Solutions to Part A: Data Types and Calculating exercises

Program to print the center of mass:

# Calculate the triangle's center of mass
#
from numpy import *

a = array([0, 1, 0])
b = array([4.3, -2.4, 0.005])
c = array([-3.2, 5.1, -3.)

print (a+b+c)/3

The output:

[0.36666666666666653, 1.2333333333333332, -0.99833333333333341]

Program to print the vectors' angles:

# Calculate a triangle's three angles
#
import numpy as np

# The vertices
a = np.array([0, 1, 0])
b = np.array([4.3, -2.4, 0.005])
c = np.array([-3.2, 5.1, -3.])

# Angle between a-b and c-b
ab = a - b
ab_modulus = np.sqrt((ab*ab).sum())
cb = c - b
cb_modulus = np.sqrt((cb*cb).sum())
abc_cos_angle = np.dot(ab,cb) / ab_modulus / cb_modulus
print "Angle a-b-c:", np.arccos(abc_cos_angle)

# Angle between c-a and b-a
ca = c - a
ca_modulus = np.sqrt((ca*ca).sum())
ba = b - a
ba_modulus = np.sqrt((ba*ba).sum())
cab_cos_angle = np.dot(ca,ba) / ca_modulus / ba_modulus
print "Angle c-a-b:", np.arccos(cab_cos_angle)

# Angle between b-c and a-c
bc = b - c 
bc_modulus = np.sqrt((bc*bc).sum())
ac = a - c 
ac_modulus = np.sqrt((ac*ac).sum())
bca_cos_angle = np.dot(bc,ac) / bc_modulus / ac_modulus
print "Angle b-c-a:", np.arccos(bca_cos_angle)

The output should be

Angle a-b-c: 0.298174974907
Angle c-a-b: 2.57187588039
Angle b-c-a: 0.271541798289

It is always good programming practice to check your programs. This often requires thinking up and then testing some constraint on the program's calculations. Here we can add up the angles---the sum must be pi. Having just run the program we can simply type the following at the prompt to do the check:

print np.arccos(abc_cos_angle) + np.arccos(cab_cos_angle) + np.arccos(bca_cos_angle)

and viola!

3.14159265359

Program to print the normal to the plane defined by the triangle:

# Calculate the vector normal to the plane in which a triangle
# lies
#
from numpy import *

# The vertices
a = array([0, 1, 0])
b = array([4.3, -2.4, 0.005])
c = array([-3.2, 5.1, -3.])

ba = b - a
ca = c - a
print "Normal to the triangle's plane:", np.cross(ba,ca)

Which generates:

Normal to the triangle's plane: [ 10.1795  12.884    6.75  ]

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