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part 3
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‎Original Approach/cfg.py‎

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# coordinates of the projections of Mars on the ecliptic plane
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xMars = [-1.45297367276038, 1.195672782788594, 1.0738853142069973, -1.6323045900130566, -1.5537673314861347]
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yMars = [0.8655335301531039, -0.686856634618109, 1.051106927548351, -0.14854179871578327, 0.6248989852957586]
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x2Mars = [0.39910431516406286, -0.29101951204595994, -0.7833937646297272, -0.9969302054095404, -0.8252535512519825, -0.057254869012341184, 0.9517725336021033, 0.6752324411578464, -0.04302706623321199, -0.6240752924965628, -0.9531154076579537, -0.9475455013657242]
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y2Mars = [0.9168436823883284, 0.9563575585908708, 0.6208484101394284, 0.07466793029864548, -0.564710366795997, -0.9980316866413974, -0.3043595904252963, 0.7376045179660177, 0.9988159644223529, 0.7808334890653823, 0.3014308227503241, -0.3192423211998812]
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z2Mars = [0.010649304298148211, 0.02622715643451466, 0.02900795005869142, 0.023555588020852912, 0.0076666667543403184, -0.025585785782483234, -0.03865596974309206, 0.0008517535642629537, 0.0227011186157881, 0.02879395160986412, 0.0266548078416554, 0.01555195153317052]
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# radius of the best fit circle of the projections of Mars on the ecliptic plane
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r = 1.57732091444
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# parameters a and b for the best fit Mars orbital plane defined by the equation
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# ax + by + z = 0
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a = 0.0236392428323
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b = -0.0221054030127

‎Original Approach/cfg.pyc‎

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‎Original Approach/ellipseGradientDescent.py‎

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""" This module runs gradient descent on the locations of Mars on it's
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orbitalplane in order to find the best-fit elliptical orbit for Mars.
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"""
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# Developed by Pulkit Singh, Niheshkumar Rathod & Rajesh Sundaresan
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# Copyright lies with the Robert Bosch Center for Cyber-Physical Systems,
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# Indian Institute of Science, Bangalore, India.
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#----------------------------------------------------------------------------#
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import numpy as np
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import math
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#----------------------------------------------------------------------------#
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def evaluateDistance(xMars, yMars, xFocus, yFocus, majorAxis):
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""" Computes the cost of fitting an ellipse with one focus at the sun,
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and the other at [xFocus, yFocus], given the locations of Mars and
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the length of the major axis.
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Cost is caculated using the following formula:
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(sum of distances from focii - length of major axis)^2
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Parameters:
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xMars (float): list of x-coordinates of Mars locations
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yMars (float): list of y-coordinates of Mars locations
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xFocus (float): x-coordinate of second focus
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yFocus (float): y-coordinate of second focus
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majorAxis (float): length of the major axis
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Returns:
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squareDist (float): cost of fitting the ellipse (sum of square
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distances)
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"""
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# Finding the distance of each point from the origin
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dist = []
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# calculating (distance to origin + distance to focus2
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# - major axis length)^2 for each (x, y) pair
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for i in range(len(xMars)):
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distOrigin = math.sqrt(math.pow(xMars[i], 2) + math.pow(yMars[i], 2))
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distFocus = math.sqrt(math.pow((xMars[i] - xFocus), 2)
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+ math.pow((yMars[i] - yFocus), 2))
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dist.append(math.pow((distOrigin + distFocus - majorAxis), 2))
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# adding up all the square distances
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squareDist = sum(dist)
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return squareDist
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#----------------------------------------------------------------------------#
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def computeGradient(xMars, yMars, xFocus, yFocus, majorAxis):
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""" Computes the gradient vector with respect to the x-y coordinates of
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the second focus andthe length of the major axis.
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gradient vector = [df/d(xFocus), df/d(yFocus), df/d(majorAxis)]
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Parameters:
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xMars (float): list of x-coordinates of Mars locations
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yMars (float): list of y-coordinates of Mars locations
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xFocus (float): x-coordinate of second focus
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yFocus (float): y-coordinate of second focus
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majorAxis (float): length of the major axis
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Returns:
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gradient (float list): list of required gradients
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"""
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dxFocus = []
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dyFocus = []
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dmajorAxis = []
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# df/d(xFocus) = (-2 * (xMars - xFocus) * evaluateDistance) / distFocus
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# df/d(yFocus) = (-2 * (yMars - yFocus) * evaluateDistance) / distFocus
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# df/d(majorAxis) = -2 * evaluateDistance
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# computing partial derivatives for each set of coordinates
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for i in range(len(xMars)):
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distOrigin = math.sqrt(math.pow(xMars[i], 2) + math.pow(yMars[i], 2))
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distFocus = math.sqrt(math.pow((xMars[i] - xFocus), 2)
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+ math.pow((yMars[i] - yFocus), 2))
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dist = distOrigin + distFocus - majorAxis
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xDiff = xMars[i] - xFocus
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yDiff = yMars[i] - yFocus
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dxFocus.append((-2 * xDiff * dist) / distFocus)
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dyFocus.append((-2 * yDiff * dist) / distFocus)
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dmajorAxis.append(-2 * dist)
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# returning gradient vector
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gradient = [float(sum(dxFocus)),
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float(sum(dyFocus)),
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float(sum(dmajorAxis))]
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return gradient
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#----------------------------------------------------------------------------#
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# Takes x-y coordinate matrix of different Mars locations, x-y coordinates of
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# the second focus and the length of the major axis as input, and runs
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# gradient descent to find the best fit ellipse. Returns the x-y coordinates
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# of the found focus, and the length of the major axis of the ellipse.
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def findEllipse(xMars, yMars, xf, yf, axis):
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""" Finds the best-fit ellipse for the Mars Orbit using gradient
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descent. Returns the x-y coordinates of the found focus and the
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length of the major axis.
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Parameters:
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xMars (float): list of x-coordinates of Mars locations
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yMars (float): list of y-coordinates of Mars locations
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xFocus (float): x-coordinate of second focus
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yFocus (float): y-coordinate of second focus
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majorAxis (float): length of the major axis
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Returns:
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xf (float): x-coordinate of the found focus
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yf (float): y-coordinate of the found focus
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axis (float): length of the major axis
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cost (float list): list of costs in each gradient descent
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iteration.
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"""
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# initialising alpha as the step value
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alpha = 0.001
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# initialising array to keep track of cost values in gradient descent
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cost = []
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# running gradient descent
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for i in range (10000):
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# finding cost for given parameters
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squareDist = evaluateDistance(xMars, yMars, xf, yf, axis)
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#print("square dist"), squareDist
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# adding current cost to list of previous costs
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cost.append(squareDist)
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# finding gradient with parameter values a & b
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delta = computeGradient(xMars, yMars, xf, yf, axis)
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# updating parameter values
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xf = xf - (alpha * delta[0])
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yf = yf - (alpha * delta[1])
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axis = axis - (alpha * delta[2])
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return xf, yf, axis, cost
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#----------------------------------------------------------------------------#

