Lissajous Curves
Curves traced by two perpendicular oscillations.
Key idea
Curves traced by two perpendicular oscillations.
In depth
x = A sin(at + δ), y = B sin(bt) traces closed loops when a/b is rational. Different frequency ratios and phase give circles, figure-eights and knots, and are seen on oscilloscopes.
Worked example
a = b with δ = 90° gives an ellipse; a:b = 1:2 gives a figure-eight.
Formula
Real-world uses
- Displaying two signals on an oscilloscope
- Checking phase between two audio channels
Common mistake
Expecting a closed curve when the ratio is irrational.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Lissajous Curves?
Curves traced by two perpendicular oscillations. It is a topic at the expert level in Trigonometry.
What is the formula for Lissajous Curves?
The key formula is x = sin(at), y = sin(bt).
Where is Lissajous Curves used in real life?
Real-world examples: Displaying two signals on an oscilloscope; Checking phase between two audio channels.
What should I learn before Lissajous Curves?
Make sure you are comfortable with Phasors, the previous topic in Trigonometry.
What is a common mistake with Lissajous Curves?
Expecting a closed curve when the ratio is irrational.
How can I practise Lissajous Curves?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Lissajous Curves?
Move on to Parametric Unit Circle next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).