## Wednesday, November 15, 2017

### How to Rotate Graphs

How to Rotate Graphs

Introduction

The key is to use parametric equations in our rotation.  Using the rotation angle θ, the rotation matrix is:

R = [ [ cos θ, -sin θ ] [ sin θ, cos θ ] ]

With the equations x(t), y(t) set as the matrix:

M = [ [ x(t) ] [ y(t) ] ]

The rotated graph is:

[ [ x’(t) ] [ y’(t) ] ] = R * M

Where:

x’(t) = x(t) * cos θ – y(t) * sin θ
y’(t) = x(t) * sin θ + y(t) * cos θ

Rotation the Function y = f(x)

Let x = t and set the parametric functions:

x(t) = t
y(t) = f(t)

Rotating the Polar Equation r = f(t)  (where t = θ)

1.  Solve for t.
2.  Substitute r and t in the following equations:
x(t) = r * cos t
y(t) = r * sin t
3.  Simplify as needed.

Some trigonometric properties:
sin^2 ϕ + cos^2 ϕ = 1
sin(2*ϕ) = 2 * cos ϕ * sin ϕ
cos(2*ϕ) = 2 * cos^2 ϕ – 1
sin(acos ϕ) = cos(asin ϕ) = √(1 – ϕ^2)

Rotation Matrices for Certain Angles

Angle 30°, π/6:  R = [ [ √3/2, -1/2 ] [ 1/2, √3/2 ] ]

Angle 45°, π/4:  R = [ [ √2/2, -√2/2 ] [ √2/2, √2/2 ] ]

Angle 60°, π/3:  R = [ [ 1/2, -√3/2 ] [ √3/2, 1/2 ] ]

Angle 90°, π/2:  R = [ [ 0, -1 ] [ 1, 0 ] ]

Angle 120°, 2π/3:  R = [ [ -1/2, -√3/2 ] [ √3/2, -1/2 ] ]

Angle 135°, 3π/4:  R = [ [ -√2/2, -√2/2 ] [ √2/2, -√2/2 ] ]

Angle 150°, 5π/6:  R = [ [ -√3/2, -1/2 ] [ 1/2, -√3/2 ] ]

Angle 180°, π:  R = [ [ -1, 0 ] [ 0, -1 ] ]

Angle 210°, 7π/6:  R = [ [ -√3/2, 1/2 ] [ -1/2, -√3/2 ] ]

Angle 225°, 5π/4:  R = [ [ -√2/2, √2/2 ] [ -√2/2, -√2/2 ] ]

Angle 240°, 4π/3:  R = [ [ -1/2, √3/2 ] [ -√3/2, -1/2 ] ]

Angle 270°, 3π/2:  R = [  [ 0, 1 ] [ -1, 0 ] ]

Angle 300°, 5π/3:  R = [ [ 1/2, √3/2 ] [ -√3/2, 1/2 ] ]

Angle 315°, 7π/4:  R = [ [ √2/2, √2/2 ] [ -√2/2, √2/2  ] ]

Angle 330°, 11π/6:  R = [ [ √3/2, 1/2 ] [ -1/2, √3/2  ] ]

Examples

Each example is followed by a graph of the original equation (blue) and the rotated equations (red).  I used a Casio fx-CG 50 for the screen shots.

Example 1:  y = 3*x^2, rotate 90°

We have a function in the form of y = f(x).  Let’s transfer the function to parametric form, first by assigning x = t and y = 3*t^2.  Angle mode is in radians.

With 90°, the rotation matrix is:  R = [ [ 0, -1 ] [ 1, 0 ] ]

The transformed equations are:

[ [ 0, -1 ] [ 1, 0 ] ] * [ [ t ] [ 3*t^2 ] ] = [ [ -3*t^2 ] [ t ] ]

Rotated Equations:  x’(t) = -3*t^2, y’(t) = t

Example 2:  x = t^3, y = 2*t – 1, rotate 270°

We have the equations in parametric form.  We’ll need the rotation matrix, where:

R = [  [ 0, 1 ] [ -1, 0 ] ]

[  [ 0, 1 ] [ -1, 0 ] ] * [ [ t^3 ] [ 2*t – 1 ] ] = [ [ 2*t – 1 ] [ -t^3 ] ]

Rotated Equations:  x’(t) = 2*t – 1, y’(t) = -t^3

Example 3:  x = sin t, y = e^t, rotate 135°

The rotation matrix is R = [ [ -√2/2, -√2/2 ] [ √2/2, -√2/2 ] ]

Rotated Equations:  x’(t) = -√2/2 * (sin t + e^t), y’(t) = √2/2 * (sin t – e^t)

Example 4:  r = 2 θ, rotate 60°

The parametric form is x(t) = 2 * t * cos t, y(t) = 2 * t * sin t

The rotation matrix is R = [ [ 1/2, -√3/2 ] [ √3/2, 1/2 ] ]

Rotated Equations: x’(t) = t * cos t - √3 * t * sin t, y’(t) = t * sin t + √3 * t * cos t

Eddie

This blog is property of Edward Shore, 2017