Sketching curves the cardioid r = a (1+cosθ)

Sketching curves the cardioid r = a (1+cosθ)

Understanding the Cardioid

• A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius, without slipping.
• It gets its name from the Greek word “kardia”, meaning heart, due to its heart-shaped appearance.

The Equation for a Cardioid

• The polar equation for a cardioid is r = a(1 + cosθ) where ‘a’ is the radius of the circle.
• ‘θ’ is the angle made with the positive x-axis.
• ‘r’ denotes the distance from the origin to the point on the curve.

Sketching the Cardioid

• Start by making several accurate plots, starting with θ=0, π/2, π, 3π/2, 2π, as well as θ=π/3, 2π/3, 4π/3, 5π/3. This will give you key points to join up.
• When plotting, remember that for any negative value of r, you must rotate the angle by π (180 degrees) and use the absolute value of r. This is because of the nature of the polar coordinate system.
• Remember that since θ represents the angle, the graph will repeat every 2π.

Features of a Cardioid

• The cardioid has a cusp, a pointed end, at the coordinate origin, which is the point (0, 0).
• At θ=0 and θ=2π the curve hits a (2a, 0) which is the farthest point on the cardioid from the origin.
• The cardioid curve does not cross itself and it is symmetric about the x-axis.

Applications of Cardioids

• Understanding how to sketch cardioids is not only important for solving maths problems but also has several real-world applications.
• For example, cardioids are frequently used in physics and engineering, such as in the design of directional microphones and antennas.