Welcome to the 2d math primer

The underlying repository does use Python but this math can be definitely implemented with other languages. I hope you might find it usefull. It's to say - when reading correctly - that you need a quite up-to-date jupyter version to work with this document. Let's start with 2d vector.

The 2d vector

Please check the repository for concept/math/vector.py containing the Python class wrapping the math logic and simplifying the usage.

A vector basically provides two information:

direction
length

The vector looks like $\vec{v} = \begin{pmatrix}v_x\\v_y \end{pmatrix}$ and the length of the vector is calculated by $|\vec{v}| = \sqrt{v_x \cdot v_x + v_y \cdot v_y}$



In [5]:

    
import math
len_vector = lambda vector: math.sqrt(vector[0]**2 + vector[1]**2)
vector = [3, 4]
print("vector is %s" % vector)
print("vector length is %g" % len_vector(vector))









    



vector is [3, 4]
vector length is 5

Sum and difference

The sum of two vectors: $\vec{a} + \vec{b} = \begin{pmatrix}a_x + b_x\\a_y + b_y\end{pmatrix}$. For the difference: $\vec{a} - \vec{b} = \begin{pmatrix}a_x - b_x\\a_y - b_y\end{pmatrix}$



In [5]:

    
sum_vector = lambda a, b: [a[0]+b[0], a[1]+b[1]]
diff_vector = lambda a, b: [a[0]-b[0], a[1]-b[1]]

vector_a = [1, 2]
vector_b = [3, 4]
print("%s + %s = %s" % (vector_a, vector_b, sum_vector(vector_a, vector_b)))
print("%s - %s = %s" % (vector_a, vector_b, diff_vector(vector_a, vector_b)))









    



[1, 2] + [3, 4] = [4, 6]
[1, 2] - [3, 4] = [-2, -2]

Products

There are different kind of products:

Scale a vector: $f \cdot \vec{v} = \begin{pmatrix}f \cdot v_x\\f \cdot v_y\end{pmatrix}$.
The scalar product of two vectors is $\vec{a} \cdot \vec{b} = a_x \cdot b_x + a_y \cdot b_y$
The cross product of two vectors is $\vec{a} x \vec{b} = a_x \cdot b_y - a_y \cdot b_x$



In [10]:

    
scale_vector = lambda f, v: [f * v[0], f * v[1]]
scalar_product = lambda a, b: a[0]*b[0] + a[1]*b[1]
cross_product = lambda a, b: a[0]*b[1] - a[1]*b[0]

print("%g * %s = %s" % (3, vector_a, scale_vector(3, vector_a)))
print("%s * %s = %s (scalar product)" % (vector_a, vector_b, scalar_product(vector_a, vector_b)))
print("%s x %s = %s (cross product)" % (vector_a, vector_b, cross_product(vector_a, vector_b)))









    



3 * [1, 2] = [3, 6]
[1, 2] * [3, 4] = 11 (scalar product)
[1, 2] x [3, 4] = -2 (cross product)

Orthogonal vector

For a current vector there are two orthogonal vectors: $\vec{v} = \begin{pmatrix}v_x\\v_y\end{pmatrix}$, $\vec{v_{left}} = \begin{pmatrix}-v_y\\v_x\end{pmatrix}$, $\vec{v_{right}} = \begin{pmatrix}v_y\\-v_x\end{pmatrix}$.



In [2]:

    
turned_Left = lambda v: [-v[1], v[0]]
turned_right = lambda v: [v[1], -v[0]]
vector = [0, 1]
print("%s rotated 90 degree counter clockwise: %s" % (vector, turned_Left(vector)))
print("%s rotated 90 degree clockwise: %s" % (vector, turned_right(vector)))









    



[0, 1] rotated 90 degree counter clockwise: [-1, 0]
[0, 1] rotated 90 degree clockwise: [1, 0]

Angle between vectors

The angle between two vectors is calculated as: $atan2(u_y, u_x) - atan2(v_y, v_x)$



In [6]:

    
vector_angle = lambda u, v: math.atan2(u[1], u[0]) - math.atan2(v[1], v[0])
vector_a = [0, 1]
vector_b = [1, 0]
print("Angle between %s and %s is %g degree" \
      % (vector_a, vector_b, vector_angle(vector_a, vector_b) * 180.0/math.pi))









    



Angle between [0, 1] and [1, 0] is 90 degree

Rotation of a vector

The rotation of vector is calculated using an angle $\alpha$ with $\vec{v}_{\alpha} =\begin{pmatrix}v_x \cdot \cos{\alpha} - v_y \cdot \sin{\alpha}\\v_x \cdot \sin{\alpha} + v_y \cdot\cos{\alpha}\end{pmatrix}$



In [11]:

    
vector_rotated = lambda v, a: [round(v[0] * math.cos(a) - v[1] * math.sin(a), 5),\
                               round(v[0] * math.sin(a) + v[1] * math.cos(a), 5)]
vector = [1, 0]
print("Angle %s rotated 90 degree to the left is %s" \
      %(vector, vector_rotated(vector, 90 * math.pi / 180.0)))
print("Angle %s rotated 90 degree to the right is %s" \
      %(vector, vector_rotated(vector, -90 * math.pi / 180.0)))









    



Angle [1, 0] rotated 90 degree to the left is [0.0, 1.0]
Angle [1, 0] rotated 90 degree to the right is [0.0, -1.0]



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