Introduction
A cardioid is a distinctive heart-shaped curve in the family of limaçons. Its shape is determined not only by its scale, but also by which trigonometric function appears in its polar equation. For the two standard forms, that choice tells us where the cusp lies and which way the curve bulges.
We will analyze the equations \(r=a(1+\cos\theta)\) and \(r=a(1+\sin\theta)\), assuming \(a>0\). The parameter \(a\) sets the scale: changing \(a\) stretches or shrinks the curve without changing its orientation or basic shape.
Reading the Cosine Cardioid
Begin with \(r=a(1+\cos\theta)\). Since \(-1\leq\cos\theta\leq 1\), its radius satisfies \(0\leq r\leq 2a\). The radius is zero when \(\cos\theta=-1\), which occurs at \(\theta=\pi\). Thus the curve reaches the pole at that angle. Its radius is largest when \(\cos\theta=1\), at \(\theta=0\) (equivalently, \(2\pi\)).
To see where these polar values put the curve in the plane, use \(x=r\cos\theta\) and \(y=r\sin\theta\). The following points are enough to establish its main landmarks and tracing order.
Worked Example: Finding the landmarks of a cosine cardioid
Let \(r=2(1+\cos\theta)\). Evaluate the radius at five angles and convert each polar point to Cartesian coordinates.
At \(\theta=0\), \(r=2(1+\cos 0)=2(1+1)=4\), so \((x,y)=(4\cos 0,4\sin 0)=(4,0)\).
At \(\theta=\pi/2\), \(r=2(1+\cos(\pi/2))=2\), so \((x,y)=(2\cdot0,2\cdot1)=(0,2)\).
At \(\theta=\pi\), \(r=2(1+\cos\pi)=2(1-1)=0\), so \((x,y)=(0,0)\).
At \(\theta=3\pi/2\), \(r=2(1+\cos(3\pi/2))=2\), so \((x,y)=(2\cdot0,2\cdot(-1))=(0,-2)\).
At \(\theta=2\pi\), \(r=2(1+\cos(2\pi))=4\), so \((x,y)=(4,0)\). These values check the landmarks: the cusp is at the pole, the farthest point is \((4,0)\), and increasing \(\theta\) traces from the rightmost point through the upper part, then the cusp, then the lower part, and back to the rightmost point.
The cosine equation is symmetric about the \(x\)-axis. Indeed, replacing \(\theta\) by \(-\theta\) leaves the radius unchanged because \(\cos(-\theta)=\cos\theta\); the corresponding point is reflected across the \(x\)-axis. This symmetry is also visible in the paired landmarks at \((0,2)\) and \((0,-2)\).
Why the Pole Is a Cusp
Finding \(r=0\) locates a point on the curve, but by itself does not explain why that point is a cusp. For the cosine cardioid, examine the curve near \(\theta=\pi\). Write \(\theta=\pi+u\), where \(u\) is close to zero. Then \(r=a(1-\cos u)\). Using the Taylor expansions of sine and cosine near zero gives \(r=\frac{a}{2}u^2+O(u^4)\). The Cartesian coordinates therefore satisfy:
As \(u\) approaches zero from either side, both coordinates approach the pole. The leading term of \(x\) is proportional to \(u^2\), while the leading term of \(y\) is proportional to \(u^3\). The two branches meet with the same tangent direction, along the \(x\)-axis, but separate on opposite sides of that tangent. More explicitly, these leading terms imply \(y^2\) is approximately \(\frac{2}{a}(-x)^3\), the local shape of a cusp.
Reading the Sine Cardioid
For \(r=a(1+\sin\theta)\), the same radius bounds apply: \(0\leq r\leq2a\). The radius is zero when \(\sin\theta=-1\), at \(\theta=3\pi/2\), and it is largest when \(\sin\theta=1\), at \(\theta=\pi/2\). At the latter angle the point is \((0,2a)\), so the curve bulges upward.
The sine equation is symmetric about the \(y\)-axis. Replacing \(\theta\) by \(\pi-\theta\) preserves \(\sin\theta\), and the corresponding polar points are reflections across the \(y\)-axis. The sine cardioid is also the cosine cardioid rotated counterclockwise by \(\pi/2\), since \(\sin\theta=\cos(\theta-\pi/2)\).
