What is a Polar Plotter?
Quick Answer: A graphing tool that draws curves defined in polar coordinates r = f(θ). Instead of x and y, it uses distance from origin and angle. It visualizes circles, cardioids, roses, spirals, and other fascinating shapes instantly.
Theory Behind Polar Coordinates
In the polar system, a point is described by a distance r from the origin and an angle θ from the positive x‑axis. The relationship r = f(θ) generates a curve. This plotter converts polar to Cartesian for drawing: x = r cos(θ), y = r sin(θ). The tool evaluates the expression using Math.js over a range of θ values, then plots the points on a polar grid (concentric circles and radial lines). Negative r values are handled by plotting in the opposite direction (adding π to θ). It supports all standard functions and trigonometric operations.
Key formulas: x = r * cos(θ), y = r * sin(θ), with r = given expression. View transform uses a square canvas with world coordinates centered.
Step-by-Step Examples
Example 1: Circle (off-center)
- Enter
2 * sin(theta). Set θ from 0 to 3.14 (π). - You'll see a circle of diameter 2 passing through the origin, centered at (0,1) in Cartesian.
- Use the trace slider to see the point move as θ varies.
Example 2: Cardioid
- Type
2 * (1 + cos(theta)). Range 0 to 6.28 (2π). - A heart‑shaped curve appears. Its cusp is at the origin, and it bulges to the right.
- Click Animate to watch the drawing sequence.
Example 3: Three‑Petal Rose
- Input
3 * cos(3 * theta). Range 0 to 3.14 (π) to get all 3 petals. - Notice how the curve loops and crosses the origin.
- Experiment with different coefficients: cos(2*theta) gives 4 petals.
Frequently Asked Questions
What variable must I use in the expression?
Use theta as the angle variable. Example: 2 + sin(3*theta). The engine evaluates radians.
How are negative r values handled?
Negative radii are plotted in the opposite direction: a point with r = -3 at angle θ is the same as r = 3 at angle θ + π. This can create interesting inner loops.
Why does the curve stop suddenly at the range end?
The plot only draws between θ min and θ max you set. Extend the max slider to see more of the periodic pattern.
Can I zoom and pan?
Yes, use the mouse wheel to zoom. Hold Shift and drag to pan the view. Reset View returns to the default square [-5,5] range.
What's the difference between polar and Cartesian plotting?
Polar plotting uses distance from origin and angle, great for circular and symmetric patterns. Cartesian uses x and y coordinates. Many curves (like circles) have simpler equations in polar form.