‎Original Approach/ellipseGradientDescent.pyc‎

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‎Original Approach/mars3.py‎

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# E0 259: Data Analytics. Module 1: Mars Orbit. Question 3.
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# Pulkit Singh, July 20 2018
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# importing required modules
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import math
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import numpy as np
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import matplotlib.pyplot as plt
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from mpl_toolkits.mplot3d import Axes3D
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from matplotlib.patches import Ellipse
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# importing file containing required values calculated in other modules
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import cfg
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# importing custom module to run gradient descent on elliptical mars orbit
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import ellipseGradientDescent
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#-------------------------------------------------------------------------------#
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# Part 1: Finding 5 different locations of Mars on it's orbital plane.
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# retrieving x-y coordinates of mars computed in Part 1
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xMars, yMars = cfg.xMars, cfg.yMars
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# x-y coordinates of Mars will be the same as the projections
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# to calculate z coordinates, we must use the formula as follows:
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# zMars = (-a * xMars) + (-b * yMars), where a and b are the parameters
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# of the best-fit plane calculated in Part 2.
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zMars = []
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for i in range(len(xMars)):
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zMars.append((-1 * cfg.a * xMars[i]) + (-1 * cfg.b * yMars[i]))
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print("\nCalculating 5 different locations of Mars on it's orbital plane.\n")
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#-------------------------------------------------------------------------------#
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# Part 2: Finding the best fit circle on Mars's orbital plane
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# Finding the distance of each point from the origin
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rMars = []
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for i in range(len(xMars)):
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sqDist = math.pow(xMars[i], 2) + math.pow(yMars[i], 2) + math.pow(zMars[i], 2)
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rMars.append(math.sqrt(sqDist))
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# Finding the average radius, which is the radius of the best fit circle
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r = sum(rMars) / len(rMars)
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# Computing the sum of losses
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loss = 0.0
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for radius in rMars:
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loss = loss + math.pow((r - radius), 2)
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print("Finding the best fit circle on the orbital plane.")
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print "Radius of circle", r
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print "Sum of losses =", loss
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print
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#-------------------------------------------------------------------------------#
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# Part 3: Finding the best fit ellipse on Mars's orbital plane
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# initialising parameters for x-y coordinates of focus and length of major axis
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xf1, yf1, axis1 = 0.0, 0.0, 0.0
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# finding the best fit ellipse
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xf, yf, axis, cost = ellipseGradientDescent.findEllipse(xMars, yMars, xf1, yf1, axis1)
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print "Finding best fit ellipse on the orbital plane:"
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print "Running gradient to find coordinates of second focus and length of major axis."
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print "x-coordinate of focus:", xf
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print "y-coordinate of focus:", yf
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print "Length of Major Axis:", axis
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print "Length of Semi-Major Axis", axis/2.0
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# calculating distance between focii
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interFocii = math.sqrt(math.pow(xf, 2) + math.pow(yf, 2))
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print "Distance between Focii:", interFocii
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# calculating values to plot ellipse
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centerX = xf / 2.0
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centerY = yf / 2.0
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minorAxis = math.sqrt(math.pow(axis, 2) - math.pow(interFocii, 2))
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rotationAngle = 360 - math.degrees(math.atan(yf / abs(xf)))
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# plotting cost as a function of number of iterations
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#plt.plot(cost)
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#plt.show()
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# creating a plot
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fig, ax = plt.subplots()
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# plotting mars locations and the focii
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ax.plot(xMars, yMars, "ro")
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ax.plot(xf, yf, "yo")
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ax.plot(0.0, 0.0, "yo")
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# adding best fit circle
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fit_c = plt.Circle((0,0), r, color='c', fill=False)
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ax.add_artist(fit_c)
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# adding best fit ellipse
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fit_e = Ellipse((centerX, centerY), axis, minorAxis, rotationAngle, color='b', fill=False)
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ax.add_artist(fit_e)
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# setting dimensions of the plot
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ax.set_xlim(-2.2, 2.2)
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ax.set_ylim(-2.2, 2.2)
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ax.set_aspect('equal')
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# updating legend
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ax.legend([fit_c, fit_e], ['Best-fit Circle', 'Best-fit Ellipse'], fontsize='x-small')
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plt.show()
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#-------------------------------------------------------------------------------#
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