Cardioid

Cardioid Meaning

There are numerous two dimensional and three dimensional geometric figures in Mathematics. The two dimensional shapes include triangles, quadrilaterals, circles and other polygons. The three dimensional figures include prisms, pyramids, spheres etc. Cardioid is one of the most important two dimensional plane geometric shapes. A cardioid has a heart shaped curve. This forms a basis for its name. The word cardioid is derived from the Greek word which means ‘heart’. Cardioid can be understood by comparing its shape to that of the cross section of an apple without stalk. This shape is formed when a point on the boundary of the circle is traced and rolled on to another circle of the same radius. 


How to Draw a Cardioid:

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A cardioid is a plane figure that has a heart shaped curve. The heart shaped curve is defined by the locus of a point lying on the circumference of a circle which rolles from outside without a slip on the boundary of another circle of radius. The figure above gives a clear idea about how to draw a cardioid. The cardioid curve is in the form of a spiral sinusoid. A parabola whose focus is at the center of inversion can be inverted  to form a cardioid curve. A cardioid can have accurately 3 parallel tangents with a specific gradient. The two branches of the cardioid graph intersect to form a cusp. The arc of the cardioid passing through the cusp measures ‘4a’ where ‘a’ is the radius of the curve. 


Cardioid Equation:

The cardioid equation can be described in both polar and cartesian coordinate system as explained below. 

  1. Polar Form of Representing Cardioid Equation:

The polar equation of a cardioid graph is represented with respect to ‘r’ and ‘θ’ coordinates as follows:


r = a (1 + Cos θ)


In the above equation, ‘a’ is the radius of the circle that is traced and θ is the polar angle.


  1. Cartesian Form of Representing Cardioid Equation:

The equation of a cardioid in the cartesian coordinate system with respect to X and Y axes of a cartesian plane is given as follows. 


(x2 + y2 + ax)2 = a2 (x2 + y2)


The parametric equations of ‘x’ and ‘y’ in the above cardioid equation is given as:

x = a Cos t (1 - Cos t)

y = a Sin t ( 1 - Cos t)

where ‘a’ is the radius of the circle which is traced. 


Cardioid Graph:

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The figure above represents a cardioid graph which has two perpendicular axes named X axis and Y axis.  It has a heart shaped curve that has a cusp formed by the intersection of the two branches. This graph is obtained by tracing out the path of a circle rolling over another fixed circle. However, the radii of both the circles remains the same. A cardioid is most commonly represented in the polar coordinate system. However, the equation of cardioid obtained in polar form can be transformed into a cartesian form.

The region enclosed by a cardioid graph is called the area of a cardioid and is calculated using the formula:


Area of Cardioid = 6 π a2


Total Length of the Arc of a Cardioid is Given by the Formula:

Length of the arc = 16 a


In both the above mentioned formula, ‘a’ stands for the radius of the circle whose path is traced to form the cardioid.


Cardioid Equation Example Problems:

  1. The equation of a cardioid is given as r = 3 (2 + 2Cos θ). Find the total length of the arc and the area of the cardioid.

Solution:

The equation of a cardioid in the given problem is

r = 3 (2 + 2 Cos θ)

If ‘2’ is taken as common, the above equation becomes

r = 6 (1 + Cos θ)

The value of ‘a’ in the above equation is a = 6.

Area of cardioid = 6 π a2 = 6 x 3.14 x (6)2 = 678.24 square units

Length of the arc = 16 a = 16 x 6 = 96 units

  1. What is the length of the arc and area of the cardioid whose equation is defined as r = 7 (1 + Cos θ)?

Solution:

The equation of a cardioid in the given problem is

r = 7 (1 + Cos θ)

The value of ‘a’ in the above equation is a = 7

Area of cardioid = 6 π a2 = 6 x \[\frac{22}{7}\] x (7)2 = 924 square units

Length of the arc = 16 a = 16 x 7 = 114 units


Cardioid Graph Fun Facts:

  • A cardioid is a special case of limacon. It belongs to a class of curves studied by Etinne Pascal, the father of Blaise Pascal. 

  • The word cardioid is also used to name the microphones and other sonic devices which are employed to extract the sound from front and sides only and not from the back side of the device.

FAQ (Frequently Asked Questions)

1. What is a Cardioid?

A cardioid is a two dimensional plane geometric shape that is in the form of a heart shape. The shape of a cardioid can be understood better by analysing the shape of the cross section of an apple without stalk. The cardioid is a plane figure obtained as a path followed by a circle of certain radius over the circumference of a fixed circle of the same radius. A cardioid can be represented in both polar and cartesian coordinate systems. A critical analysis of cardioid is very helpful in analytical geometry and other branches of Mathematics.

2. What are Limacons?

All the equations that can be represented in the form of any one of the following is represented as a limacon.

  • r = a + b Sin θ

  • r = a - b Sin θ

  • r = a + b Cos θ

  • r = a - b Cos θ

The different types of limacons are:

  1. Inner loop Limacon: The value of ‘a’ is less than that of ‘b’ in the above equations.

  2. Dimpled Limacon: The value of ‘a’ is greater than that of ‘b’

  3. Convex Limacon: The value of ‘a’ is greater than or equal to ‘2b’

  4. Cardioid: The value of ‘a’ is equal to that of ‘b’.