Cardioid

A cardioid is an important mathematical principle, especially about curves and associated shapes. Cardioid – is one of the type of geometrical figure which closed in shape, looks like a heart. From a geometrical standpoint, it is defined as a set of points each of which has been stated at equal distances from a point known as focus and a straight line termed as directrix. It has been used in the application of physics and engineering purposes along with the different areas of computer graphics.

A cardioid is a specific type of mathematical curve that resembles the shape of a heart. It is a plane curve generated by a point on the circumference of a circle that rolls around a fixed circle of equal radius. The name “cardioid” is derived from the Greek word “kardia,” meaning heart, due to its heart-like shape.

Cardioids can be classified as a type of curve referred to as epicycloids, which are defined as curves that are generated when a point is moving along a path which is the circumference of one circle while another circle is also rolling without slipping on the first circle. Cardioid is the part of the epicycloid when the two circles have the same radius.

Definition of Cardioid

Mathematically, a cardioid can be defined as the locus of points traced by a fixed point on the circumference of a circle as it rolls around another fixed circle of the same radius.

The equation of a cardioid can be represented in both polar and Cartesian coordinate systems which are as follows:

• Polar Equation of Cardioid
• Cartesian Equation of Cardioid
• Parametric Equation of Cardioid

Polar Equation of Cardioid

The polar equation of a cardioid in the horizontal direction is 

r = 2a (1+cos⁡(?))

And in the vertical direction, it is 

r = 2a (1+sin⁡(?))

Here, ? is the radius of the tracing circle, ? is the angle, and cos⁡(?) and sin⁡(?) represent the cosine and sine of the angle, respectively.

Cartesian Equation of Cardioid

In Cartesian coordinates, the equation of a cardioid is given by 

(x² + y²)² + 4ax(x² + y²) – 4a²y² = 0

Parametric Equation of Cardioid

The parametric equations for a cardioid are 

x = 2a cos t (1 – cos t) and y = 2a sin t(1 – cos t)

Where t is the parameter.

A cardioid is another type of graph that resembles that of a heart whereby the graph can either be symmetric along the x-axis or the y-axis depending on the equation chosen. The curve begins at the origin and a loop is formed at this point and then drawn again towards the axis at the point of origin. In fact, the shape of the cardioid can be described where the figure appears to look like a cross-section of an Apple without the stem.

Cardioid can best be described as a curve formed by the trace of a point on the circumference of one circle as the other circle touches and moves along it. This process results in the generation of a sinusoidal spiral whose curve has a (cusp) at the point where the rolling circle come to be in contact with the stationary circle. The cardioid graph has several interesting properties, such as having exactly three parallel tangents with any given gradient and having a cusp where the two branches of the curve intersect.

The area enclosed by a cardioid can be calculated using the formula 

A = 6πa²

Where ? is the radius of the cardioid.

For example, let’s consider a cardioid with a radius of 4 units. Substituting this value into the formula, we get:

A = 6π × 4² = 6π × 16 = 96π

Therefore, the area of a cardioid with a radius of 4 units is 96π square units.

The length of the arc of a cardioid can be calculated using the formula 

? = 16?

Where a is the radius of the tracing circle.

For instance, if the radius of a cardioid is 3 units, the length of the arc would be:

L = 16 × 3 = 48

Therefore, the length of the arc of a cardioid with a radius of 3 units is 24 units.

The properties of Cardioid are as follows:

Question 1. Find the perimeter of the cardioid represented by the equation r =2(1+cos⁡(?)).

Solution:

Given the equation of the cardioid is r = 2(1+cos⁡(?)).

Comparing this with the general equation of a cardioid parallel to the positive x-axis, we have 

r = 2a (1+cos⁡(?)), where 2a = 2,

so a = 1.

The perimeter of the cardioid is given by 

L = 16a

Substituting a = 1, we get 

L = 16 × 1 = 16 units.

Question 2. Determine the area enclosed by the cardioid with the equation r = 3(1+cos⁡(?)).

Solution:

The given cardioid equation is r = 3(1+cos⁡(?)) = 3(1+cos⁡(?)).

Comparing this with the general equation of a cardioid parallel to the positive x-axis, we find that 

2a = 3, so a =3/2.

The area of the cardioid is calculated using the formula A = 6πa².

Substituting a = 3/2​, we get 

A = 6π(3/2)² = 6π × 9/4 = (27/2) π square units.

Question 3. Calculate the length of the arc passing through the cusp of the cardioid given by the equation ?=4(1+cos⁡(?)).

Solution:

For a cardioid with the equation r = 4(1+cos⁡(?)), we have 

2a = 4, so a = 2.

The length of the arc of a cardioid is determined by the formula 

L = 16a.

Substituting a = 2, we find 

L = 16 × 2 = 32 units.

Question 1: Find the area of the cardioid given by the equation r = 3(1+sin⁡(?)).

Question 2: Calculate the length of the arc of the cardioid r = 5(1+cos⁡(?)) between ? = 0 and ? = ?.

Question 3: Determine the equation of the cardioid that passes through the points (0, 0), (2, 0), and (1, √3).

Question 4: Sketch the graph of the cardioid r = 4(1−cos⁡(?)). Identify the cusp and any points of intersection with the coordinate axes.

What is a cardioid?

A cardioid is a geometrical figure shaped like a heart and produced by a point sliding on the circumference of one circle that rolls around the other, similar circle.

What is the polar equation of a cardioid?

The geometrical representation for a cardioid can be given in parametric or polar form, with the equation for the polar form being ? = ?(1+cos ?) or ? = ?(1+sin ?) where ? is a constant.

What is the length of the cardioid?

The length of a cardioid is 16?, where ? is the constant in its polar equation.

What is the volume of a cardioid?

A cardioid is a figure that does not possess a volume because it is a 2D figure but its area is equal to 6??².

What is a cardioid mic?

A cardioid mic is a microphone with a polar pattern that resembles the heart shape that gives the best pickup of the sound in front and from the side of this microphone, and at the same time has the minimum response to the sounds coming from behind that microphone.