving the stylus to the model are not present.
	Best method for handling each set of data? Justify.
		For pivot_calibration_0 (the provided data set) RANSAC is the best method for handling the data. Looking at the two spheres and the data points, it's qualitatively obvious that the sphere generated by RANSAC is a better fit to the inlier points. However, RANSAC does have a higher standard deviation. This is possibly because the outliers have a high deviation from the sphere and therefore disproportiantely affect the standard deviation.
		For the two collected data sets there were fewer outliers in the data, therefore the sphere fit with and without RANSAC resulted in very similar values and standard deviations, although the method without RANSAC has marginally smaller standard deviations. Qualitatively, we were not able to see significant improvements in the sphere fit with either method when looking at the graphs of the sphere and points.
	Recommendations for pivot calibration?
		One recommendation for pivot calibration would be to start and end data collection while the stylus is in place. This will minimize the collection of outliers. Another recommendation would be to ensure the stylus is rotated in many directions to maximize the spherical surface covered by the points.
		
Ellipsoid axis discussion:
	We believe that the ellipsoidal axis with the shortest ellipsoid radius corresponds with the axis of the stylus that has the least variation. The stylus is less likely to move upwards or downwards from its pivot point, whereas side-to-side slippage is more common. Therefore, the longest ellipsoidal radius corresponds with the axis of the stylus that has the most variation (i.e. side to side).