PreCalc*A Cardiods and Limacons
By: Ethan Saddler & Charles Witmer
Intro to Equations
r=a+bsin(θ)
r=a+bcos(θ)
These are the two equations we will be focusing on. Both when graphed produce the same shape, the difference comes with the cosine and sine, with the cosine graph being symmetric across the x-axis and the sine graph symmetric across the y.
In a cardioid or limacon: r is the Radius, a, b are variables, and θ is the angle of rotation.
*note when discussing properties of the functions, we will be using the sine function. All properties are the same in cosine, just rotated about the x-axis*
Sine vs Cosine
The difference in sine vs cosine, as mentioned above is the rotation along the coordinate plane. It is rotated -pi/2 radians when converting from sine to cosine. This is because, as seen with the cos(θ) = r vs sin(θ) = r, the sine version of the function is on the y-axis, and the cosine is on the x-axis
The Graphs
Cardioid
A Cardioid is formed when in the sine version of the equation above, a=b, creating this nice shape. If b is negative, it reflects the curve about the y-axis.
The curve at first glance looks odd. However, upon further studying, you may realize that it is just a circle with a curve in, like a heart. This occurs because when the angle gets to the point where the y value is negative, it becomes subtracting from the 1 instead of adding, until it reaches 270 degrees, where it is 1-1, making it 0.
Limacons
A limacon is a cardioid but when a does not equal b. This will cause an inner loop to form as seen on the right. In this case, a=1 and b=5. This can be seen as a is causing a variation of + or - 1 on the point on the y-axis (other than the origin). This is because the b is going to be multiplied by 1, so it will be 5, then it will be either + 1 or -1, depending on the angle. We will discuss later the different types of Limacons.
"squashed" curve
3+4sin(theta)
Transformations
Cardioids
When cardioids a and b scale, since they have to be the same value (to be a cardioid) it simply scales up. This is because it is being scaled up by the b value, then moved up by the a value, causing it to look like it was just stretched out. The orientation is determined by b, when b is positive, the graph will face "upwards" (most y-values will be positive). The a value on the other hand has no impact whether it is positive or negative.
Limacons
A Limacon's length from the point where the inner loop and outer loop intersect to the furthest point cab be determined by adding a and b. When a=3 and b=4 (see curve to right), this would result in a Limacon that has a longest internal distance of 3+4=7. In addition, the length of the inner loop will be b - a, which gives a length of 1 in the example above. The same rules about a and b still apply. a being negative does not affect the graph but b being negative reflects it. As b approaches +infinity, the inner curve becomes bigger and bigger; however the a value does not allow it to touch, with the gap in between being 2a. When a grows larger, it becomes more and more like a circle. At first it has a bit of a squashed shape, but as it approaches +infinity it becomes closer and closer to being a perfect circle
Transformation into Cartesian format
Using the standard equation
r = a + b sin θ is the standard equation for a cardioid/limacon. First, turn sin θ into y/r. Then multiply by r on both sides to get r^2 = a*r + by/r. Convert r^2 into x^2 + y^2, and r into √(x^2 + y^2), which yields x^2 + y^2 = a*√(x^2 + y^2) + by/√(x^2 + y^2). Technically, this is in cartesian format.
Below is a more accessible explanation of how to convert the equation to a Cartesian graph. Notice how the graph of r = a + b sin θ and the graph of x^2 + y^2 = a*√(x^2 + y^2) + by/√(x^2 + y^2) are the same. This shows that they are the same equation.
Different Classifications of the Curves w/ Possible Variable Values
Cardioids = r=a+b sin(θ) Dimpled Limacons = r=a+b sin(θ)
|a| = |b| b < a < 2b
Looped = r=a+b sin(θ) Convex Limacons = r=a+b sin(θ)
a < b 2b < a
Looped Dimpled Convex
What Causes Differences Seen in Limacons?
Differences between a and b
One main difference between Limacons and Cardioids is the presence or absence of an inner loop. The size of the loop, and its point of inflection can be determined by observing the equation. Begin with 3 + 4 cos θ. The sum of a and b is 7, so this limacon will be seven units long from the origin to its furthest point of inflection on the larger loop of the curve. In addition, the difference between a and b is 1, which gives the length of the inner loop. The inner loop is one unit long, e.g. its point of inflection is located at (1,0). The graph is included below in black.
There is a different case for where b is less than a. Begin with 3 + 2 cos θ. The same rule applies, as stated above, that the cardioid will be a + b units long from the origin to the furthest inflection point. Thus, this cardioid will be 5 units long from the origin. However, because b is now less than a, the loop of the limacon is flipped outward. The difference between a and b here is 1, and a is now greater. This results in a graph where the point of inflection on the dimple is now in the opposite direction from the point of inflection on the larger curve. It will be one unit from the origin, as 1 is the difference between a and b. The graph is included below in red.
So this is why when we look at the looped, dimpled and convex limacons, we can notice a trend that as the difference between a and b increases and a increases, the position of the dimple moves further away, coming closer and closer to becoming a circle.
