What is the Cartesian equation of the cardioid $r=2a(1-\cos φ)$?

$(x^2+y^2)^2+4ax(x^2+y^2)-4a^2y^2=0$.

What is the length of a chord through the cusp of a cardioid with parameter $a$?

$4a$, for points at opposite polar angles $φ$ and $φ+π$.

What is the evolute of a standard cardioid?

Another cardioid, one third as large and oppositely oriented.

What is the pickup pattern of a cardioid microphone?

A directional sensitivity pattern resembling a cardioid in a two-dimensional slice.

CalculusLesson 2 of 2 in the course Parametric and polar curves

Cardioid Geometry, Constructions, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

A cardioid is a distinctive plane curve whose name comes from the Greek word for heart. This lesson introduces its geometric origins, shape, and basic equations.

A cardioid is the path traced by a point on the perimeter of one circle as that circle rolls around the outside of a fixed circle of the same radius. It is also an epicycloid with one cusp. The cusp is the sharp point where the curve meets itself in a limiting direction.

All flashcards

What is the Cartesian equation of the cardioid $r=2a(1-\cos φ)$ ?: $(x^2+y^2)^2+4ax(x^2+y^2)-4a^2y^2=0$ .
What is the length of a chord through the cusp of a cardioid with parameter $a$ ?: $4a$ , for points at opposite polar angles $φ$ and $φ+π$ .
What polar curve is the pedal curve of a circle centered at $(2a,0)$ with radius $2a$ , using the origin as the fixed point?: $r=2a(1+\cos φ)$ .
What is the evolute of a standard cardioid?: Another cardioid, one third as large and oppositely oriented.
What is the pickup pattern of a cardioid microphone?: A directional sensitivity pattern resembling a cardioid in a two-dimensional slice.

Quiz questions

Which construction traces a cardioid using rolling circles?Answer: A point on one circle rolls around the outside of a fixed circle of equal radius.
Reflecting the cardioid $r=2a(1-\cos φ)$ across the $y$ -axis gives which polar equation?Answer: $r=2a(1+\cos φ)$
What is an envelope of a continuously varying family of curves?Answer: A curve tangent to each member of the family.
What produces a cardioid catacaustic inside a circular mirror?Answer: Reflected rays from a point source on the mirror's circumference.
Which boundary of the Mandelbrot set is cardioid-shaped?Answer: The boundary of its central period-1 component.

Based on the Wikipedia article Cardioid, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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