Polar coordinates & curves
Predict firstWhat stays unchanged if you change only the angle?
Locate points, explain signed radii, construct polar curves, and build their area from sectors.
Prerequisites: Know horizontal and vertical coordinates. An angle measures a turn; a radius measures distance from a chosen origin.Find a point by turning and travelling
Predict firstWhat stays unchanged if you change only the angle?
A Cartesian address gives horizontal x and vertical y. A polar address (r, θ) instead uses a fixed origin, a reference direction (positive x), an angle θ and a signed distance r. Start at the origin, turn anticlockwise for positive angles, then travel r units. Drag the point on the disc; or use Left/Right for angle and Up/Down for radius while the canvas has focus. Holding r fixed traces a circle.
How to observe
- Drag a point, compare Cartesian and polar addresses, and reverse the radius sign.
- Play the pen to grow a circle, rose, cardioid or spiral.
- Build area from finer sectors and compare the sum with the exact value.
Model notes
Ideal plane geometry with fixed unit spacing. Colour distinguishes signed radii and area tiles. Dragging snaps to 0.05 units and 1°; curves use continuous equations. Integer rose k only. Spiral tracing shows two turns; area uses the first turn, closed radially. Sector sums are midpoint approximations; exact areas use analytic integrals.