Calculus

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Limacon and r as a function of theta.

Limaçon curve formation illustrated by polar function r = 2cos⁡(θ)−1, showing variation of r with respect to θ. Includes intermediate steps with negative r values and the complete curve with inner and outer loops.

This diagram illustrates the construction and behavior of the Limaçon curve defined by the polar equation r = 2cos⁡(θ)−1. The Limaçon is part of a broader family of curves that includes cardioids and Pascal’s snail, depending on parameter values.

The images provide a step-by-step exploration of how the radial distance r changes with respect to the angle θ, giving a detailed breakdown of the curve’s formation. The diagrams depict individual steps of the curve’s formation, including positive and negative r values, and visualize the full Limaçon shape with inner and outer loops.

This illustration is ideal for calculus textbooks and online learning materials.

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