AP® Precalculus · Unit 3 · Topic 3.14
Reading a Polar Curve from Its r-Graph
The r-graph (r plotted against theta like an ordinary function) is a blueprint: its peaks become the tips farthest from the pole, its zeros become the moments the polar curve passes through the pole.
Plot r as a function of theta the normal way, with theta on the horizontal axis and r on the vertical axis. That picture is just a wave, nothing polar about it yet. But every point on it carries an instruction: at this angle theta, go out this far, r. Read the wave left to right and you are reading the polar curve in the order it gets traced.
Where the r-graph peaks, the polar curve reaches its farthest point from the pole at that angle, a rounded tip. Where the r-graph crosses zero, the polar curve passes through the pole itself, then heads back out in a new direction. Where the r-graph dips below zero, something stranger happens: a negative r does not mean no point, it means plot the point in the opposite direction from theta, reflected through the pole. That reflection is what pulls part of the curve back through the center, drawing an inner loop.
Take r = 1 + 2 cos(theta). As a rectangular graph it is a cosine wave shifted up by 1, with max value 3 at theta = 0 and min value -1 at theta = π. Where that wave is positive, the polar curve bulges outward, reaching 3 at theta = 0. Where it dips negative, near theta = π, the points reflect through the pole and trace a small loop inside the larger outer loop. This is the limacon with an inner loop, and it connects directly to the next idea: classifying limacons, cardioids, and roses just by comparing a and b, or by counting the zeros of r(theta).
Key terms
- r-graph
- The ordinary rectangular graph of r as a function of theta, used as a blueprint for the polar curve.
- pole
- The origin of the polar coordinate system, where r = 0.
- limacon
- A polar curve of the form r = a + b cos(theta) or r = a + b sin(theta), shaped like a rounded curve that may have a dimple or an inner loop.
- inner loop
- A small loop traced inside a larger polar curve when r(theta) becomes negative and the points reflect through the pole.
Watch it explained
- How it worksHow do the peaks and zeros of r(theta) become the tips and the returns of a polar curve?
- Worked exampleGraph r = 1 + 2 cos(theta) as a rectangular curve first and then as a limacon with an inner loop.
- Ties togetherHow does a stretch where r(theta) goes negative end up drawing that inner loop?
Check yourself
End of topic.