Worked Example: Locating the cusp and axis of a sine cardioid
Let \(r=3(1+\sin\theta)\). At \(\theta=0\), \(r=3(1+0)=3\), giving \((x,y)=(3\cdot1,3\cdot0)=(3,0)\).
At \(\theta=\pi/2\), \(r=3(1+1)=6\), giving \((x,y)=(6\cdot0,6\cdot1)=(0,6)\), the farthest point.
At \(\theta=\pi\), \(r=3(1+0)=3\), giving \((x,y)=(3\cdot(-1),3\cdot0)=(-3,0)\).
At \(\theta=3\pi/2\), \(r=3(1-1)=0\), giving the pole \((0,0)\), where the cusp occurs.
At \(\theta=2\pi\), \(r=3(1+0)=3\), giving \((x,y)=(3,0)\) again. Thus increasing \(\theta\) traces from the right point through the top and left points, reaches the cusp at the bottom-facing angle, and returns to the right. The paired points \((3,0)\) and \((-3,0)\), together with the equation’s reflection symmetry, confirm the vertical symmetry axis.
Orientation and Scale
The sine and cosine forms have the same shape, but their landmarks are rotated. For the cosine form, the farthest point is on the positive \(x\)-axis, and the cusp occurs at the negative \(x\)-axis angle. For the sine form, the farthest point is on the positive \(y\)-axis, and the cusp occurs at the negative \(y\)-axis angle. In both cases, the farthest radius is \(2a\), while the cusp is at the pole.
The parameter \(a\) changes distances from the pole proportionally. For example, if \(a\) doubles, every radius and every Cartesian coordinate doubles. It does not move the cusp away from the pole, change the symmetry axis, or reverse the tracing direction.
Worked Example: Checking the orientation and scale
Consider \(r=4(1+\sin\theta)\). The radius is largest when \(\sin\theta=1\), so at \(\theta=\pi/2\) it is \(r=4(1+1)=8\). The Cartesian point is \((8\cos(\pi/2),8\sin(\pi/2))=(0,8)\).
The radius is zero when \(\sin\theta=-1\), so at \(\theta=3\pi/2\) the point is \((0,0)\). The radius at \(\theta=0\) is \(4(1+0)=4\), giving \((4,0)\), and at \(\theta=\pi\) it is also \(4\), giving \((-4,0)\).
These substitutions verify an upward-bulging curve with a vertical symmetry axis and a cusp at the pole. The maximum distance from the pole is \(8=2a\), as expected for \(a=4\).
A Common Misreading
A common mistake is to identify the angle of maximum radius as the cusp angle. These are different landmarks. For the cosine cardioid, \(r\) is largest at \(\theta=0\), but it is zero at \(\theta=\pi\). The former gives the farthest point, \((2a,0)\); the latter gives the cusp, the pole.
For the sine cardioid, the distinction is just as important: the maximum occurs at \(\theta=\pi/2\), giving \((0,2a)\), while the cusp occurs at \(\theta=3\pi/2\), where \(r=0\). Checking the radius at both the maximum and minimum of the trigonometric function prevents confusing the broad end of the curve with its pointed end.
Why These Landmarks Matter
A polar equation describes a curve through changing radius and angle, so a sketch can be misleading if it is based only on the equation’s appearance. Exact landmark calculations anchor the sketch: the zero radius identifies the pole, the maximum radius identifies the farthest point, symmetry determines paired points, and increasing-angle values establish the tracing order. Together, these facts give a reliable description without needing to plot many points.
Check Your Understanding
Use exact values and explain your reasoning in each case.
- For \(r=2(1+\cos\theta)\), find the cusp angle, the symmetry axis, and the farthest point.
- For \(r=5(1+\sin\theta)\), calculate the maximum radius and the Cartesian coordinates of the farthest point.
- At what angle does \(r=a(1+\sin\theta)\) reach the pole? What is its symmetry axis?
- For the cosine cardioid, what order do the upper point, cusp, and lower point occur in as \(\theta\) increases from \(0\) to \(2\pi\)?
- Explain why the maximum-radius point is not the cusp for either standard cardioid.
