Cardioid
Lesson notes
Overview
A heart-shaped curve traced by a point on one circle rolling around another of equal size. This topic is rated beginner and is number 38 of 245 in Mathematics. Core ideas and first intuitions. No prior knowledge needed.
What you are seeing
The panel above shows a path traced by a parametric curve as its parameters change. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
The cardioid, named for its heart shape, is the path of a point on one circle that rolls around another circle of the same size. Its polar equation r = a(1 − cos θ) has a single cusp at the origin. It appears as the envelope of reflected rays inside a coffee cup, as the shape of the main bulb of the Mandelbrot set, and in the pickup pattern of cardioid microphones, which are most sensitive to sound from the front and reject sound from behind.
Reading the relationship
The expression "r = a(1 − cos θ)" is a compact summary of the idea. It uses the symbols r, a, θ; name what each one stands for before you start. Notice that it contains a trigonometric function, so the result repeats as the angle goes round. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.
Try this
Start by only moving one slider at a time. Say aloud what you expect, then check.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Polar Roses”, the topic before it in Mathematics. Once you are comfortable, continue with “Lemniscate of Bernoulli”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Cardioid” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Cardioid?
A heart-shaped curve traced by a point on one circle rolling around another of equal size. It is a beginner topic in Mathematics.
How do I use the Cardioid simulation?
The panel at the top shows a path traced by a parametric curve as its parameters change. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression r = a(1 − cos θ) packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Core ideas and first intuitions. No prior knowledge needed. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Polar Roses”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.